January 08, 2026
The search for invariant metrics in complex Finsler geometry has revealed a fundamental dichotomy. Zhong [1] proved that no \(\operatorname{Aut}(B_n)\)-invariant strongly pseudoconvex complex Finsler metric exists on the open unit ball \(B_n \subset \mathbb{C}^n\) except for constant multiples of the Bergman metric. However, on the unit polydisk \(P_n\) (\(n \geq 2\)), there exist infinitely many non-Hermitian quadratic \(\operatorname{Aut}(P_n)\)-invariant Kähler-Finsler metrics.
Subsequent work has generalized these findings. Lin and Zhong [2] characterized all such invariant metrics which are strongly pseudconvex on the reducible bounded symmetric domain \(P_n\), showing that they are all Kähler-Berwald metrics. For irreducible bounded symmetric domains \(\mathfrak{R}_A\) with \(\operatorname{rank}(\mathfrak{R}_A) \geq 2\), Ge and Zhong [3] also constructed infinitely many \(\operatorname{Aut}(\mathfrak{R}_A)\)-invariant Kähler-Berwald metrics. Moreover, Zhong [4] characterized all holomorphically invariant metrics which are strongly pseudoconvex on the classical domains, proving that they are all Kähler-Berwald metrics and they share key holomorphic sectional curvature properties with the Bergman metric. These results underscore the central role of Kähler-Berwald manifolds in complex Finsler geometry.
The purpose of this paper is to give a geometric characterization of strongly convex Kähler-Berwald metrics. Given a strongly convex complex Finsler metric \(F: T^{1,0}M \rightarrow [0, +\infty)\) on a complex manifold \(M\), with the canonical complex structures \(J\) (on \(M\)) and \(\pmb{J}\) (on \(T^{1,0}M\)). Our method is via investigating different levels of parallelism of the canonical complex structure \(\pmb{J}\) with respect to the Cartan connection \(\nabla\) associated to \(F\). We find that such parallelism exhibits strong rigidity, which leads to the study of a special class of complex Finsler metrics, namely the so-called Kähler-Berwald metrics.
Let’s firstly take a look of the Hermitian case. A Hermitian metric on a complex manifold \(M\) is a Riemannian metric \(g\) on the underling smooth manifold that is \(J\)-invariant: \(g(JX,JY)=g(X,Y)\) for all real tangent vectors \(X\) and \(Y\). Locally \(g\) can be represented by \(g=2\text{Re}(g_{\alpha\bar{\beta}}dz^\alpha\otimes d\bar{z}^\beta)\) with the associated real \((1,1)\)-form \(\omega=ig_{\alpha\bar{\beta}}dz^\alpha\wedge d\bar{z}^\beta\). Denote \(\nabla\) the Levi-Civita connection of \(g\). In [5], N. Mok gave the following geometric definition of a Hermitian manifold to be a Kähler manifold.
Definition 1. [5]A Hermitian manifold \((M,g)\) is said to be Kähler iff the types of complexified tangent vector fields are preserved under parallel transport.
Moreover, N. Mok proved the following
Proposition 1 ([5]). Let \((M,g)\) be a Hermitian manifold such that \(g\) is given by \(2\text{Re}\left(g_{\alpha\bar{\beta}}dz^\alpha\otimes d\bar{z}^\beta\right)\) in local holomorphic coordinates \((z^\alpha)\). Then, \((M,g)\) is Kähler iff one of the following equivalent conditions is satisfied:
types of complexified tangent vectors are preserved under parallel transport;
for any parallel (real) vector field \(\eta\) along a smooth curve \(\gamma\), \(J\eta\) is also parallel;
\(\nabla J\equiv 0\), i.e., the almost complex structure \(J\) is parallel;
\(\nabla\omega\equiv 0\), i.e., the Hermitian form \(\omega\) is parallel;
\(d\omega\equiv 0\), i.e., the Hermitian form \(\omega\) is closed;
locally there exists a potential function \(\varphi\) such that \(g_{\alpha\bar{\beta}}=\frac{\partial^2\varphi}{\partial z^\alpha\partial \bar{z}^\beta}\);
at every point \(p\in M\) there exists complex geodesic coordinates \((z^\alpha)\) in the sense that the Hermitian metric \(g\) is represented by the Hermitian matrix \((g_{\alpha\bar{\beta}})\) satisfying \(g_{\alpha\bar{\beta}}(p)=\delta_{\alpha\beta}\) and \(dg_{\alpha\bar{\beta}}(p)=0\).
The study of Kähler conditions in complex Finsler geometry originated with Royden’s work [6]. Let \(\triangle(r)\) denote the open disk of radius \(r\) in \(\mathbb{C}\) centered at the origin which is endowed with the Poincaré metric. Under the assumption that for any \((z;v) \in T^{1,0}M\) there exists a holomorphic map \(\varphi: \triangle(r) \rightarrow M\) with \(\varphi(0)=z\), \(\varphi_*(0)=v\), and such that the curve \(\gamma(t) = \varphi(e^{i\theta}t)\) is a geodesic tangent to \(\mathbb{C} \cdot v\) at \(z\) for each \(\theta \in \mathbb{R}\) (a condition later termed Royden’s condition in [7]), Royden proved that \(F\) must be a weakly Kähler-Finsler metric along such maps \(\varphi\). For a clear reformulation as well as a characterization of Kähler-Finsler metrics, we refer to Theorem 2.3.10 in [8].
In their work [9], Chen, Liu, and Zhao investigated the weak parallelism of the canonical complex structure \(J\) on a complex manifold \(M\) with respect to the real Berwald connection \(\breve{\nabla}\) of a strongly convex weakly Kähler-Finsler metric \(F\). They proved that the real fundamental tensor of \(F\) is \(J\)-invariant iff \(F\) comes from a Hermitian metric. Furthermore, they investigated the relationship between \(J\) and both the real and complex Ricci curvatures of \(F\), they also showed that for strongly convex Kähler-Berwald metrics, the real and complex notions of an Einstein metric coincide.
A further inspiration for this work comes from the classical study by M. Abate and G. Patrizio on the Cartan and Chern-Finsler connections for a strongly convex complex Finsler metric [8]. In particular, they inquire about the existence of a canonical connection on real Finsler manifolds that agrees with the Chern-Finsler connection in the Kähler-Finsler case (see [8], p. 122). Moreover, they observe that the Cartan connection includes elements deemed geometrically non-essential, since these do not contribute to the first and second variation formulas (see [8], p. 125).
It is crucial to distinguish the base manifold’s canonical complex structure \(J\) (a \((1,1)\)-tensor on \(M\)) from the complex structure \(\pmb{J}\) (a \((1,1)\)-tensor on \(T^{1,0}M\)), on which the geometry of a strongly convex complex Finsler metric \(F\) fundamentally resides. In this paper, we derive rigidity results by investigating the parallelism of \(\pmb{J}\) with respect to the associated Cartan connection \(\nabla\), an approach grounded in the concept of parallelism relative to a real Finsler connection of \(F\). Our first main result is
Theorem 2. Let \(F: T^{1,0}M\rightarrow [0,+\infty)\) be a strongly convex complex Finsler metric on a complex manifold \(M\). Then \(\nabla \pmb{J}=0\) iff the Cartan connection \(\nabla\) associated to \(F\) coincides with the Chern-Finsler connection \(D\) associated to \(F\).
In Hermitian geometry, if \(h\) is a Hermitian metric on a complex manifold \(M\) and \(g=2\text{Re}\,h\) is the associated Riemannian metric, and if the canonical complex structure \(J\) on \(M\) is parallel with respect to the Levi-Civita connection of \(g\), then \(h\) must be a Kähler metric. Theorem 2 generalizes this classical result to the Finsler setting.
Our second main result is
Theorem 3. Let \(F: T^{1,0}M\rightarrow [0,+\infty)\) be a strongly convex weakly Kähler-Finsler metric on a complex manifold \(M\) and \(\sigma:[0,1]\rightarrow M\) a smooth regular curve in \(M\). Then the following assertions are equivalent:
\(F\) is a strongly convex Kähler-Berwald metric;
The types of complexified vectors in \(T_{\mathbb{R}}M\) are preserved under parallel transport along \(\sigma\) with respect to \(\nabla\);
For any parallel real vector field \(V\) along \(\sigma\) with respect to \(\nabla\), \(JV\) is also parallel along \(\sigma\) with respect to \(\nabla\);
\(\nabla_X \pmb{J}\equiv 0\) for all \(X\in\mathcal{H}_{\mathbb{R}}\);
Remark 4. There are lots of nontrivial (non-Hermitian quadratic) strongly convex Kähler-Berwald metrics [1], [2], [4]. So Theorem 3 gives a geometric characterization of strongly convex Kähler-Berwald manifolds, which partially generalizing Proposition 1 to Finsler setting.
As an application of Theorem 2, we obtain the following
Theorem 5. Let \(F: T^{1,0}M\rightarrow [0,+\infty)\) be a strongly convex complex Finsler metric on a complex manifold \(M\). If \(\nabla \pmb{J}=0\) and \(F\) has constant holomorphic sectional curvature \(c\neq 0\), then \(F\) is necessary a Kähler-Einstein metric on \(M\).
Using Theorem 5, we immediately obtain the following
Theorem 6. Let \(F: T^{1,0}M\rightarrow[0,+\infty)\) be a complete strongly convex complex Finsler metric on a simply connected complex manifold \(M\). If \(\nabla\pmb{J}=0\) and \(F\) has constant holomorphic sectional curvature \(c\in \mathbb{R}\). Then
if \(c<0\), \((M,F)\) is a Kähler manifold which is holomorphically isometric to the open unit ball \(B_n\) in \(\mathbb{C}^n\) with a constant multiple of the Bergman metric, namely \[F^2(z;v)=-\frac{4}{c}\frac{(1-\|z\|^2)\|v\|^2+\vert\langle z,v\rangle\vert^2}{(1-\|z\|^2)^2}\]
if \(c=0\), \((M,F)\) is locally a complex Minkowski space which is locally holomorphic isometric to \(\mathbb{C}^n\) with some complex Minkowski metric \(F(z;v)=f(v)\);
if \(c>0\), \((M,F)\) is a Kähler manifold which is holomorphically isometric to the complex projective space \(\mathbb{CP}^n\) with a constant multiple of the Fubini-Study metric, namely \[F^2(z;v)=\frac{4}{c}\frac{(1+\|z\|^2)\|v\|^2-\vert\langle z,v\rangle\vert^2}{(1+\|z\|^2)^2}.\]
In [7], under the assumptions that \((M,F)\) is a simply connected and complete complex manifold modeled on a complex Minkowski space \((\mathbb{C}^n,f)\), that \(F\) is Kähler-Finsler and satisfies the Royden condition, and that \((M,F)\) has constant holomorphic sectional curvature \(c\), Aikou [7] actually outlined the proof of the above Theorem 5. In contrast, our hypothesis \(\nabla\pmb{J}=0\) simple and purely a geometric condition. Furthermore, our proof of Theorem 6 is mainly based on the rigidity Theorem 32, which is essentially different from the approach of Aikou [7].
This section recalls some necessary notations and definitions, which can be found in [8], [10].
Let \(M\) be a complex manifold of complex dimension \(n (\geq 2)\), and let \(\pi:T_{\mathbb{R}}M\rightarrow M\) be its real tangent bundle. Denote by \(J:T_{\mathbb{R}}M\rightarrow T_{\mathbb{R}}M\) the canonical complex structure on \(M\), and throughout this paper we denote by \(i=\sqrt{-1}\) the imaginary unit. We also use Einstein summation convention, lowercase greek indices will run from \(1\) to \(n\), whereas lowercase roman indices will run from \(1\) to \(2n\), and \(\alpha^\ast=\alpha+n,\beta^\ast=\beta+n,\) etc.
Let \(z=(z^1,\cdots,z^n)\) be the local holomorphic coordinates on an open subset \(U\subset M\). Setting \(z^\alpha=x^\alpha+ix^{\alpha^\ast}\) for \(\alpha=1,\cdots,n\), we obtain the corresponding real coordinates \(x=(x^1,\cdots,x^{2n})\) on \(U\). The canonical complex structure \(J\) on \(M\) satisfies \[J\left(\frac{\partial}{\partial x^{\alpha}}\right) =\frac{\partial }{\partial x^{\alpha^\ast}},\quad J\left(\frac{\partial}{\partial x^{\alpha^\ast}}\right)=-\frac{\partial}{\partial x^\alpha},\quad \alpha=1,\cdots,n.\]
Extending \(J\) by complex linearity to the complexified tangent bundle \(T_{\mathbb{C}}M\), we obtain the decomposition \(T_{\mathbb{C}}M = T^{1,0}M \oplus T^{0,1}M\), where \(T^{1,0}M\) and \(T^{0,1}M\) are the holomorphic and anti-holomorphic tangent bundle, respectively. Local frames for \(T^{1,0} M\) and \(T^{0,1}M\) over \(U\) are given by \[\left\{\frac{\partial}{\partial z^1},\cdots,\frac{\partial}{\partial z^n}\right\}\quad \text{and}\quad \left\{\frac{\partial}{\partial \bar{z}^1},\cdots,\frac{\partial}{\partial \bar{z}^n}\right\}\] respectively, where \[\frac{\partial}{\partial z^{\alpha}}=\frac{1}{2}\left(\frac{\partial}{\partial x^{\alpha}}-i\frac{\partial}{\partial x^{\alpha^\ast }}\right),\quad \frac{\partial}{\partial \bar{z}^{\alpha}}=\frac{1}{2}\left(\frac{\partial}{\partial x^{\alpha}}+i\frac{\partial}{\partial x^{\alpha^\ast }}\right).\]
There exists an explicit \(\mathbb{R}\)-isomorphism between \(T_{\mathbb{R}}M\) and \(T^{1,0}M\), defined by \[_o:T_{\mathbb{R}}M \rightarrow T^{1,0}M, \;y\mapsto v=\frac{1}{2}(y-iJy),\quad \forall y=y^j\frac{\partial}{\partial x^j}\] and \[^o:T^{1,0}M \rightarrow T_{\mathbb{R}}M, \;v\mapsto y=v+\bar{v},\quad \forall v=v^\alpha\frac{\partial}{\partial z^\alpha}.\] It is easy to verify that \(y_o=v\) and \(v^o=y\) if we set \(v^\alpha=y^\alpha+iy^{\alpha^\ast}\) for \(\alpha=1,\cdots,n\).
A vector \(v\in T_p^{1,0}M\) can be written as \(v=v^{\alpha}\frac{\partial}{\partial z^{\alpha}}\vert_p\). This induces holomorphic coordinates \((z;v)=(z^1,\cdots,z^n;v^1,\cdots,v^n)\) on \(\pi^{-1}(U)\subset T^{1,0}M\), showing that \(T^{1,0}M\) is a complex manifold of complex dimension \(2n\). Setting \(v^{\alpha}=y^{\alpha}+\sqrt{-1}y^{\alpha^\ast }\) for \(\alpha=1,\cdots,n\), we define \[\frac{\partial }{\partial v^{\alpha}}=\frac{1}{2}\left(\frac{\partial}{\partial y^{\alpha}}-i\frac{\partial}{\partial y^{\alpha^\ast }}\right),\quad\frac{\partial }{\partial \bar{v}^{\alpha}}=\frac{1}{2}\left(\frac{\partial}{\partial y^{\alpha}}+i\frac{\partial}{\partial y^{\alpha^\ast }}\right).\]
In the following, we denote by \(\pmb{J}\) the canonical complex structure on the complex manifold \(T^{1,0}M\), namely \[\label{CS-T} \begin{align} &&\pmb{J}\left(\frac{\partial}{\partial x^\alpha}\right)=\frac{\partial}{\partial x^{\alpha^\ast }},\quad \pmb{J}\left(\frac{\partial}{\partial x^{\alpha^\ast }}\right)=-\frac{\partial}{\partial x^{\alpha}},\\ && \pmb{J}\left(\frac{\partial}{\partial y^\alpha}\right)=\frac{\partial}{\partial y^{\alpha^\ast }},\quad \pmb{J}\left(\frac{\partial}{\partial y^{\alpha^\ast }}\right)=-\frac{\partial}{\partial y^\alpha}, \end{align}\tag{1}\] for \(\alpha=1,\cdots,n\).
The real (resp. complex) Finsler metrics considered in this paper are smooth outside the zero section of the real (resp. holomorphic) tangent bundle of \(M\). In the following, we denote by \(\tilde{M}\) either \(T_{\mathbb{R}}M\setminus\{0\}\) or \(T^{1,0}M\setminus\{0\}\) when no confusion arises. A local real frame for \(T_{\mathbb{R}}\tilde{M}\) is given by \[\left\{\frac{\partial}{\partial x^1},\cdots,\frac{\partial}{\partial x^{2n}},\frac{\partial}{\partial y^1},\cdots,\frac{\partial}{\partial y^{2n}}\right\};\] Correspondingly, a local holomorphic frame for \(T^{1,0}\tilde{M}\) is \[\left\{\frac{\partial}{\partial z^1},\cdots,\frac{\partial}{\partial z^n},\frac{\partial}{\partial v^1},\cdots,\frac{\partial}{\partial v^n}\right\}.\]
Similar to \(T_{\mathbb{R}}M\) and \(T^{1,0}M\), there is a natural \(\mathbb{R}\)-isomorphism between \(T_{\mathbb{R}}\tilde{M}\) and \(T^{1,0}\tilde{M}\), defined by \[_o:T_{\mathbb{R}}\tilde{M}\rightarrow T^{1,0}\tilde{M},\quad U\mapsto U_o=\frac{1}{2}(U-i\pmb{J}U),\quad\forall U\in T_{\mathbb{R}}\tilde{M}\] and \[^o:T^{1,0}\tilde{M}\rightarrow T_{\mathbb{R}}\tilde{M},\quad X\mapsto X^o=X+\overline{X},\quad\forall X\in T^{1,0}\tilde{M}.\]
It is easy to check that \[\frac{\partial}{\partial z^\alpha}=\left(\frac{\partial}{\partial x^\alpha}\right)_o,\quad i\frac{\partial}{\partial z^\alpha}=\left(\frac{\partial}{\partial x^{\alpha^\ast }}\right)_o,\quad\frac{\partial}{\partial v^\alpha}=\left(\frac{\partial}{\partial y^\alpha}\right)_o,\quad i\frac{\partial}{\partial v^\alpha}=\left(\frac{\partial}{\partial y^{\alpha^\ast }}\right)_o\] or equivalently, \[\left(\frac{\partial}{\partial z^\alpha}\right)^o=\frac{\partial}{\partial x^\alpha},\quad \left(i\frac{\partial}{\partial z^\alpha}\right)^o=\frac{\partial}{\partial x^{\alpha^\ast }},\quad\left(\frac{\partial}{\partial v^\alpha}\right)^o=\frac{\partial}{\partial y^\alpha},\quad \left(i\frac{\partial}{\partial v^\alpha}\right)^o=\frac{\partial}{\partial y^{\alpha^\ast }}.\]
Definition 2. [10] A real (strongly convex) Finsler metric on a smooth manifold \(M\) of real dimension \(n\) is a function \(F:T_{\mathbb{R}}M\rightarrow [0,+\infty)\) satisfying
\(F\) is positive and \(C^\infty\) on \(\tilde{M}=T_{\mathbb{R}}M\setminus\{0\}\);
\(F(x;\lambda y)=\lambda F(x;y)\) for all \((x;y)\in T_{\mathbb{R}}M\) and \(\lambda>0\);
The real fundamental tensor \[(g_{jk}):=\left(\frac{1}{2}\frac{\partial^2F^2}{\partial y^j\partial y^k}\right)\] is positive definite on \(\tilde{M}\).
Definition 3. [8] A strongly pseudoconvex complex Finsler metric on a complex manifold \(M\) of complex dimension \(n\) is a function \(F:T^{1,0}M\rightarrow [0,+\infty)\) satisfying
\(F\) is positive and \(C^\infty\) on \(\tilde{M}=T^{1,0}M\setminus\{0\}\);
\(F(z;\lambda v)=\vert \lambda\vert F(z;v)\) for all \((z;v)\in T^{1,0}M\) and \(\lambda\in\mathbb{C}\);
The complex fundamental tensor \[(G_{\alpha\bar{\beta}}):=\left(\frac{\partial^2F^2}{\partial v^\alpha \partial\bar{v}^\beta}\right)\] is Hermitian positive definite on \(\tilde{M}\).
Let \(F:T^{1,0}M\rightarrow [0,+\infty)\) be a strongly convex complex Finsler metric. It induces a real Finsler metric \(F_{\mathbb{R}}\) on \(M\) via \[F_{\mathbb{R}}(y):=F(y_o)=F(v).\label{RaC}\tag{2}\] Clearly, \(F_{\mathbb{R}}\) is \(J\)-invariant, namely \(F_{\mathbb{R}}(Jy)=F_{\mathbb{R}}(y)\).
Because of equation 2 , for simplicity, we will make no distinction between notations \(F\) and \(F_{\mathbb{R}}\) in the following if it causes no confusion.
Let \(F:T^{1,0}M\rightarrow [0,+\infty)\) be a strongly convex complex Finsler metric on a complex manifold \(M\), and \(\sigma:[a,b]\rightarrow M\) be a smooth curve. According to the proof of Corollary 1.5.2 in [8], \(\sigma\) is a real geodesic for \(F\) iff it satisfies the following differential equation \[\ddot{\sigma}^k+\hat{\Gamma}_j^k(\sigma;\dot{\sigma})\dot{\sigma}^j=0,\label{geodesic-1}\tag{3}\] where the coefficients \(\hat{\Gamma}_j^k\) are the Christoffel symbols of the real non-linear connection \(\tilde{\nabla}:\mathcal{X}(T_{\mathbb{R}}M)\rightarrow \mathcal{X}(T_{\mathbb{R}}^\ast M\otimes T_{\mathbb{R}}M)\) associated to \(F\). More precisely, \[\hat{\Gamma}_j^k(x;y):=\frac{\partial \hat{\mathbb{G}}^k}{\partial y^j},\label{geodesic-2}\tag{4}\] where \[\hat{\mathbb{G}}^k:=\frac{1}{4}g^{kl}\left(\frac{\partial^2F^2}{\partial x^a\partial y^l}y^a-\frac{\partial F^2}{\partial x^l}\right)\label{geodesic-3}\tag{5}\] are called the real geodesic spray coefficients of \(F\).
Equations 3 5 show that the real non-linear connection coefficients \(\hat{\Gamma}_j^k\) completely determine the real geodesics of \(F\). The real non-linear connection \(\tilde{\nabla}\) induces a direct sum decomposition of the tangent bundle of the slit tangent bundle \(\tilde{M}\): \[T_{\mathbb{R}}\tilde{M}=\mathcal{H}_{\mathbb{R}}\oplus\mathcal{V}_{\mathbb{R}},\] where \(\mathcal{V}_{\mathbb{R}}=\ker d\pi\) is the real vertical bundle, and \(\mathcal{H}_{\mathbb{R}}\) is the real horizontal bundle determined by \(\tilde{\nabla}\). This decomposition is locally characterized by adapted frames \[\mathcal{H}_{\mathbb{R}}=\text{span}\left\{\frac{\delta}{\delta x^1},\cdots,\frac{\delta}{\delta x^{2n}}\right\},\quad \mathcal{V}_{\mathbb{R}}=\text{span}\left\{\frac{\partial}{\partial y^1},\cdots,\frac{\partial}{\partial y^{2n}}\right\},\] where \(\frac{\delta}{\delta x^j}:=\frac{\partial}{\partial x^j}-\hat{\Gamma}_j^k\frac{\partial}{\partial y^k}\) for \(j,k=1,\cdots,2n\). The dual bundles \(\mathcal{H}_{\mathbb{R}}^\ast\) and \(\mathcal{V}_{\mathbb{R}}^\ast\) are correspondingly spanned by \(\{dx^1,\cdots,dx^{2n}\}\) and \(\{\delta y^1,\cdots,\delta y^{2n}\}\), respectively, with \(\delta y^j:=dy^j+\hat{\Gamma}^j_k dx^k\).
Let \(F:T^{1,0}M\rightarrow [0,+\infty)\) be a strongly convex complex Finsler metric on a complex manifold \(M\), and let \(g=\langle\cdot\vert \cdot\rangle\) be the natural Riemannian metric induced by \(F\) on the vertical bundle \(\mathcal{V}_{\mathbb{R}}\). In local coordinates, this metric is expressed as \[g=g_{jk}(x;y)\delta y^j\otimes \delta y^k\] so that for any vertical vector fields \(X=X^j\frac{\partial}{\partial y^j}\) and \(Y=Y^k\frac{\partial}{\partial y^k}\) in \((\mathcal{V}_{\mathbb{R}})_{(x;y)}\), \[\langle X\vert Y\rangle=g_{jk}(x;y)X^jY^k\quad\text{where}\quad g_{jk}(x;y)=g\left(\frac{\partial}{\partial y^j},\frac{\partial}{\partial y^k}\right).\]
Theorem 7. [8] Let \(F\) be a real Finsler metric on \(M\), and let \(\langle\cdot\vert \cdot\rangle\) be the Riemannian structure on \(\mathcal{V}_{\mathbb{R}}\) induced by \(F\). Then there exists a unique real vertical connection \[\nabla : \mathcal{X}(\mathcal{V}_{\mathbb{R}}) \rightarrow \mathcal{X}(T_{\mathbb{R}}^*\tilde{M}\otimes\mathcal{V}_{\mathbb{R}})\] satisfying the following properties:
\(\nabla\) is good;
\(\nabla\) is compatible with the metric: for any \(X \in T_{\mathbb{R}}\tilde{M}\) and \(V,W\in \mathcal{X}(\mathcal{V}_{\mathbb{R}})\), \[X\langle V\vert W\rangle =\langle \nabla_X V\vert W\rangle+\langle V\vert \nabla_X W\rangle;\]
The vertical torsion vanishes: \(\theta(V,W)=0\) for all \(V,W\in \mathcal{V}_{\mathbb{R}}\);
The horizontal torsion is vertical: \(\theta(H,K)\in \mathcal{V}_{\mathbb{R}}\) for all \(H,K\in \mathcal{H}_{\mathbb{R}}\).
In local coordinates, the connection \(1\)-form \(\hat{\omega}_j^{\;k}\) of the Cartan connection is given by \[\hat{\omega}_j^{k}=\hat{\Gamma}_{j;l}^kdx^l+\hat{\Gamma}^k_{jl}\delta y^l,\] where the coefficients are defined as \[\begin{align} \hat{\Gamma}_{j;l}^k=\frac{g^{ks}}{2}\left(\frac{\delta g_{sj}}{\delta x^l}-\frac{\delta g_{jl}}{\delta x^s}+\frac{\delta g_{ls}}{\delta x^j}\right),\quad \hat{\Gamma}^k_{jl}=\frac{1}{4}g^{ks}\frac{\partial^2F^2}{\partial y^j\partial y^l\partial y^s}. \end{align}\] These coefficients are symmetric in their lower indices: \(\hat{\Gamma}_{j;l}^k=\hat{\Gamma}_{l;j}^k\) and \(\hat{\Gamma}_{jl}^k=\hat{\Gamma}_{lj}^k\). Moreover, since the Cartan connection is good, it follows from Lemma 1.2.2 in [8] that the non-linear connection coefficients satisfy \[\hat{\Gamma}^k_j=\hat{\Gamma}^k_{l;j}y^l.\label{hcp}\tag{6}\]
Using the horizontal Cartan connection coefficients, one can introduce the parallelism of a real vector field \(V\in\mathcal{X}(T_{\mathbb{R}}M)\) along a smooth curve \(\sigma:[0,1]\rightarrow M\). In real Finsler geometry, however, there are several different definitions for the parallel transport of a real vector field along a curve. Different definitions stem from distinct geometric interpretations of "parallelism" and the use of different connections. In general, these definitions are not equivalent on a real Finsler manifold, we refer to [8],[10] for more details.
Definition 4. Let \(V(t)=V^k(t)\frac{\partial}{\partial x^k}\) be a real vector field defined along a smooth curve \(\sigma:[0,1]\rightarrow M\). We say \(V\) is parallel along \(\sigma\) with respect to the Cartan connection \(\nabla\) if locally \[\frac{\pmb{D}V^k}{dt}:=\frac{dV^k}{dt}+V^l\hat{\Gamma}_{l;j}^k(\sigma(t);\dot{\sigma}(t))\dot{\sigma}^j(t)=0,\quad\forall k=1,\cdots,2n.\label{parallel}\qquad{(1)}\]
In particular, \(\sigma\) itself is a geodesic iff \(\dot{\sigma}\) is parallel along \(\sigma\) with respect to \(\nabla\), which is precisely the equation 3 since we always have \[\dot{\sigma}^l(t)\hat{\Gamma}_{l;j}^k(\sigma(t);\dot{\sigma}(t))=\hat{\Gamma}_{j}^k(\sigma(t);\dot{\sigma}(t)).\]
Remark 8. By the above definition, if \(V_1\) and \(V_2\) are two real vector fields defined along \(\sigma\), which are parallel along \(\sigma\) with respect to the Cartan connection \(\nabla\), then \(aV_1+bV_2\) are also parallel along \(\sigma\) with respect to \(\nabla\) for any \(a,b\in\mathbb{R}\).
The torsion \(\theta\in\mathcal{X}(\wedge^2 T_{\mathbb{R}}^\ast \tilde{M}\otimes T_{\mathbb{R}}\tilde{M})\) of \(\nabla\) has the local expression: \[\begin{align} \theta=\theta^{j}\otimes \frac{\delta}{\delta x^j}+\dot{\theta}^{a}\otimes \frac{\partial}{\partial y^a}, \end{align}\] where \[\begin{align} \theta^j&=&\hat{\Gamma}^j_{lc}\delta y^c\wedge dx^l,\\ \dot{\theta}^a&=&\frac{1}{2}\left[\frac{\delta}{\delta x^j}(\hat{\Gamma}^a_k)-\frac{\delta}{\delta x^k}(\hat{\Gamma}^a_j)\right]dx^j\wedge dx^k+\left[\frac{\partial}{\partial y^b}(\hat{\Gamma}^a_k)-\hat{\Gamma}^a_{b;k}\right]\delta y^b \wedge dx^k. \end{align}\]
The curvature operator \(\hat{\Omega}\in\mathcal{X}(\wedge^2T_{\mathbb{R}}^\ast \tilde{M}\otimes T_{\mathbb{R}}^\ast\tilde{M}\times T_{\mathbb{R}}\tilde{M})\) of \(\nabla\) is given by \[\hat{\Omega}=\hat{\Omega}_j^k\otimes \left[dx^j\otimes \frac{\delta}{\delta x^k}+\delta y^j\otimes \frac{\partial}{\partial y^k}\right],\] where \[\hat{\Omega}_{j}^{k} = d\hat{\omega}_{j}^{k} - \hat{\omega}_{j}^{l} \wedge \hat{\omega}_{l}^{k}.\]
The horizontal flag curvature of \(\nabla\) at \(y\in\tilde{M}\) is given by \[R_y(H,K)=\langle \hat{\Omega} (\hat{\chi}(y),H)K\vert \hat{\chi}(y)\rangle,\quad \forall H,K \in (\mathcal{H}_{\mathbb{R}})_y.\]
The study of complex Finsler geometry hinges on the introduction of a complex horizontal subbundle \(\mathcal{H}_{\mathbb{C}}\subset T_{\mathbb{C}}\tilde{M}\) that is \(\pmb{J}\)-invariant, conjugation invariant, and satisfies the decomposition: \[T_{\mathbb{C}}\tilde{M}=\mathcal{H}_{\mathbb{C}}\oplus\mathcal{V}_{\mathbb{C}}, \quadand dually\quad T_{\mathbb{C}}^\ast\tilde{M}=\mathcal{H}_{\mathbb{C}}^\ast\oplus\mathcal{V}_{\mathbb{C}}^\ast.\] Such a bundle \(\mathcal{H}_{\mathbb{C}}\) allows one to define a complex non-linear connection \(\tilde{D}_{\mathcal{H}_{\mathbb{C}}}\) on \(M\). This connection, in turn, determines a complex horizontal map \(\Theta^{\tilde{D}_{\mathcal{H}_{\mathbb{C}}}}\), whose image is precisely the original bundle \(\mathcal{H}_{\mathbb{C}}\). This establishes a one-to-one correspondence among complex horizontal bundles, complex non-linear connections and complex horizontal maps [8]. We note that a complex horizontal bundle is completely determined by its \((1,0)\)-part \(\mathcal{H}^{1,0}\), yielding the decomposition \(T^{1,0}\tilde{M}=\mathcal{H}^{1,0}\oplus\mathcal{V}^{1,0}\), where \(\mathcal{V}^{1,0}=\ker d\pi\subset T^{1,0}\tilde{M}\). In general, \(\mathcal{H}^{1,0}\) may not comes from a strongly pseudoconvex complex Finsler metric.
However, for a given strongly pseudoconvex complex Finsler metric \(F:T^{1,0}M\rightarrow[0,+\infty)\), there is a canonical way to construct \(\mathcal{H}^{1,0}\) from \(F\), giving rise to the associated complex horizontal bundle \(\mathcal{H}_{\mathbb{C}}=\mathcal{H}^{1,0}\oplus\mathcal{H}^{0,1}\). This bundle is called the complex horizontal bundle associated to the Chern-Finsler connection of \(F\). More precisely, \(\mathcal{H}^{1,0}=Span_{\mathbb{C}}\{\delta_1,\cdots,\delta_n\}\) where \(\delta_\mu=\frac{\partial}{\partial z^\mu}-\Gamma_{;\mu}^\alpha\frac{\partial}{\partial v^\alpha}\) for \(\mu=1,\cdots,n\) and the coefficients \[\Gamma_{;\mu}^\alpha=G^{\bar{\tau}\alpha}\frac{\partial^2F^2}{\partial z^\mu\partial\bar{v}^\tau}\label{HVB}\tag{7}\] are called the complex non-linear connection coefficients of the Chern-Finsler connection associated to \(F\).
Let \(\langle \cdot,\cdot\rangle\) denote the Hermitian inner product induced by \(F\) on the holomorphic vertical subbundle \(\mathcal{V}^{1,0}\cong\mathcal{V}_{\mathbb{R}}\). For any \(V=V^\alpha\frac{\partial}{\partial v^\alpha}\), \(W=W^\beta\frac{\partial}{\partial v^\beta}\) in \((\mathcal{V}^{1,0})_{(z;v)}\), \[\langle V, W\rangle=G_{\alpha\bar{\beta}}(z;v)V^\alpha \overline{W^\beta}.\]
Theorem 9. [8]Let \(F:T^{1,0}M\rightarrow [0,+\infty)\) be a strongly pseudoconvex complex Finsler metric on a complex manifold \(M\), and let \(\langle\cdot,\cdot\rangle\) be the induced Hermitian structure on \(\mathcal{V}^{1,0}\). Then there is a unique complex vertical connection \[D:\mathcal{X}(\mathcal{V}^{1,0})\rightarrow \mathcal{X}(T_{\mathbb{C}}^\ast\tilde{M}\otimes \mathcal{V}^{1,0})\] such that
For all \(X\in T^{1,0}\tilde{M}\) and \(V,W\in\mathcal{X}(\mathcal{V}^{1,0})\), \[X\langle V,W\rangle=\langle D_XV,W\rangle+\langle V,D_{\overline{X}}W\rangle;\label{MC}\qquad{(2)}\]
\(D\) is good.
The connection \(D\) in Theorem 9 is called the Chern-Finsler connection of \(F\). Its connection \(1\)-forms \(\omega_\beta^\alpha\) are given by \[\omega_\beta^\alpha=\Gamma_{\beta;\mu}^\alpha dz^\mu+\Gamma_{\beta\gamma}^\alpha\psi^\gamma,\] where \(\psi^\gamma=dv^\gamma+\Gamma_{;\mu}^\gamma dz^\mu\) and the coefficients \[\Gamma_{\beta;\mu}^\alpha=G^{\bar{\tau}\alpha}\delta_\mu(G_{\beta\bar{\tau}}),\quad \Gamma_{\beta\gamma}^\alpha=G^{\bar{\tau}\alpha}\frac{\partial}{\partial v^\gamma}(G_{\beta\bar{\tau}})=\Gamma_{\gamma\beta}^\alpha.\label{HVC}\tag{8}\] are called the horizontal and vertical connection coefficients of \(D\), respectively.
Several important classes of complex Finsler metrics are defined via the horizontal connection coefficients \(\Gamma_{\beta;\mu}^\alpha\):
\(F\) is called a Kähler-Finsler metric (cf. [8], [11]) if \[\Gamma_{\beta;\mu}^\alpha-\Gamma_{\mu;\beta}^\alpha=0,\label{KFD}\tag{9}\] which is equivalent to the condition \((\Gamma_{\beta;\mu}^\alpha-\Gamma_{\mu;\beta}^\alpha)v^\beta=0\) .
\(F\) is called a weakly Kähler-Finsler metric [8] if \[G_\alpha(\Gamma_{\beta;\mu}^\alpha-\Gamma_{\mu;\beta}^\alpha)v^\beta=0;\]
\(F\) is called a complex Berwald metric [12] if each \(\Gamma_{\beta;\mu}^\alpha\) depends only on the base point \(z\);
\(F\) is called a complex locally Minkowski metric if each \(\Gamma_{\beta;\mu}^\alpha\) depends locally only on the fiber coordinate \(v\).
A complex Berwald manifold \((M,F)\) is also characterized by being modeled on a complex Minkowski space [7].
Definition 5. Let \(F:T^{1,0}M\rightarrow [0,+\infty)\) be a strongly convex complex Finsler metric on a complex manifold \(M\). If \(F\) is both a Kähler-Finsler metric and a complex Berwald metric, then \(F\) is called a strongly convex Kähler-Berwald metric, and \((M,F)\) is called a strongly convex Kähler-Berwald manifold.
Remark 10. While Kähler metrics represent the optimal symmetry in Hermitian geometry, our recent work [1]–[4] reveals a similar phenomenon in complex Finsler geometry: the only invariant, strongly pseudoconvex metrics admissible on symmetric complex manifolds are Kähler-Berwald metrics. This fact establishes Kähler-Berwald metrics as the natural counterpart to Kähler metrics in Finsler setting and naturally raises the problem of characterizing them among all complex Finsler metrics.
Let \(\langle\langle\cdot,\cdot\rangle\rangle\) denote the symmetric product induced by \(F\) on \(\mathcal{V}^{1,0}\), defined for \(V=V^\alpha\frac{\partial}{\partial v^\alpha},W=W^\beta\frac{\partial}{\partial v^\beta}\in(\mathcal{V}^{1,0})_{(z;v)}\) by \[\langle\langle V,W\rangle\rangle=G_{\alpha\beta}(z;v)V^\alpha W^\beta.\]
Proposition 11. [8]Let \(F:T^{1,0}M\rightarrow [0,+\infty)\) be a strongly convex complex Finsler metric on a complex manifold \(M\). Then \[\langle V^o\vert W^o\rangle=Re\,[\langle V, W\rangle+\langle\langle V,W\rangle\rangle],\quad\forall V,W\in \mathcal{V}^{1,0}.\label{VW-a}\qquad{(3)}\]
Remark 12. In applications of the above identity ?? , the presence of the symmetric product \(\langle\langle V,W\rangle\rangle\) possibly causes inconvenience. Fortunately, this drawback can be overcome by the following Lemma 1.
More precisely, extending the real inner product \(\langle \cdot\vert \cdot\rangle\) to the complexification \(\mathcal{V}_{\mathbb{C}}=\mathcal{V}_{\mathbb{R}}\otimes \mathbb{C}\) in the following natural way: \[\langle X_1+iY_1\vert X_2+iY_2\rangle=\langle X_1\vert X_2\rangle-\langle Y_1\vert Y_2\rangle+i(\langle X_1\vert Y_2\rangle+\langle Y_1\vert X_2\rangle).\label{CLE}\tag{10}\] This extension (still denoted \(\langle\cdot\vert \cdot\rangle\)) is \(\mathbb{C}\)-bilinear. In particular, it is \(\mathbb{C}\)-bilinear on \(\mathcal{V}^{1,0}\) and \(\mathcal{V}^{1,0}\), since \(\mathcal{V}_{\mathbb{C}}=\mathcal{V}^{1,0}\oplus\mathcal{V}^{0,1}\).
The following lemma plays a crucial role in the proof of Theorem 23 and Theorem 24.
Lemma 1. Let \(F:T^{1,0}M\rightarrow [0,+\infty)\) be a strongly convex complex Finsler metric on a complex manifold \(M\). Then \[\label{eq-rh} \langle V, W\rangle=2\langle V\vert \overline{W}\rangle,\quad \forall V,W\in\mathcal{V}^{1,0}.\qquad{(4)}\]
Proof. Let \(V,W\in\mathcal{V}^{1,0}\) be written as \[V=X_o=\frac{1}{2}(X-i\pmb{J}X),\quad W=Y_o=\frac{1}{2}(Y-i\pmb{J}Y)\label{VW}\tag{11}\] for real vertical vector fields \(X, Y\in \mathcal{V}_{\mathbb{R}}\). Identifying \(X=V^o\), \(Y=W^o\), \(\pmb{J}X=(iV)^o\), and \(\pmb{J}Y=(iW)^o\), and substituting into ?? , we obtain \[\begin{align} \langle X\vert Y\rangle&=&\text{Re}\,[\langle V,W\rangle+\langle\!\langle V,W \rangle\!\rangle],\\ \langle \pmb{J}X\vert \pmb{J}Y\rangle&=&\text{Re}\,[\langle V,W\rangle-\langle\!\langle V,W \rangle\!\rangle],\\ \langle X\vert \pmb{J}Y\rangle&=&\text{Im}\,[\langle V,W\rangle-\langle\!\langle V,W \rangle\!\rangle],\\ \langle \pmb{J}X\vert Y\rangle&=&-\text{Im}\,[\langle V,W\rangle+\langle\!\langle V,W \rangle\!\rangle]. \end{align}\] From these, it follows that \[\begin{align} \langle X\vert Y\rangle+\langle \pmb{J}X\vert \pmb{J}Y\rangle&=&2\text{Re}[\langle V,W\rangle],\\ \langle X\vert \pmb{J}Y\rangle-\langle \pmb{J}X\vert Y\rangle&=&2\text{Im}[\langle V,W\rangle]. \end{align}\] Therefore, \[\langle X\vert Y\rangle+\langle \pmb{J}X\vert JY\rangle+i\langle X\vert \pmb{J}Y\rangle-i\langle \pmb{J}X\vert Y\rangle=2\langle V,W\rangle.\] On the other hand, using the extension 10 , we compute \[\langle V\vert \overline{W}\rangle=\frac{1}{4}\left\{\langle X\vert Y\rangle+\langle \pmb{X}\vert \pmb{J}Y\rangle+i\langle X\vert \pmb{J}Y\rangle-i\langle \pmb{J}X\vert Y\rangle\right\}.\] Comparing the two expressions yields the desired identity ?? . ◻
Remark 13. Via the horizontal map \(\hat{\Theta}:\mathcal{V}_{\mathbb{R}}\rightarrow\mathcal{H}_{\mathbb{R}}\), the inner product \(\langle\cdot\vert \cdot\rangle\) can also be extended to the complexification \(\hat{\mathcal{H}}_{\mathbb{C}}:=\mathcal{H}_{\mathbb{R}}\otimes \mathbb{C}\). In general, however, \(\hat{\mathcal{H}}_{\mathbb{C}}\neq \mathcal{H}^{1,0}\oplus \mathcal{H}^{0,1}\).
In this section, we mainly investigate the properties of strongly convex complex Finsler manifolds \((M,F)\) under the condition that the canonical complex structure \(\pmb{J}\) on the complex manifold \(T^{1,0}M\) satisfies some different extent of parallelism with respect to the Cartan connection \(\nabla\) associated to \(F\).
Firstly, we obtain the following necessary and sufficient condition for the parallelism of \(\pmb{J}\) with respect to \(\nabla\).
Theorem 14. Let \(F: T^{1,0}M\rightarrow [0,+\infty)\) be a strongly convex complex Finsler metric on a complex manifold \(M\). Then \(\nabla \pmb{J}=0\) iff \[\begin{align} \hat{\Gamma}_{\alpha^\ast ;j}^\beta&=&-\hat{\Gamma}_{\alpha; j}^{\beta^\ast },\quad\hat{\Gamma}_{\alpha^\ast ;j}^{\beta^\ast }=\Gamma_{\alpha;j}^\beta,\label{eq-ab}\\ \hat{\Gamma}_{\alpha^\ast j}^\beta&=&-\hat{\Gamma}_{\alpha j}^{\beta^\ast }, \quad \hat{\Gamma}_{\alpha^\ast j}^{\beta^\ast }=\hat{\Gamma}_{\alpha j}^\beta,\label{eq-abc} \end{align}\] {#eq: sublabel=eq:eq-ab,eq:eq-abc} for all \(\alpha,\beta=1,\cdots,n\) and \(j=1,\cdots,2n\).
Proof. Since \[(\nabla_X \pmb{J} )(Y)=\nabla_X( \pmb{J} Y)- \pmb{J} (\nabla_XY)\] for all \(X\in T_{\mathbb{R}}\tilde{M}\) and \(Y\in\mathcal{X}(\mathcal{V}_{\mathbb{R}})\), the condition that \(\nabla\pmb{J} =0\) is equivalent to \[\nabla_X( \pmb{J} Y)= \pmb{J} (\nabla_XY)\label{pc}\tag{12}\] for all \(X\in T_{\mathbb{R}}\tilde{M}\) and \(Y\in\mathcal{X}(\mathcal{V}_{\mathbb{R}})\).
Taking \(X=\frac{\delta}{\delta x^j}, Y=\frac{\partial}{\partial y^k}\) and substituting them into 12 and using 1 , we obtain \[\begin{align} \nabla_{\frac{\delta}{\delta x^j}}\left( \pmb{J} \frac{\partial}{\partial y^k}\right)&=&\left\{ \begin{array}{ll} \hat{\Gamma}_{\alpha^\ast ;j}^\beta\frac{\partial}{\partial y^\beta}+\hat{\Gamma}_{\alpha^\ast ;j}^{\beta^\ast }\frac{\partial}{\partial y^{\beta^\ast }}, & k=\alpha, \\ -\hat{\Gamma}_{\alpha; j}^\beta\frac{\partial}{\partial y^\beta}-\hat{\Gamma}_{\alpha ;j}^{\beta^\ast }\frac{\partial}{\partial y^{\beta^\ast }}, & k=\alpha^\ast \end{array} \right. \label{eq-ha} \end{align}\tag{13}\] and \[\begin{align} \pmb{J}\left(\nabla_{\frac{\delta}{\delta x^j}}\frac{\partial}{\partial y^k}\right) =\left\{ \begin{array}{ll} \hat{\Gamma}_{\alpha; j}^\beta\frac{\partial}{\partial y^{\beta^\ast }}-\hat{\Gamma}_{\alpha; j}^{\beta^\ast }\frac{\partial}{\partial y^\beta}, & k=\alpha, \\ \hat{\Gamma}_{\alpha^\ast ; j}^\beta\frac{\partial}{\partial y^{\beta^\ast }}-\hat{\Gamma}_{\alpha^\ast ; j}^{\beta^\ast }\frac{\partial}{\partial y^\beta}, & k=\alpha^\ast . \end{array} \right. \label{eq-hb} \end{align}\tag{14}\] Comparing 13 and 14 , we see that 12 holds for all \(X\in\mathcal{H}_{\mathbb{R}}\) and \(Y\in\mathcal{X}(\mathcal{V}_{\mathbb{R}})\) iff the equalities in ?? hold.
Taking \(X=\frac{\partial}{\partial y^j}, Y=\frac{\partial}{\partial y^k}\) and substituting them into 12 and using 1 , we obtain \[\begin{align} \nabla_{\frac{\partial }{\partial y^j}}\left( \pmb{J} \frac{\partial}{\partial y^k}\right)&=&\left\{ \begin{array}{ll} \hat{\Gamma}_{\alpha^\ast j}^\beta\frac{\partial}{\partial y^\beta}+\hat{\Gamma}_{\alpha^\ast j}^{\beta^\ast }\frac{\partial}{\partial y^{\beta^\ast }}, & k=\alpha, \\ -\hat{\Gamma}_{\alpha j}^\beta\frac{\partial}{\partial y^\beta}-\hat{\Gamma}_{\alpha j}^{\beta^\ast }\frac{\partial}{\partial y^{\beta^\ast }}, & k=\alpha^\ast \end{array} \right. \label{eq-hc} \end{align}\tag{15}\] and \[\begin{align} \pmb{J}\left(\nabla_{\frac{\partial}{\partial y^j}}\frac{\partial}{\partial y^k}\right) =\left\{ \begin{array}{ll} \hat{\Gamma}_{\alpha j}^\beta\frac{\partial}{\partial y^{\beta^\ast }}-\hat{\Gamma}_{\alpha j}^{\beta^\ast }\frac{\partial}{\partial y^\beta}, & k=\alpha, \\ \hat{\Gamma}_{\alpha^\ast j}^\beta\frac{\partial}{\partial y^{\beta^\ast }}-\hat{\Gamma}_{\alpha^\ast j}^{\beta^\ast }\frac{\partial}{\partial y^\beta}, & k=\alpha^\ast . \end{array} \right. \label{eq-hd} \end{align}\tag{16}\] Comparing 15 and 16 , we see that 12 holds for all \(X\in\mathcal{V}_{\mathbb{R}}\) and \(Y\in\mathcal{X}(\mathcal{V}_{\mathbb{R}})\) iff ?? hold. ◻
Corollary 1. Let \(F: T^{1,0}M\rightarrow [0,+\infty)\) be a strongly convex complex Finsler metric on a complex manifold \(M\). If \(\pmb{J}\) is horizontal parallel with respect to \(\nabla\), namely \(\nabla_X \pmb{J}=0\) for all \(X\in\mathcal{H}_{\mathbb{R}}\), then \[\begin{align} \hat{\Gamma}_{\alpha^\ast }^\beta=-\hat{\Gamma}_{\alpha}^{\beta^\ast },\quad \hat{\Gamma}_{\alpha^\ast }^{\beta^\ast }=\hat{\Gamma}_{\alpha}^\beta \label{eq-cc} \end{align}\qquad{(5)}\] for \(\alpha=1,\cdots,n\).
Proof. By Theorem 14, \(\nabla_X \pmb{J}=0\) for all \(X\in\mathcal{H}_{\mathbb{R}}\) iff the equalities in ?? hold. Since the Cartann connection coefficients are symmetric with respect to lower indices, namely \(\hat{\Gamma}_{j;l}^k=\hat{\Gamma}_{l;j}^k\), and they satisfy \(\hat{\Gamma}_{j;l}^ky^j=\hat{\Gamma}_{l}^k\). Contracting the equalities in ?? with \(y^j\), we immediately obtain ?? . ◻
Definition 6. Let \(\pmb{J}\) be the canonical structure on the complex manifold \(T^{1,0}M\). The real horizontal bundle \(\mathcal{H}_{\mathbb{R}}\) is called \(\pmb{J}\)-invariant if \[\pmb{J}\left(\frac{\delta}{\delta x^\alpha}\right)=\frac{\delta}{\delta x^{\alpha^\ast }},\quad \pmb{J}\left(\frac{\delta}{\delta x^{\alpha^\ast }}\right)=-\frac{\delta}{\delta x^\alpha},\quad\alpha=1,\cdots,n.\]
Proposition 15. [8] \(\mathcal{H}_{\mathbb{R}}\) is \(\pmb{J}\)-invariant iff the equalities in ?? hold.
Proof. Indeed, \[\begin{align} \pmb{J}\left(\frac{\delta}{\delta x^\alpha}\right)&=&\frac{\partial}{\partial x^{\alpha^\ast }}-\hat{\Gamma}_{\alpha}^\beta\frac{\partial}{\partial x^{\beta^\ast }}+\hat{\Gamma}_{\alpha}^{\beta^\ast }\frac{\partial}{\partial x^\beta},\quad\frac{\delta}{\delta x^{\alpha^\ast }} =\frac{\partial}{\partial x^{\alpha^\ast }}-\hat{\Gamma}_{\alpha^\ast }^{\beta^\ast }\frac{\partial}{\partial x^{\beta^\ast }}-\hat{\Gamma}_{\alpha^\ast }^\beta\frac{\partial}{\partial x^\beta},\\ \pmb{J}\left(\frac{\delta}{\delta x^{\alpha^\ast }}\right)&=&-\frac{\partial}{\partial x^{\alpha}}-\hat{\Gamma}_{\alpha^\ast }^\beta\frac{\partial}{\partial x^{\beta^\ast }}+\hat{\Gamma}_{\alpha^\ast }^{\beta^\ast }\frac{\partial}{\partial x^\beta},\quad-\frac{\delta}{\delta x^\alpha} =-\frac{\partial}{\partial x^\alpha}+\hat{\Gamma}_{\alpha}^{\beta^\ast }\frac{\partial}{\partial x^{\beta^\ast }}+\hat{\Gamma}_{\alpha}^{\beta}\frac{\partial}{\partial x^\beta}. \end{align}\] Thus \(\mathcal{H}_{\mathbb{R}}\) is \(\pmb{J}\)-invariant iff the equalities in ?? hold. ◻
Let \(\widetilde{\mathcal{H}}_{\mathbb{C}}=\mathcal{H}_{\mathbb{R}}\otimes\mathbb{C}\) be the complexified horizontal bundle of \(\mathcal{H}_{\mathbb{R}}\). Formally, we have \[\begin{align} \left(\frac{\delta}{\delta x^\alpha}\right)_o=\frac{1}{2}\left(\frac{\delta}{\delta x^\alpha}-i \pmb{J}\frac{\delta}{\delta x^\alpha}\right),\, \left(\frac{\delta}{\delta x^{\alpha^\ast }}\right)_o=\frac{1}{2}\left(\frac{\delta}{\delta x^{\alpha^\ast }}-i \pmb{J}\frac{\delta}{\delta x^{\alpha^\ast }}\right), \,\alpha=1,\cdots,n. \end{align}\] In general, \(\left(\frac{\delta}{\delta x^\alpha}\right)_o\neq \delta_\alpha\) and \(\left(\frac{\delta}{\delta x^{\alpha^\ast }}\right)_o\neq i\delta_\alpha\), as pointed out on page 114 in [8]. However, if \(\mathcal{H}_{\mathbb{R}}\) is \(\pmb{J}\)-invariant, then we are able to obtain a complex horizontal bundle, denoted by \(\widetilde{\mathcal{H}}^{1,0}\).
For this purpose, we define \[\mathcal{N}_{;\alpha}^\beta:=\hat{\Gamma}_{\alpha}^\beta+i\hat{\Gamma}_{\alpha}^{\beta^\ast }\] and \[\frac{\delta}{\delta z^\alpha}:=\frac{\partial}{\partial z^\alpha}-\mathcal{N}_{;\alpha}^\beta\frac{\partial}{\partial v^\beta},\quad \alpha=1,\cdots,n.\]
Proposition 16. If \(\mathcal{H}_{\mathbb{R}}\) is \(\pmb{J}\)-invariant, then \[\left(\frac{\delta}{\delta x^\alpha}\right)_o=\frac{\delta}{\delta z^\alpha}\quad\text{and}\quad \left(\frac{\delta}{\delta x^{\alpha^\ast }}\right)_o=i\frac{\delta}{\delta z^\alpha},\] or equivalently \[\left(\frac{\delta}{\delta z^\alpha}\right)^o=\frac{\delta}{\delta x^\alpha}\quad\text{and}\quad \left(i\frac{\delta}{\delta z^\alpha}\right)^o=\frac{\delta}{\delta x^{\alpha^\ast }}.\]
Proof. Direct computation shows \[\begin{align} \left(\frac{\delta}{\delta x^\alpha}\right)_o&=&\frac{1}{2}\left(\frac{\delta}{\delta x^\alpha}-i \pmb{J}\frac{\delta}{\delta x^\alpha}\right)\\ &=&\frac{1}{2}\left[\frac{\partial}{\partial x^\alpha}-\hat{\Gamma}_{\alpha}^\beta\frac{\partial}{\partial y^\beta}-\hat{\Gamma}_{\alpha}^{\beta^\ast }\frac{\partial}{\partial y^{\beta^\ast }}-i\left(\frac{\partial}{\partial x^{\alpha^\ast }}-\hat{\Gamma}_{\alpha}^\beta\frac{\partial}{\partial y^{\beta^\ast }}+\hat{\Gamma}_{\alpha}^{\beta^\ast }\frac{\partial}{\partial y^\beta}\right)\right]\\ &=&\frac{1}{2}\left[\left(\frac{\partial}{\partial x^\alpha}-i\frac{\partial}{\partial x^{\alpha^\ast }}\right)-\left(\hat{\Gamma}_{\alpha}^\beta+i\hat{\Gamma}_{\alpha}^{\beta^\ast }\right)\frac{\partial}{\partial y^\beta}+i\left(\hat{\Gamma}_{\alpha}^\beta+i\hat{\Gamma}_{\alpha}^{\beta^\ast }\right)\frac{\partial}{\partial y^{\beta^\ast }}\right]\\ &=&\frac{1}{2}\left[\left(\frac{\partial}{\partial x^\alpha}-i\frac{\partial}{\partial x^{\alpha^\ast }}\right)-\left(\hat{\Gamma}_{\alpha}^\beta+i\hat{\Gamma}_{\alpha}^{\beta^\ast }\right)\left(\frac{\partial}{\partial y^\beta}-i\frac{\partial}{\partial y^{\beta^\ast }}\right)\right]\\ &=&\frac{\partial}{\partial z^\alpha}-\left(\hat{\Gamma}_{\alpha}^\beta+i\hat{\Gamma}_{\alpha}^{\beta^\ast }\right)\frac{\partial}{\partial v^\beta}\\ &=&\frac{\delta}{\delta z^\alpha} \end{align}\] and \[\begin{align} \left(\frac{\delta}{\delta x^{\alpha^\ast }}\right)_o&=&\frac{1}{2}\left(\frac{\delta}{\delta x^{\alpha^\ast }}-i \pmb{J}\frac{\delta}{\delta x^{\alpha^\ast }}\right)\\ &=&\frac{1}{2}\left[\frac{\partial}{\partial x^{\alpha^\ast }}-\hat{\Gamma}_{\alpha^\ast }^\beta\frac{\partial}{\partial y^\beta}-\hat{\Gamma}_{\alpha^\ast }^{\beta^\ast }\frac{\partial}{\partial y^{\beta^\ast }}-i\left(-\frac{\partial}{\partial x^\alpha}-\hat{\Gamma}_{\;\alpha^\ast }^\beta\frac{\partial}{\partial y^{\beta^\ast }}+\hat{\Gamma}_{\alpha^\ast }^{\beta^\ast }\frac{\partial}{\partial y^{\beta}}\right)\right]\\ &=&\frac{1}{2}\left[i\left(\frac{\partial}{\partial x^\alpha}-i\frac{\partial}{\partial x^{\alpha^\ast }}\right)-\left(\hat{\Gamma}_{\alpha^\ast }^\beta+i\hat{\Gamma}_{\alpha^\ast }^{\beta^\ast }\right)\frac{\partial}{\partial y^\beta}-\left(\hat{\Gamma}_{\alpha^\ast }^{\beta^\ast }-i\hat{\Gamma}_{\alpha^\ast }^\beta\right)\frac{\partial}{\partial y^{\beta^\ast }}\right]\\ &=&\frac{1}{2}\left[i\left(\frac{\partial}{\partial x^\alpha}-i\frac{\partial}{\partial x^{\alpha^\ast }}\right)-i\left(\hat{\Gamma}_{\alpha^\ast }^{\beta^\ast }-i\hat{\Gamma}_{\alpha^\ast }^\beta\right)\frac{\partial}{\partial y^\beta}-\left(\hat{\Gamma}_{\alpha^\ast }^{\beta^\ast }-i\hat{\Gamma}_{\;\alpha^\ast }^\beta\right)\frac{\partial}{\partial y^{\beta^\ast }}\right]\\ &=&i\frac{1}{2}\left[\left(\frac{\partial}{\partial x^\alpha}-i\frac{\partial}{\partial x^{\alpha^\ast }}\right)-\left(\hat{\Gamma}_{\alpha^\ast }^{\beta^\ast }-i\hat{\Gamma}_{\alpha^\ast }^\beta\right)\left(\frac{\partial}{\partial y^\beta}-i\frac{\partial}{\partial y^{\beta^\ast }}\right)\right]\\ &=&i\left[\frac{\partial}{\partial z^\alpha}-\left(\hat{\Gamma}_{\alpha^\ast }^{\beta^\ast }-i\hat{\Gamma}_{\alpha^\ast }^\beta\right)\frac{\partial}{\partial v^\beta}\right]\\ &=&i\left[\frac{\partial}{\partial z^\alpha}-\left(\hat{\Gamma}_{\alpha}^{\beta}+i\hat{\Gamma}_{\alpha}^{\beta^\ast }\right)\frac{\partial}{\partial v^\beta}\right]\\ &=&i\frac{\delta}{\delta z^\alpha}. \end{align}\] ◻
Let \(\hat{\Gamma}_{j;l}^k\) and \(\hat{\Gamma}_{jl}^k\) be the horizontal and vertical Cartan connection coefficients of \(F\). Define \[\mathcal{N}_{\alpha;\gamma}^\beta:=\hat{\Gamma}_{\alpha;\gamma}^\beta+i\hat{\Gamma}_{\alpha;\gamma}^{\beta^\ast } \quad \text{and }\quad \mathcal{N}_{\alpha\gamma}^\beta:=\hat{\Gamma}_{\alpha\gamma}^\beta+i\hat{\Gamma}_{\alpha\gamma}^{\beta^\ast },\quad \alpha,\beta,\gamma=1,\cdots,n.\]
Remark 17. Since \(\hat{\Gamma}_{j;l}^k=\hat{\Gamma}_{l;j}^k\) and \(\hat{\Gamma}_{jl}^k=\hat{\Gamma}_{lj}^k\), it is clear that \[\mathcal{N}_{\alpha;\gamma}^\beta=\mathcal{N}_{\gamma;\alpha}^\beta \quad and\quad \mathcal{N}_{\alpha\gamma}^\beta=\mathcal{N}_{\gamma\alpha}^\beta.\label{SN}\qquad{(6)}\]
Proposition 18. If \(\nabla_X\pmb{J}=0\) for all \(X\in\mathcal{H}_{\mathbb{R}}\), then \[\mathcal{N}_{\alpha;\gamma}^\beta v^\alpha=\mathcal{N}_{;\gamma}^\beta;\] if \(\nabla_Y\pmb{J}=0\) for all \(Y\in\mathcal{V}_{\mathbb{R}}\), then \[\mathcal{N}_{\alpha\gamma}^\beta v^\alpha=0.\]
Proof. Indeed, since \(v^\alpha=y^\alpha+iy^{\alpha^\ast}\), \[\begin{align} \mathcal{N}_{\alpha;\gamma}^\beta v^\alpha &=&(\hat{\Gamma}_{\alpha;\gamma}^\beta y^\alpha-\hat{\Gamma}_{\alpha;\gamma}^{\beta^\ast }y^{\alpha^\ast })+i(\hat{\Gamma}_{\alpha;\gamma}^\beta y^{\alpha^\ast }+\hat{\Gamma}_{\alpha;\gamma}^{\beta^\ast }y^\alpha). \end{align}\] By assumption \(\nabla_X\pmb{J}=0\) for all \(X\in\mathcal{H}_{\mathbb{R}}\), thus we have ?? . So that \[\begin{align} \mathcal{N}_{\alpha;\gamma}^\beta v^\alpha&=&(\hat{\Gamma}_{\alpha;\gamma}^\beta y^\alpha+\hat{\Gamma}_{\alpha^\ast ;\gamma}^{\beta}y^{\alpha^\ast })+i(\hat{\Gamma}_{\alpha^\ast ;\gamma}^{\beta^\ast } y^{\alpha^\ast }+\hat{\Gamma}_{\alpha;\gamma}^{\beta^\ast }y^\alpha) =\hat{\Gamma}_\gamma^\beta+i\hat{\Gamma}_\gamma^{\beta^\ast}=\mathcal{N}_{;\gamma}^\beta, \end{align}\] since the horizontal Cartan connection coefficients satisfy \(\hat{\Gamma}_{j;l}^ky^j=\hat{\Gamma}_{l}^k\).
Similarly, \[\begin{align} \mathcal{N}_{\alpha\gamma}^\beta v^\alpha &=&(\hat{\Gamma}_{\alpha\gamma}^\beta y^\alpha-\hat{\Gamma}_{\alpha\gamma}^{\beta^\ast }y^{\alpha^\ast })+i(\hat{\Gamma}_{\alpha\gamma}^\beta y^{\alpha^\ast }+\hat{\Gamma}_{\alpha\gamma}^{\beta^\ast }y^\alpha). \end{align}\] By assumption \(\nabla_Y\pmb{J}=0\) for all \(Y\in\mathcal{V}_{\mathbb{R}}\), thus we have ?? . So that \[\begin{align} \mathcal{N}_{\alpha\gamma}^\beta v^\alpha&=&(\hat{\Gamma}_{\alpha\gamma}^\beta y^\alpha+\hat{\Gamma}_{\alpha^\ast \gamma}^{\beta}y^{\alpha^\ast })+i(\hat{\Gamma}_{\alpha^\ast \gamma}^{\beta^\ast } y^{\alpha^\ast }+\hat{\Gamma}_{\alpha\gamma}^{\beta^\ast }y^\alpha)=\hat{\Gamma}_{j\gamma}^\beta y^j+i\hat{\Gamma}_{j\gamma}^{\beta^\ast}y^j=0, \end{align}\] since the vertical Cartan connection coefficients satisfy \(\hat{\Gamma}_{jl}^ky^j=0\). ◻
Proposition 19. Let \(F: T^{1,0}M\rightarrow [0,+\infty)\) be a strongly convex complex Finsler metric on a complex manifold \(M\). Let \(U_A\) and \(U_B\) be local holomorphic coordinate neighborhoods on \(M\) with \(U_A\cap U_B\neq \emptyset\), such that \((z_A;v_A)=(z_A^1,\cdots, z_A^n; v_A^1,\cdots, v_A^n)\) and \((z_B;v_B)=(z_B^1,\cdots, z_B^n;v_B^1\), \(\cdots\), \(v_B^n)\) are the induced holomorphic coordinates on \(\pi^{-1}(U_A)\) and \(\pi^{-1}(U_B)\subset T^{1,0}M\), respectively. If \(\nabla \pmb{J}=0\), then \[\begin{align} \left(\mathcal{N}_{\alpha;\gamma}^\beta\right)_B&=&\frac{\partial z_A^\mu}{\partial z_B^\alpha}\frac{\partial z_A^\nu}{\partial z_B^\gamma}\left(\mathcal{N}_{\mu;\nu}^\delta\right)_A\frac{\partial z_B^\beta}{\partial z_A^\delta} -\frac{\partial z_A^\mu}{\partial z_B^\alpha}\frac{\partial z_A^\nu}{\partial z_B^\gamma}\frac{\partial^2 z_B^\beta}{\partial z_A^\mu\partial z_A^\nu},\label{cbc-a}\\ \left(\mathcal{N}_{;\gamma}^\beta\right)_B&=&\frac{\partial z_A^\nu}{\partial z_B^\gamma}\left(\mathcal{N}_{;\nu}^\delta\right)_A\frac{\partial z_B^\beta}{\partial z_A^\delta} -\frac{\partial z_A^\nu}{\partial z_B^\gamma}\frac{\partial^2 z_B^\beta}{\partial z_A^\mu\partial z_A^\nu}v_A^\mu,\label{cbc-b}\\ \left(\mathcal{N}_{\alpha\gamma}^\beta\right)_B&=&\frac{\partial z_A^\mu}{\partial z_B^\alpha}\frac{\partial z_A^\nu}{\partial z_B^\gamma}\left(\mathcal{N}_{\mu\nu}^\delta\right)_A\frac{\partial z_B^\beta}{\partial z_A^\delta}.\label{cbc-c} \end{align}\] {#eq: sublabel=eq:cbc-a,eq:cbc-b,eq:cbc-c}
Proof. Let \[\begin{align} z_A^\alpha=x_A^\alpha+ix_A^{\alpha^\ast },\quad z_B^\alpha=x_B^\alpha+ix_B^{\alpha^\ast },\quad v_A^\alpha=y_A^\alpha+iy_A^{\alpha^\ast },\quad v_B^\alpha=y_B^\alpha+iy_B^{\alpha^\ast } \end{align}\] for \(\alpha=1,\cdots,n\). Then \(x_A=(x_A^1,\cdots, x_A^{2n})\) and \(x_B=(x_B^1,\cdots, x_B^{2n})\) are local real coordinates on \(U_A\) and \(U_B\), respectively, and \((x_A,y_A)=(x_A^1,\cdots, x_A^{2n};y_A^1,\cdots, y_A^{2n}),(x_B;y_B)=(x_B^1,\cdots,x_B^{2n};y_B^1,\cdots,y_B^{2n})\) are the induced real coordinates on \(\pi^{-1}(U_A)\) and \(\pi^{-1}(U_B)\), respectively. Under a change of coordinates on \(\pi^{-1}(U_A)\cap \pi^{-1}(U_B)\neq \emptyset\), the horizontal connection coefficients transform as \[\label{eq-clc1} \left(\hat{\Gamma}_{a;c}^b\right)_B=\sum_{h,k=1}^{2n}\frac{\partial x_A^h}{\partial x_B^a}\frac{\partial x_A^k}{\partial x_B^c}\left[\sum_{j=1}^{2n}\left(\hat{\Gamma}_{h;k}^j\right)_A\frac{\partial x_B^b}{\partial x_A^j}-\frac{\partial^2x_B^b}{\partial x_A^h\partial x_A^k}\right].\tag{17}\] By definition, \(\mathcal{N}_{\alpha;\gamma}^\beta=\hat{\Gamma}_{\alpha;\gamma}^\beta+i\hat{\Gamma}_{\alpha;\gamma}^{\beta^\ast }\), which together with 17 yield \[\left(\mathcal{N}_{\alpha;\gamma}^\beta\right)_B =\sum_{h,k=1}^{2n}\frac{\partial x_A^h}{\partial x_B^\alpha}\frac{\partial x_A^k}{\partial x_B^\gamma}\left[\sum_{j=1}^{2n}\left(\hat{\Gamma}_{h;k}^j\right)_A\frac{\partial z_B^\beta}{\partial x_A^j}-\frac{\partial^2z_B^\beta}{\partial x_A^h\partial x_A^k}\right].\label{eq-b}\tag{18}\]
Since \(z_B^\beta\) is holomorphic in \(z_A^\delta=x_A^\delta+ix_A^{\delta^\ast }\), we have \[\label{eq-zhc} \frac{\partial z_B^\beta}{\partial x_A^\delta}=\frac{\partial z_B^\beta}{\partial z_A^\delta}\quad \text{and } \quad \frac{\partial z_B^\beta}{\partial x_A^{\delta^\ast}}=i\frac{\partial z_B^\beta}{\partial z_A^\delta}.\tag{19}\] Thus \[\begin{align} \sum_{j=1}^{2n}\left(\hat{\Gamma}_{h;k}^j\right)_A\frac{\partial z_B^\beta}{\partial x_A^j} &=&\sum_{\delta=1}^{n}\left[\left(\hat{\Gamma}_{h;k}^\delta\right)_A\frac{\partial z_B^\beta}{\partial x_A^\delta} +\left(\hat{\Gamma}_{h;k}^{\delta^\ast }\right)_A\frac{\partial z_B^\beta}{\partial x_A^{\delta^\ast }}\right]\\ &=&\sum_{\delta=1}^n\left[\left(\hat{\Gamma}_{h;k}^\delta\right)_A+i\left(\hat{\Gamma}_{h;k}^{\delta^\ast }\right)_A\right]\frac{\partial z_B^\beta}{\partial z_A^\delta}. \end{align}\] Using 13 , a direct computation shows that \[\begin{align} &&\sum_{h,k=1}^{2n}\frac{\partial x_A^h}{\partial x_B^\alpha}\frac{\partial x_A^k}{\partial x_B^\gamma}\sum_{\delta=1}^n\left[\left(\hat{\Gamma}_{h;k}^\delta\right)_A+i\left(\hat{\Gamma}_{h;k}^{\delta^\ast }\right)_A\right]\\ &=&\sum_{\mu,\nu=1}^n\frac{\partial z_A^\mu}{\partial x_B^\alpha}\frac{\partial z_A^\nu}{\partial x_B^\gamma}\sum_{\delta=1}^n\left[\left(\hat{\Gamma}_{\mu;\nu}^\delta\right)_A+i\left(\hat{\Gamma}_{\mu;\nu}^{\delta^\ast }\right)_A\right]\\ &=&\frac{\partial z_A^\mu}{\partial x_B^\alpha}\frac{\partial z_A^\nu}{\partial x_B^\gamma}\left(\mathcal{N}_{\mu;\nu}^\delta\right)_A. \end{align}\] Similarly, \[\sum_{h,k=1}^{2n}\frac{\partial x_A^h}{\partial x_B^\alpha}\frac{\partial x_A^k}{\partial x_B^\gamma}\frac{\partial^2z_B^\beta}{\partial x_A^h\partial x_A^k}=\sum_{\mu,\nu=1}^n\frac{\partial z_A^\mu}{\partial x_B^\alpha}\frac{\partial z_A^\nu}{\partial z_B^\gamma}\frac{\partial^2z_B^\beta}{\partial z_A^\mu\partial z_A^\nu}.\] This proves ?? . Contracting ?? with \(v_B^\alpha=v_A^\gamma\frac{\partial z_B^\alpha}{\partial z_A^\gamma}\) yields ?? .
The vertical connection coefficients \(\hat{\Gamma}_{jk}^i\) of the Cartan connection transform as follows [8] \[\left(\hat{\Gamma}_{ac}^b\right)_B=\sum_{h,k=1}^{2n}\frac{\partial x_A^h}{\partial x_B^a}\frac{\partial x_A^k}{\partial x_B^c}\left[\sum_{j=1}^{2n}\left(\hat{\Gamma}_{hk}^j\right)_A\frac{\partial x_B^b}{\partial x_A^j}\right].\] Using the definition of \(\mathcal{N}_{\alpha\gamma}^\beta\) in ?? , we obtain ?? by a similar argument. ◻
If \(\nabla_X \pmb{J}=0\) for all \(X\in\mathcal{H}_{\mathbb{R}}\), then the transformation rule ?? shows that \(\mathcal{N}_{;\gamma}^\beta\) are coefficients of a complex non-linear connection. In other words, \(\widetilde{\mathcal{H}}^{1,0}\) is a complex horizontal bundle, which is spanned by \(\{\frac{\delta}{\delta z^1},\cdots,\frac{\delta}{\delta z^n}\}\). Now we define \[\delta v^\alpha=dv^\alpha+\mathcal{N}_{;\beta}^\alpha dz^\beta,\quad \alpha=1,\cdots,n.\label{hb}\tag{20}\] Then \(\{\delta v^\alpha\}\) is a local frame for the dual bundle \((\mathcal{V}^{1,0})^\ast\). Thus we obtain the following decompositions \[\begin{align} T_{\mathbb{C}}\tilde{M}&=&\widetilde{\mathcal{H}}_{\mathbb{C}}\oplus\mathcal{V}_{\mathbb{C}}=\widetilde{\mathcal{H}}^{1,0}\oplus\widetilde{\mathcal{H}}^{0,1}\oplus\mathcal{V}^{1,0}\oplus\mathcal{V}^{0,1},\\ T_{\mathbb{C}}^\ast\tilde{M}&=&\widetilde{\mathcal{H}}_{\mathbb{C}}^\ast\oplus\mathcal{V}_{\mathbb{C}}^\ast=(\widetilde{\mathcal{H}}^{1,0})^\ast\oplus(\widetilde{\mathcal{H}}^{0,1})^\ast\oplus(\mathcal{V}^{1,0})^\ast\oplus(\mathcal{V}^{0,1})^\ast. \end{align}\]
Let \(\Theta_{\mathbb{C}}:\mathcal{V}_{\mathbb{C}}\rightarrow \widetilde{\mathcal{H}}_{\mathbb{C}}\) be the complex horizontal map associated to \(\widetilde{\mathcal{H}}_{\mathbb{C}}=\widetilde{\mathcal{H}}^{1,0}\oplus\widetilde{\mathcal{H}}^{0,1}\). Locally, \[\Theta_{\mathbb{C}}\left(\frac{\partial}{\partial v^\alpha}\right)=\frac{\delta}{\delta z^\alpha}\quad\text{and}\quad\Theta_{\mathbb{C}}\left(\frac{\partial}{\partial \bar{v}^\alpha}\right)=\frac{\delta}{\delta\bar{z}^\alpha},\quad \alpha=1,\cdots,n.\] Let \(\hat{\Theta}:\mathcal{V}_{\mathbb{R}}\rightarrow \mathcal{H}_{\mathbb{R}}\) be the real horizontal map. Since \(^o:\mathcal{V}^{1,0}\rightarrow\mathcal{V}_{\mathbb{R}}\) is an \(\mathbb{R}\)-isomorphism and \(\widetilde{\mathcal{H}}^{1,0}\) is a complex horizontal bundle, we define an \(\mathbb{R}\)-isomorphism \(^\sharp:\widetilde{\mathcal{H}}^{1,0}\rightarrow\mathcal{H}_{\mathbb{R}}\) by \[\label{eq-hrc} H^\sharp=\hat{\Theta}((\Theta_{\mathbb{C}}^{-1}(H))^o),\quad \forall H\in\widetilde{\mathcal{H}}^{1,0}\tag{21}\] with inverse \(_\sharp:\mathcal{H}_{\mathbb{R}}\rightarrow \widetilde{\mathcal{H}}^{1,0}\) given by \[H_\sharp=\Theta_{\mathbb{C}}((\hat{\Theta}^{-1}(H))_o),\quad \forall H\in\mathcal{H}_{\mathbb{R}}.\]
On the other hand, \(\mathcal{H}^{1,0}\) is a complex horizontal bundle, as in [8] we denote \(\Theta:\mathcal{V}_{\mathbb{C}}\rightarrow \mathcal{H}_{\mathbb{C}}\) the complex horizontal map associated to \(\mathcal{H}_{\mathbb{C}}=\mathcal{H}^{1,0}\oplus\mathcal{H}^{0,1}\). Then locally \[\Theta\left(\frac{\partial}{\partial v^\alpha}\right)=\delta_\alpha\quad\text{and}\quad\Theta\left(\frac{\partial}{\partial \bar{v}^\alpha}\right)=\delta_{\bar{\alpha}},\quad \alpha=1,\cdots,n.\] Similarly, we can define an \(\mathbb{R}\)-isomorphism \(^o:\mathcal{H}^{1,0}\rightarrow\mathcal{H}_{\mathbb{R}}\) by \[\label{mcgunpfv} H^o=\hat{\Theta}((\Theta^{-1}(H))^o),\quad \forall H\in\mathcal{H}^{1,0}\tag{22}\] with inverse \(_o:\mathcal{H}_{\mathbb{R}}\rightarrow \mathcal{H}^{1,0}\) given by \[H_o=\Theta((\hat{\Theta}^{-1}(H))_o),\quad \forall H\in\mathcal{H}_{\mathbb{R}}.\]
The following proposition shows that if \(\pmb{J}\) is horizontal parallel with respect to \(\nabla\), then \(\pmb{J}\) commutes with \(\hat{\Theta}\), and \(_\sharp=_o, ^\sharp=^o\).
Proposition 20. Let \(F: T^{1,0}M\rightarrow [0,+\infty)\) be a strongly convex complex Finsler metric on a complex manifold \(M\). If \(\nabla_X \pmb{J}=0\) for all \(X\in\mathcal{H}_{\mathbb{R}}\), then \[\hat{\Theta}\circ \pmb{J}=\pmb{J} \circ \hat{\Theta},\quad\text{and}\quad H^\sharp=H^o,\quad \forall H\in\widetilde{\mathcal{H}}^{1,0}.\]
Proof. Indeed, \[\begin{align} &&\hat{\Theta}\left(\pmb{J}\frac{\partial}{\partial y^\alpha}\right) = \hat{\Theta}\left(\frac{\partial}{\partial y^{\alpha^\ast }}\right)=\frac{\delta}{\delta x^{\alpha^\ast }}=\pmb{J}\left( \hat{\Theta}\left(\frac{\partial}{\partial y^\alpha}\right)\right),\\ &&\hat{\Theta}\left(\pmb{J}\frac{\partial}{\partial y^{\alpha^\ast }}\right) = -\hat{\Theta}\left(\frac{\partial}{\partial y^{\alpha}}\right)=-\frac{\delta}{\delta x^{\alpha}}=\pmb{J}\left( \hat{\Theta}\left(\frac{\partial}{\partial y^{\alpha^\ast }}\right)\right). \end{align}\] Thus \(\hat{\Theta}\) commutes with \(\pmb{J}\).
Next, since \[\begin{align} \left(\frac{\delta}{\delta z^\alpha}\right)^\sharp&=&\hat{\Theta}\left(\left(\Theta_{\mathbb{C}}^{-1}\left(\frac{\delta}{\delta z^\alpha}\right)\right)^o\right)=\hat{\Theta}\left(\left(\frac{\partial}{\partial v^\alpha}\right)^o\right) =\hat{\Theta}\left(\frac{\partial}{\partial y^\alpha}\right)=\frac{\delta}{\delta x^\alpha},\\ \left(i\frac{\delta}{\delta z^\alpha}\right)^\sharp&=&\hat{\Theta}\left(\left(\Theta_{\mathbb{C}}^{-1}\left(i\frac{\delta}{\delta z^\alpha}\right)\right)^o\right)=\hat{\Theta}\left(\left(i\frac{\partial}{\partial v^\alpha}\right)^o\right) =\hat{\Theta}\left(\frac{\partial}{\partial y^{\alpha^\ast }}\right) =\frac{\delta}{\delta x^{\alpha^\ast }}. \end{align}\] or equivalently \[\begin{align} \left(\frac{\delta}{\delta x^\alpha}\right)_\sharp&=&\Theta_{\mathbb{C}}\left(\left(\hat{\Theta}^{-1}\left(\frac{\delta}{\delta x^\alpha}\right)\right)_o\right)=\Theta_{\mathbb{C}}\left(\left(\frac{\partial}{\partial y^\alpha}\right)_o\right) =\Theta_{\mathbb{C}}\left(\frac{\partial}{\partial v^\alpha}\right)=\frac{\delta}{\delta z^\alpha},\\ \left(\frac{\delta}{\delta x^{\alpha^\ast }}\right)_\sharp&=&\Theta_{\mathbb{C}}\left(\left(\hat{\Theta}^{-1}\left(\frac{\delta}{\delta x^{\alpha^\ast }}\right)\right)_o\right) =\Theta_{\mathbb{C}}\left(\left(\frac{\partial}{\partial y^{\alpha^\ast }}\right)_o\right)=\Theta_{\mathbb{C}}\left(i\frac{\partial}{\partial v^\alpha}\right)=i\frac{\delta}{\delta z^\alpha}. \end{align}\] Hence \(^\sharp\) and \(_\sharp\) are actually the restrictions of \(^o\) and \(_o\) to \(\widetilde{\mathcal{H}}^{1,0}\) and \(\mathcal{H}_{\mathbb{R}}\), respectively. ◻
Proposition 21. Let \(F: T^{1,0}M\rightarrow [0,+\infty)\) be a strongly convex complex Finsler metric on a complex manifold \(M\). If \(\nabla_X \pmb{J}=0\) for all \(X\in\mathcal{H}_{\mathbb{R}}\), then \[\frac{\partial }{\partial\bar{v}^\gamma}\mathcal{N}_{;\alpha}^\beta\equiv0,\quad \forall\alpha,\beta,\gamma=1,\cdots,n.\]
Proof. Using 4 , the equalities in ?? become \[\hat{\Gamma}_{\alpha}^\beta=\frac{\partial {\hat{\mathbb{G}}}^\beta}{\partial y^\alpha}=\frac{\partial {\hat{\mathbb{G}}}^{\beta^\ast }}{\partial y^{\alpha^\ast }}=\hat{\Gamma}_{\alpha^\ast }^{\beta^\ast },\quad \hat{\Gamma}_{\alpha}^{\beta^\ast }=\frac{\partial {\hat{\mathbb{G}}}^{\beta^\ast }}{\partial y^\alpha}=-\frac{\partial {\hat{\mathbb{G}}}^\beta}{\partial y^{\alpha^\ast }}=-\hat{\Gamma}_{\alpha^\ast }^\beta,\] which implies \[\frac{\partial}{\partial y^\alpha}(\hat{\mathbb{G}}^\beta+i\hat{\mathbb{G}}^{\beta^\ast })=-i\frac{\partial}{\partial y^{\alpha^\ast }}(\hat{\mathbb{G}}^\beta+i\hat{\mathbb{G}}^{\beta^\ast }),\label{hnlc}\tag{23}\] or equivalently \[\frac{\partial}{\partial\bar{v}^\alpha}(\hat{\mathbb{G}}^\beta+i\hat{\mathbb{G}}^{\beta^\ast })=0.\label{pa}\tag{24}\] From 23 , we have \[\mathcal{N}_{;\alpha}^\beta=\hat{\Gamma}_{\alpha}^\beta+i\hat{\Gamma}_{\alpha}^{\beta^\ast }=\frac{\partial}{\partial y^\alpha}(\hat{\mathbb{G}}^\beta+i\hat{\mathbb{G}}^{\beta^\ast })=-i\frac{\partial}{\partial y^{\alpha^\ast }}(\hat{\mathbb{G}}^\beta+i\hat{\mathbb{G}}^{\beta^\ast }),\] so \[\mathcal{N}_{;\alpha}^\beta=\frac{1}{2}\left(\frac{\partial}{\partial y^\alpha}-i\frac{\partial}{\partial y^{\alpha^\ast }}\right)\left(\hat{\mathbb{G}}^\beta+i\hat{\mathbb{G}}^{\beta^\ast }\right)=\frac{\partial}{\partial v^\alpha}(\hat{\mathbb{G}}^\beta+i\hat{\mathbb{G}}^{\beta^\ast }).\label{pb}\tag{25}\] Then 24 and 25 give \[\frac{\partial }{\partial\bar{v}^\gamma}\mathcal{N}_{;\alpha}^\beta=\frac{\partial^2}{\partial v^\alpha\partial\bar{v}^\gamma}(\hat{\mathbb{G}}^\beta+i\hat{\mathbb{G}}^{\beta^\ast })\equiv0.\] ◻
Theorem 22. Let \(F: T^{1,0}M\rightarrow [0,+\infty)\) be a strongly convex complex Finsler metric on a complex manifold \(M\). If \(\nabla\pmb{J}=0\), then the extension \(\nabla: \mathcal{X}(\mathcal{V}_{\mathbb{C}})\rightarrow \mathcal{X}(T_{\mathbb{C}}^\ast\tilde{M}\otimes \mathcal{V}_{\mathbb{C}})\) is a good complex vertical connection of type \((1,0)\), and we have \[\begin{align} \nabla_{\frac{\delta}{\delta z^\gamma}}\frac{\partial}{\partial v^\alpha}&=&\mathcal{N}_{\alpha;\gamma}^\beta\frac{\partial}{\partial v^\beta},\quad\nabla_{\frac{\delta}{\delta \bar{z}^\gamma}}\frac{\partial}{\partial v^\alpha}=0 \end{align}\] and \[\begin{align} \nabla_{\frac{\partial}{\partial v^\gamma}}\frac{\partial}{\partial v^\alpha} &=&\mathcal{N}_{\alpha\gamma}^\beta\frac{\partial}{\partial v^\beta},\quad \nabla_{\frac{\partial}{\partial\bar{v}^\gamma}}\frac{\partial}{\partial v^\alpha}=0 \end{align}\] and its conjugations.
Proof. Note that \(\nabla\pmb{J}=0\) implies that \(\nabla_X \pmb{J}=0\) for any \(X\in\mathcal{X}(\mathcal{H}_{\mathbb{R}})\). Thus by Corollary 1, we have \[\hat{\Gamma}_{\alpha^\ast }^\beta=-\hat{\Gamma}_{\alpha}^{\beta^\ast },\quad \hat{\Gamma}_{\alpha^\ast }^{\beta^\ast }=\hat{\Gamma}_{\alpha}^\beta\] for any \(\alpha,\beta=1,\cdots,n\). This shows that \(\mathcal{H}_{\mathbb{R}}\) is \(\pmb{J}\)-invariant.
Extending \(\nabla\) to \(\mathcal{X}(\mathcal{V}_{\mathbb{C}})\) (still denote by \(\nabla\)) as follows: \[\nabla_{X+iY}(V+iW):=\nabla_XV+i\nabla_XW+i\nabla_YV-\nabla_YW,\label{ECC}\tag{26}\] for any \(X,Y\in T_{\mathbb{R}}\tilde{M}\) and \(V,W\in\mathcal{X}(\mathcal{V}_{\mathbb{C}})\). Then by Proposition 16, we have \[\begin{align} \nabla_{\frac{\delta}{\delta z^\gamma}}\frac{\partial}{\partial v^\alpha} &=&\frac{1}{4}\left(\hat{\Gamma}_{\alpha;\gamma}^\beta-i\hat{\Gamma}_{\alpha^\ast ;\gamma}^\beta-i\hat{\Gamma}_{\alpha;\gamma^\ast }^\beta-\hat{\Gamma}_{\alpha^\ast ;\gamma^\ast }^\beta\right)\frac{\partial}{\partial y^\beta}\\ &&+\frac{1}{4}\left(\hat{\Gamma}_{\alpha;\gamma}^{\beta^\ast }-i\hat{\Gamma}_{\alpha^\ast ;\gamma}^{\beta^\ast }-i\hat{\Gamma}_{\alpha;\gamma^\ast }^{\beta^\ast }-\hat{\Gamma}_{\alpha^\ast ;\gamma^\ast }^{\beta^\ast }\right)\frac{\partial}{\partial y^{\beta^\ast }}. \end{align}\] Using \(\hat{\Gamma}^k_{j;l}=\hat{\Gamma}^k_{l;j}\) and ?? , it follows that \[\begin{align} \left(\hat{\Gamma}_{\alpha;\gamma}^\beta\frac{\partial}{\partial y^\beta}-i\hat{\Gamma}_{\alpha^\ast ;\gamma}^{\beta^\ast }\frac{\partial}{\partial y^{\beta^\ast }}\right) &=&\left(\hat{\Gamma}_{\alpha;\gamma}^\beta\frac{\partial}{\partial y^\beta}-i\hat{\Gamma}_{\alpha;\gamma}^{\beta}\frac{\partial}{\partial y^{\beta^\ast }}\right)=2\hat{\Gamma}_{\alpha;\gamma}^\beta\frac{\partial}{\partial v^\beta},\\ \left(-i\hat{\Gamma}_{\alpha^\ast ;\gamma}^\beta\frac{\partial}{\partial y^\beta}+\hat{\Gamma}_{\alpha;\gamma}^{\beta^\ast }\frac{\partial}{\partial y^{\beta^\ast }}\right) &=&\left(i\hat{\Gamma}_{\alpha;\gamma}^{\beta^\ast}\frac{\partial}{\partial y^\beta}+\hat{\Gamma}_{\alpha;\gamma}^{\beta^\ast }\frac{\partial}{\partial y^{\beta^\ast }}\right) =2i\hat{\Gamma}_{\alpha;\gamma}^{\beta^\ast}\frac{\partial}{\partial v^\beta},\\ \left(-i\hat{\Gamma}_{\alpha;\gamma^\ast }^\beta\frac{\partial}{\partial y^\beta}-\hat{\Gamma}_{\alpha^\ast ;\gamma^\ast }^{\beta^\ast }\frac{\partial}{\partial y^{\beta^\ast }}\right) &=&\left(i\hat{\Gamma}_{\alpha;\gamma }^{\beta^\ast}+\hat{\Gamma}_{\gamma;\alpha}^{\beta^\ast}\frac{\partial}{\partial y^{\beta^\ast }}\right)=2i\hat{\Gamma}_{\alpha;\gamma }^{\beta^\ast}\frac{\partial}{\partial v^\beta},\\ \left(-\hat{\Gamma}_{\alpha^\ast ;\gamma^\ast }^\beta\frac{\partial}{\partial y^\beta}-i\hat{\Gamma}_{\alpha;\gamma^\ast }^{\beta^\ast }\frac{\partial}{\partial y^{\beta^\ast }}\right) &=&\left(\hat{\Gamma}_{\alpha ;\gamma^\ast }^{\beta^\ast}\frac{\partial}{\partial y^\beta}-i\hat{\Gamma}_{\gamma;\alpha}^{\beta }\frac{\partial}{\partial y^{\beta^\ast }}\right) =2\hat{\Gamma}_{\gamma;\alpha}^{\beta }\frac{\partial}{\partial v^\beta}. \end{align}\] Thus \[\begin{align} \nabla_{\frac{\delta}{\delta z^\gamma}}\frac{\partial}{\partial v^\alpha} &=&(\hat{\Gamma}_{\alpha;\gamma}^\beta+i\hat{\Gamma}_{\alpha;\gamma}^{\beta^\ast })\frac{\partial}{\partial v^\beta}=\mathcal{N}_{\alpha;\gamma}^\beta\frac{\partial}{\partial v^\beta}. \end{align}\] Similarly, \[\begin{align} \nabla_{\frac{\delta}{\delta \bar{z}^\gamma}}\frac{\partial}{\partial v^\alpha} &=&\frac{1}{4}\left(\hat{\Gamma}_{\alpha;\gamma}^\beta-i\hat{\Gamma}_{\alpha^\ast ;\gamma}^\beta+i\hat{\Gamma}_{\alpha;\gamma^\ast }^\beta+\hat{\Gamma}_{\alpha^\ast ;\gamma^\ast }^\beta\right)\frac{\partial}{\partial y^\beta}\\ &&+\frac{1}{4}\left(\hat{\Gamma}_{\alpha;\gamma}^{\beta^\ast }-i\hat{\Gamma}_{\alpha^\ast ;\gamma}^{\beta^\ast }+i\hat{\Gamma}_{\alpha;\gamma^\ast }^{\beta^\ast }+\hat{\Gamma}_{\alpha^\ast ;\gamma^\ast }^{\beta^\ast }\right)\frac{\partial}{\partial y^{\beta^\ast }}. \end{align}\] Using \(\hat{\Gamma}^k_{j;l}=\hat{\Gamma}^k_{l;j}\) and ?? again, it follows that \[\begin{align} \hat{\Gamma}_{\alpha;\gamma}^\beta+\hat{\Gamma}_{\alpha^\ast ;\gamma^\ast }^\beta&=&0,\quad -\hat{\Gamma}_{\alpha^\ast ;\gamma}^\beta+\hat{\Gamma}_{\alpha;\gamma^\ast }^\beta=0,\\ \hat{\Gamma}_{\alpha;\gamma}^{\beta^\ast }+\hat{\Gamma}_{\alpha^\ast ;\gamma^\ast }^{\beta^\ast}&=&0,\quad-\hat{\Gamma}_{\alpha^\ast ;\gamma}^{\beta^\ast }+\hat{\Gamma}_{\alpha;\gamma^\ast }^{\beta^\ast }=0. \end{align}\] Thus \[\nabla_{\frac{\delta}{\delta \bar{z}^\gamma}}\frac{\partial}{\partial v^\alpha}=0.\]
By conjugation, we obtain \[\nabla_{\frac{\delta}{\delta \bar{z}^\gamma}}\frac{\partial}{\partial \bar{v}^\alpha}=\overline{\mathcal{N}_{\alpha;\gamma}^\beta}\frac{\partial}{\partial \bar{v}^\beta},\quad \nabla_{\frac{\delta}{\delta z^\gamma}}\frac{\partial}{\partial \bar{v}^\alpha}=0.\]
Similarly, \[\begin{align} \nabla_{\frac{\partial}{\partial v^\gamma}}\frac{\partial}{\partial v^\alpha} &=&\frac{1}{4}\left(\hat{\Gamma}_{\alpha\gamma}^\beta-i\hat{\Gamma}_{\alpha^\ast \gamma}^\beta-i\hat{\Gamma}_{\;\alpha\gamma^\ast }^\beta-\hat{\Gamma}_{\alpha^\ast \gamma^\ast }^\beta\right)\frac{\partial}{\partial y^\beta}\\ &&+\frac{1}{4}\left(\hat{\Gamma}_{\alpha\gamma}^{\beta^\ast }-i\hat{\Gamma}_{\alpha^\ast \gamma}^{\beta^\ast }-i\hat{\Gamma}_{\alpha\gamma^\ast }^{\beta^\ast }-\hat{\Gamma}_{\alpha^\ast \gamma^\ast }^{\beta^\ast }\right)\frac{\partial}{\partial y^{\beta^\ast }}. \end{align}\] Since \(\nabla\pmb{J}=0\) also implies that \(\nabla_X \pmb{J}=0\) for any \(X \in\mathcal{V}_{\mathbb{R}}\), using the equality \(\hat{\Gamma}^k_{jl}=\hat{\Gamma}^k_{lj}\) and ?? , the above equality can be simplified as
\[\begin{align} \nabla_{\frac{\partial}{\partial v^\gamma}}\frac{\partial}{\partial v^\alpha} &=&(\hat{\Gamma}_{\alpha\gamma}^{\beta}+i\hat{\Gamma}_{\alpha\gamma}^{\beta^\ast })\frac{\partial}{\partial v^\beta}=\mathcal{N}_{\alpha\gamma}^\beta\frac{\partial}{\partial v^\beta}. \end{align}\] Finally, \[\begin{align} \nabla_{\frac{\partial}{\partial \bar{v}^\gamma}}\frac{\partial}{\partial v^\alpha} &=&\frac{1}{4}\left(\hat{\Gamma}_{\alpha\gamma}^\beta-i\hat{\Gamma}_{\alpha^\ast \gamma}^\beta+i\hat{\Gamma}_{\alpha\gamma^\ast }^\beta+\hat{\Gamma}_{\alpha^\ast \gamma^\ast }^\beta\right)\frac{\partial}{\partial y^\beta}\\ &&+\frac{1}{4}\left(\hat{\Gamma}_{\alpha\gamma}^{\beta^\ast }-i\hat{\Gamma}_{\alpha^\ast \gamma}^{\beta^\ast }+i\hat{\Gamma}_{\alpha\gamma^\ast }^{\beta^\ast }+\hat{\Gamma}_{\alpha^\ast \gamma^\ast }^{\beta^\ast }\right)\frac{\partial}{\partial y^{\beta^\ast }}. \end{align}\] Since by ?? , one can check that \[\begin{align} \hat{\Gamma}_{\alpha\gamma}^\beta+\hat{\Gamma}_{\alpha^\ast \gamma^\ast }^\beta&=&0,\quad-\hat{\Gamma}_{\alpha^\ast \gamma}^\beta+\hat{\Gamma}_{\alpha\gamma^\ast }^\beta=0,\\ \hat{\Gamma}_{\alpha\gamma}^{\beta^\ast }+\hat{\Gamma}_{\alpha^\ast \gamma^\ast }^{\beta^\ast }&=&0,\quad -\hat{\Gamma}_{\alpha^\ast \gamma}^{\beta^\ast }+\hat{\Gamma}_{\alpha\gamma^\ast }^{\beta^\ast }=0. \end{align}\] So that \[\nabla_{\frac{\partial}{\partial \bar{v}^\gamma}}\frac{\partial}{\partial v^\alpha}=0.\] It is clear that \[\nabla_{\frac{\partial}{\partial \bar{v}^\gamma}}\frac{\partial}{\partial \bar{v}^\alpha} =\overline{\mathcal{N}_{\alpha\gamma}^\beta}\frac{\partial}{\partial \bar{v}^\beta}\quadand\quad \nabla_{\frac{\partial}{\partial v^\gamma}}\frac{\partial}{\partial \bar{v}^\alpha}=0.\]
Finally, that \(\nabla:\mathcal{X}(\mathcal{V}^{1,0})\rightarrow \mathcal{X}(T_{\mathbb{C}}^\ast\tilde{M}\otimes\mathcal{V}^{1,0})\) is a good complex vertical connection follows immediately from Proposition 18. This completes the proof. ◻
Theorem 23. Let \(F: T^{1,0}M\rightarrow [0,+\infty)\) be a strongly convex complex Finsler metric on a complex manifold \(M\). If \(\nabla_X \pmb{J}=0\) for \(X\in\mathcal{H}_{\mathbb{R}}\), then \(F\) must be a Kähler-Berwald metric.
Proof. Let \(V=\frac{1}{2}(X-i \pmb{J}X),W=\frac{1}{2}(Y-i \pmb{J}Y)\) with \(X,Y\in\mathcal{V}_{\mathbb{R}}\). It follows that \(V,W\in\mathcal{V}^{1,0}\) and by Lemma 1, we have \[\langle V,W\rangle=2\langle V\vert \overline{W}\rangle.\label{rc}\tag{27}\]
By Theorem 7, the metric compatibility of the Cartan connection \(\nabla\) implies that its natural extension satisfies \[Z\langle X\vert Y\rangle=\langle \nabla_ZX\vert Y\rangle +\langle X\vert \nabla_ZY\rangle\label{ETC-b}\tag{28}\] for any \(Z=A+iB\) with \(A, B\in T_{\mathbb{R}}\tilde{M}\) and \(X,Y\in \mathcal{X}(\mathcal{V}_{\mathbb{C}})\).
By assumption, \(\nabla_X\pmb{J}=0\) for all \(X\in\mathcal{H}_{\mathbb{R}}\). Thus by Corollary 1, \(\mathcal{H}_{\mathbb{R}}\) is \(\pmb{J}\)-invariant. So that by Proposition 16, \(\frac{\delta}{\delta z^\alpha}=\frac{1}{2}(\frac{\delta}{\delta x^\alpha}-i\frac{\delta}{\delta x^{\alpha^\ast}})\). Substituting \(V=\frac{\partial}{\partial v^\beta}, W=\frac{\partial}{\partial v^\gamma}\in\mathcal{V}^{1,0}\) into 27 and then using 28 with \(Z=\frac{\delta}{\delta z^\alpha}=\frac{1}{2}(\frac{\delta}{\delta x^\alpha}-i\frac{\delta}{\delta x^{\alpha^\ast}})\in\widetilde{\mathcal{H}}^{1,0}\), we obtain \[\begin{align} \frac{\delta}{\delta z^\alpha}\left\langle \frac{\partial}{\partial v^\beta},\frac{\partial}{\partial v^\gamma}\right\rangle &=&2\frac{\delta}{\delta z^\alpha}\left\langle \frac{\partial}{\partial v^\beta}\Big\vert \frac{\partial}{\partial \bar{v}^\gamma}\right\rangle\\ &=&2\left\langle \nabla_{\frac{\delta}{\delta z^\alpha}} \frac{\partial}{\partial v^\beta}\Big\vert \frac{\partial}{\partial\bar{v}^\gamma}\right\rangle+2\left\langle \frac{\partial}{\partial v^\beta}\Big\vert \nabla_{\frac{\delta}{\delta z^\alpha}}\frac{\partial}{\partial\bar{v}^\gamma}\right\rangle\\ &=&\left\langle \nabla_{\frac{\delta}{\delta z^\alpha}}\frac{\partial}{\partial v^\beta},\frac{\partial}{\partial v^\gamma}\right\rangle+\left\langle \frac{\partial}{\partial v^\alpha}, \nabla_{\frac{\delta}{\delta \bar{z}^\alpha}}\frac{\partial}{\partial v^\gamma}\right\rangle. \end{align}\] This together with Theorem 22 imply \[\frac{\delta}{\delta z^\alpha} (G_{\beta\bar{\gamma}})=\left\langle\mathcal{N}_{\beta;\alpha}^\mu\frac{\partial}{\partial v^\mu},\frac{\partial}{\partial v^\gamma}\right\rangle=\mathcal{N}_{\beta;\alpha}^\mu G_{\mu\bar{\gamma}},\] from which it follows that \[\mathcal{N}_{\beta;\alpha}^\mu=G^{\bar{\gamma}\mu}\frac{\delta}{\delta z^\alpha} (G_{\beta\bar{\gamma}}) =G^{\bar{\gamma}\mu}\left(\frac{\partial}{\partial z^\alpha}-\mathcal{N}_{;\alpha}^\nu\frac{\partial}{\partial v^\nu}\right)(G_{\beta\bar{\gamma}}).\label{ccc}\tag{29}\]
Contracting both sides of 29 with respect to \(v^{\beta}\), we obtain \[\mathcal{N}_{;\alpha}^\mu =G^{\bar{\gamma}\mu}\frac{\partial}{\partial z^\alpha}(G_{\beta\bar{\gamma}})v^\beta=G^{\bar{\gamma}\mu}\frac{\partial^2F^2}{\partial z^\alpha\partial\bar{v}^\gamma}=\Gamma_{;\alpha}^\mu,\] where we use the equality \(\frac{\partial}{\partial v ^{\nu}}(G_{\beta\bar{\gamma}})v^\beta=0\) and \(\Gamma_{;\alpha}^\mu\) are the complex non-linear connection coefficients of \(F\) defined by 7 . Substituting \(\Gamma_{;\alpha}^\mu=\mathcal{N}_{;\alpha}^\mu\) into 29 , we obtain \[\mathcal{N}_{\beta;\alpha}^\mu=G^{\bar{\gamma}\mu}\left(\frac{\partial}{\partial z^\alpha}-\Gamma_{;\alpha}^\nu\frac{\partial}{\partial v ^{\nu}}\right)\left(G_{\beta\bar{\gamma}}\right)=\Gamma_{\beta;\alpha}^\mu,\] where \(\Gamma_{\beta;\alpha}^\mu\) are exactly the horizontal connection coefficients of the Chern-Finsler connection given by 8 . By Remark 17, we alread have \(\mathcal{N}_{\beta;\alpha}^\mu=\mathcal{N}_{\alpha;\beta}^\mu\), hence \(\Gamma_{\beta;\alpha}^\mu =\Gamma_{\alpha;\beta}^\mu\). That is, \(F\) must be a Kähler-Finsler metric.
Next we show that \(F\) actually must be a Kähler-Berwald metric. By Proposition 21 and the quality \(\Gamma_{;\alpha}^\mu=\mathcal{N}_{;\alpha}^\mu\), we have \[\begin{align} 0= \frac{\partial}{\partial \bar{v}^{\beta}}\Gamma_{;\alpha}^{\mu}&=&\frac{1}{2}\left(\frac{\partial}{\partial y^{\beta}}+i\frac{\partial}{\partial y^{\beta^\ast }}\right) \Gamma_{;\alpha}^{\mu} \\ &=&\frac{1}{2}\left(\frac{\partial}{\partial y^{\beta}}+i\frac{\partial}{\partial y^{\beta^\ast }}\right)\left(\hat{\Gamma}_{\alpha}^{\mu}+i\hat{\Gamma}_{\alpha}^{\mu^\ast }\right)\\ &=&\frac{1}{2}\left[(\hat{\mathbb{G}}_{\beta\alpha}^{\mu}-\hat{\mathbb{G}}_{\beta^\ast \,\alpha}^{\mu^\ast })+i(\hat{\mathbb{G}}_{\beta\alpha}^{\mu^\ast }+\hat{\mathbb{G}}_{\beta^\ast \,\alpha}^{\mu}) \right], \end{align}\] where \(\hat{\mathbb{G}}^{k}_{jl}=\frac{\partial ^2 \hat{\mathbb{G}}^k}{\partial y^j \partial y^l}\) are the real Berwald connection coefficients associated to \(F\). This implies that \[\label{B-a} \hat{\mathbb{G}}_{\beta\alpha}^{\mu}=\hat{\mathbb{G}}_{\beta^\ast \,\alpha}^{\mu^\ast },\quad \hat{\mathbb{G}}_{\beta\alpha}^{\mu^\ast }=-\hat{\mathbb{G}}_{\beta^\ast \,\alpha}^{\mu}.\tag{30}\] Thus we have \[\begin{align} \frac{\partial }{\partial \bar{v}^\gamma}\Gamma^{\mu}_{\beta;\alpha}=\frac{\partial^2}{\partial \bar{v}^\gamma\partial v^{\beta}}\Gamma^{\mu}_{;\alpha}=\frac{\partial^2}{\partial v^{\beta}\partial \bar{v}^\gamma}\Gamma^{\mu}_{;\alpha}=0. \end{align}\] A direct computation shows \[\begin{align} \Gamma_{\beta;\alpha}^{\mu}=\frac{\partial}{\partial v^{\beta}}\Gamma_{;\alpha}^{\mu}&=&\frac{\partial}{\partial v^{\beta}}\mathcal{N}_{;\alpha}^{\mu}\\ &=&\frac{1}{2}\left(\frac{\partial}{\partial y^{\beta}}-i\frac{\partial}{\partial y^{\beta^\ast }}\right) \mathcal{N}_{;\alpha}^{\mu} \\ &=&\frac{1}{2}\left(\frac{\partial}{\partial y^{\beta}}-i\frac{\partial}{\partial y^{\beta^\ast }}\right)\left(\hat{\Gamma}_{\alpha}^{\mu}+i\hat{\Gamma}_{\alpha}^{\mu^\ast }\right)\\ &=&\frac{1}{2}\left[\hat{\mathbb{G}}_{\beta\alpha}^{\mu}+i\hat{\mathbb{G}}_{\beta\alpha}^{\mu^\ast }-i\hat{\mathbb{G}}_{\beta^\ast \,\alpha}^{\mu}+\hat{\mathbb{G}}_{\beta^\ast \,\alpha}^{\mu^\ast } \right]\\ &=&\hat{\mathbb{G}}_{\beta\alpha}^{\mu}+i\hat{\mathbb{G}}_{\beta\alpha}^{\mu^\ast }, \end{align}\] where we used 30 in the last equality. On the other hand, we have \[\Gamma_{\beta;\alpha}^\mu=\mathcal{N}_{\beta;\alpha}^\mu=\hat{\Gamma}^{\mu}_{\beta;\alpha}+i\hat{\Gamma}^{\mu^\ast }_{\beta;\alpha}.\] Thus we must have \[\hat{\Gamma}^{\mu}_{\beta;\alpha}=\hat{\mathbb{G}}^{\mu}_{\beta\alpha},\quad \text{and}\quad \hat{\Gamma}^{\mu^\ast }_{\beta;\alpha}=\hat{\mathbb{G}}^{\mu^\ast }_{\beta\alpha}.\] This implies that real Berwald connection coefficients coincide with the horizontal connection coefficients of the Cartan connection associated to \(F\). Since the horizontal Cartan connection coefficients (the same as the Chern connection coefficients) and the real Berwald connection coefficients satisfy \(\hat{\Gamma}^{k}_{j;l}=\hat{\mathbb{G}}^{k}_{jl}-\dot{A}^k_{\;jl}\) (see p. 39 in [10]), it follows that \(\dot{A}_{\;jl}^k=0\), hence \(F\) must be a real Landsberg metric. Furthermore, a strongly convex weakly Kähler-Finsler metric is a real Landsberg metric iff it is a weakly complex Berwald metric [13], and we have already showed that \(F\) is a Kähler-Finsler metric, it follows that \(F\) must be a Kähler-Berwald metric. ◻
Theorem 24. Let \(F: T^{1,0}M\rightarrow [0,+\infty)\) be a strongly convex complex Finsler metric on a complex manifold \(M\). Then \(\nabla \pmb{J}=0\) iff the Cartan connection \(\nabla\) associated to \(F\) coincides with the Chern-Finsler connection \(D\) associated to \(F\).
Proof. First we show the necessity. Suppose \(\nabla \pmb{J}=0\). Extending \(\nabla\) to \(\mathcal{V}_{\mathbb{C}}\), then by Theorem 22, we obtain a good complex vertical connection \(\nabla:\mathcal{X}(\mathcal{V}^{1,0})\rightarrow\mathcal{X}(T_{\mathbb{C}}^\ast\tilde{M}\otimes \mathcal{V}^{1,0})\), which is of type \((1,0)\) such that \[\nabla_{\frac{\delta}{\delta z^\gamma}}\frac{\partial}{\partial v^\alpha}=\mathcal{N}_{\alpha;\gamma}^\beta\frac{\partial}{\partial v^\beta},\quad \nabla_{\frac{\partial}{\partial v^\gamma}}\frac{\partial}{\partial v^\alpha}=\mathcal{N}_{\alpha\gamma}^\beta\frac{\partial}{\partial v^\beta}.\] Since \(\nabla\pmb{J}=0\) implies \(\nabla_X\pmb{J}=0\) for all \(X\in\mathcal{H}_{\mathbb{R}}\), thus by Theorem 23, we have \(\mathcal{N}_{\alpha;\gamma}^\beta=\Gamma_{\alpha;\gamma}^\beta\). Next we show \(\mathcal{N}_{\alpha\gamma}^\beta=\Gamma_{\alpha\gamma}^\beta\). Using 27 and 28 , we have \[\begin{align} \frac{\partial}{\partial v^\alpha}\left\langle \frac{\partial}{\partial v^\beta},\frac{\partial}{\partial v^\gamma}\right\rangle &=&2\frac{\partial}{\partial v^\alpha}\left\langle\frac{\partial}{\partial v^\beta}\Big\vert \frac{\partial}{\partial \bar{v}^\gamma}\right\rangle\\ &=&2\left\langle\nabla_{\frac{\partial}{\partial v^\alpha}}\frac{\partial}{\partial v^\beta}\Big\vert \frac{\partial}{\partial \bar{v}^\gamma}\right\rangle +\left\langle\frac{\partial}{\partial v^\beta}\Big\vert \nabla_{\frac{\partial}{\partial v^\alpha}}\frac{\partial}{\partial \bar{v}^\gamma}\right\rangle\\ &=&\left\langle \nabla_{\frac{\partial}{\partial v^\alpha}}\frac{\partial}{\partial v^\beta},\frac{\partial}{\partial v^\gamma}\right\rangle+\left\langle \frac{\partial}{\partial v^\alpha}, \nabla_{\frac{\partial}{\partial \bar{v}^\alpha}}\frac{\partial}{\partial v^\gamma}\right\rangle. \end{align}\] Again using Theorem 22, the above equality reduces to \[\frac{\partial}{\partial v^\alpha} (G_{\beta\bar{\gamma}})=\left\langle\mathcal{N}_{\beta\alpha}^\mu\frac{\partial}{\partial v^\mu},\frac{\partial}{\partial v^\gamma}\right\rangle=\mathcal{N}_{\beta\alpha}^\mu G_{\mu\bar{\gamma}}.\] Hence, \[\mathcal{N}_{\beta\alpha}^\mu=G^{\bar{\gamma}\mu}\frac{\partial}{\partial v^\alpha} (G_{\beta\bar{\gamma}}) =\Gamma_{\beta\alpha}^\mu,\] which are exactly the vertical connection coefficients of the Chern-Finsler connection associated to \(F\). Thus \(\nabla:\mathcal{X}(\mathcal{V}^{1,0})\rightarrow \mathcal{X}(T_{\mathbb{C}}^\ast \tilde{M}\otimes \mathcal{V}^{1,0})\) coincides with the Chern-Finsler connection \(D\) associated to \(F\).
Next we show the sufficiency. By Theorem 14, it suffices to show that ?? and ?? hold. By assumption, the Cartan connection \(\nabla\) and the Chern-Finsler connection \(D\) coincide, thus the real non-linear connection \(\tilde{\nabla}\) associated to \(\nabla\) and the complex non-linear connection \(\tilde{D}\) associated to \(D\) coincide. In other words, let \(\xi\in\mathcal{X}(T^{1,0}M)\) be a holomorphic vector field and \(w\in T_p^{1,0}M\) such that \(\eta_o=\xi\) and \(u_o=w\), then at the point \(p\), we have \[(\tilde{D}_w\xi)(p)=((\tilde{\nabla}_u\eta)_o)(p),\label{RAC}\tag{31}\] which implies \[\hat{\Gamma}_\alpha^\beta=\hat{\Gamma}_{\alpha^\ast}^{\beta^\ast},\quad \hat{\Gamma}_\alpha^{\beta^\ast}=-\hat{\Gamma}_{\alpha^\ast}^\beta\label{A-a}\tag{32}\] and \[\mathcal{N}_{;\alpha}^\beta=\Gamma_{;\alpha}^\beta=\hat{\Gamma}_\alpha^\beta+i\hat{\Gamma}_\alpha^{\beta^\ast}.\] Note that 32 implies that \(\mathcal{H}_{\mathbb{R}}\) is \(\pmb{J}\)-invariant, so that the complex horizontal bundle \(\widetilde{\mathcal{H}}^{1,0}\) spanned by \(\{\frac{\delta}{\delta z^1},\cdots,\frac{\delta}{\delta z^n}\}\) actually coincides with \(\mathcal{H}^{1,0}\), namely \(\frac{\delta}{\delta z^\alpha}=\delta_\alpha\) for \(\alpha=1,\cdots,n\). Thus by Proposition 16, we have \[\left(\frac{\delta}{\delta x^\alpha}\right)_o=\frac{\delta}{\delta z^\alpha}=\delta_\alpha\quad\text{and}\quad \left(\frac{\delta}{\delta x^{\alpha^\ast }}\right)_o=i\frac{\delta}{\delta z^\alpha}=i\delta_\alpha.\] Therefore \[\begin{align} 0&=&D_{\delta_{\bar{\gamma}}}\frac{\partial}{\partial v^\alpha}= \nabla_{\frac{\delta}{\delta \bar{z}^\gamma}}\frac{\partial}{\partial v^\alpha}\\ &=&\frac{1}{2}\left(\nabla_{\frac{\delta}{\delta x^\gamma}}\frac{\partial}{\partial v^\alpha}+i\nabla_{\frac{\delta}{\delta x^{\gamma^\ast }}}\frac{\partial}{\partial v^\alpha}\right)\\ &=&\frac{1}{4}\left(\nabla_{\frac{\delta}{\delta x^\gamma}}\frac{\partial}{\partial y^\alpha}-i\nabla_{\frac{\delta}{\delta x^\gamma}}\frac{\partial}{\partial y^{\alpha^\ast }} +i\left(\nabla_{\frac{\delta}{\delta x^{\gamma^\ast }}}\frac{\partial}{\partial y^\alpha}-i\nabla_{\frac{\delta}{\delta x^{\gamma^\ast }}}\frac{\partial}{\partial y^{\alpha^\ast }}\right)\right)\\ &=&\frac{1}{4}\left(\hat{\Gamma}_{\alpha;\gamma}^\beta-i\hat{\Gamma}_{\alpha^\ast ;\gamma}^\beta+i\hat{\Gamma}_{\alpha;\gamma^\ast }^\beta+\hat{\Gamma}_{\alpha^\ast ;\gamma^\ast }^\beta\right)\frac{\partial}{\partial y^\beta}\\ &&+\frac{1}{4}\left(\hat{\Gamma}_{\alpha;\gamma}^{\beta^\ast }-i\hat{\Gamma}_{\alpha^\ast ;\gamma}^{\beta^\ast }+i\hat{\Gamma}_{\alpha;\gamma^\ast }^{\beta^\ast }+\hat{\Gamma}_{\alpha^\ast ;\gamma^\ast }^{\beta^\ast }\right)\frac{\partial}{\partial y^{\beta^\ast }}, \end{align}\] which means that \[\begin{align} \hat{\Gamma}_{\alpha;\gamma}^\beta=-\hat{\Gamma}_{\alpha^\ast ;\gamma^\ast }^\beta,\quad \hat{\Gamma}_{\alpha^\ast ;\gamma}^\beta=\hat{\Gamma}_{\alpha;\gamma^\ast }^\beta,\tag{33}\\ \hat{\Gamma}_{\alpha;\gamma}^{\beta^\ast }=-\hat{\Gamma}_{\alpha^\ast ;\gamma^\ast }^{\beta^\ast },\quad \hat{\Gamma}_{\alpha^\ast ;\gamma}^{\beta^\ast }=\hat{\Gamma}_{\alpha;\gamma^\ast }^{\beta^\ast }.\tag{34} \end{align}\] Using 33 and 34 , we have \[\begin{align} \Gamma^\beta_{\alpha;\gamma}\frac{\partial}{\partial v^\beta}&=&D_{\delta_\gamma}\frac{\partial}{\partial v^\alpha}=\nabla_{\frac{\delta}{\delta z^\gamma}}\frac{\partial}{\partial v^\alpha}\\ &=&\frac{1}{4}\left(\nabla_{\frac{\delta}{\delta x^\gamma}}\frac{\partial}{\partial y^\alpha}-i\nabla_{\frac{\delta}{\delta x^\gamma}}\frac{\partial}{\partial y^{\alpha^\ast }} -i\nabla_{\frac{\delta}{\delta x^{\gamma^\ast }}}\frac{\partial}{\partial y^\alpha}-\nabla_{\frac{\delta}{\delta x^{\gamma^\ast }}}\frac{\partial}{\partial y^{\alpha^\ast }}\right)\\ &=&\frac{1}{4}\left(\hat{\Gamma}_{\alpha;\gamma}^\beta-i\hat{\Gamma}_{\alpha^\ast ;\gamma}^\beta-i\hat{\Gamma}_{\alpha;\gamma^\ast }^\beta-\hat{\Gamma}_{\alpha^\ast ;\gamma^\ast }^\beta\right)\frac{\partial}{\partial y^\beta}\\ &&+\frac{1}{4}\left(\hat{\Gamma}_{\alpha;\gamma}^{\beta^\ast }-i\hat{\Gamma}_{\alpha^\ast ;\gamma}^{\beta^\ast }-i\hat{\Gamma}_{\alpha;\gamma^\ast }^{\beta^\ast }-\hat{\Gamma}_{\alpha^\ast ;\gamma^\ast }^{\beta^\ast }\right)\frac{\partial}{\partial y^{\beta^\ast }}\\ &=&\frac{1}{2}\left(\hat{\Gamma}_{\alpha;\gamma}^\beta-i\hat{\Gamma}_{\alpha^\ast ;\gamma}^\beta\right)\frac{\partial}{\partial y^\beta}+\frac{1}{2}\left(\hat{\Gamma}_{\alpha;\gamma}^{\beta^\ast }-i\hat{\Gamma}_{\alpha^\ast ;\gamma}^{\beta^\ast }\right)\frac{\partial}{\partial y^{\beta^\ast }}, \end{align}\] which implies that \[\text{Re}\Gamma_{\alpha;\gamma}^\beta=\hat{\Gamma}_{\alpha;\gamma}^\beta= \hat{\Gamma}_{\alpha^\ast ;\gamma}^{\beta^\ast },\quad \text{Im}\Gamma_{\alpha;\gamma}^\beta=-\hat{\Gamma}_{\alpha^\ast ;\gamma}^\mu=\hat{\Gamma}_{\alpha;\gamma}^{\beta^\ast }.\] Thus we obtain ?? .
Similarly, we have \[\begin{align} 0&=&D_{\frac{\partial}{\partial \bar{v}^\gamma}}\frac{\partial}{\partial v^\alpha}=\nabla_{\frac{\partial}{\partial \bar{v}^\gamma}}\frac{\partial}{\partial v^\alpha}\\ &=&\frac{1}{4}(\hat{\Gamma}_{\alpha\gamma}^\beta-i\hat{\Gamma}_{\alpha^\ast \gamma}^\beta+i\hat{\Gamma}_{\alpha\gamma^\ast }^\beta+\hat{\Gamma}_{\alpha^\ast \gamma^\ast }^\beta)\frac{\partial}{\partial y^\beta}\\ && +\frac{1}{4}(\hat{\Gamma}_{\alpha\gamma}^{\beta^\ast }-i\hat{\Gamma}_{\alpha^\ast \gamma}^{\beta^\ast }+i\hat{\Gamma}_{\alpha\gamma^\ast }^{\beta^\ast }+\hat{\Gamma}_{\alpha^\ast \gamma^\ast }^{\beta^\ast })\frac{\partial}{\partial y^{\beta^\ast }}. \end{align}\] It follows that \[\begin{align} \hat{\Gamma}_{\alpha\gamma}^\beta=-\hat{\Gamma}_{\alpha^\ast \gamma^\ast }^\beta,\quad \hat{\Gamma}_{\alpha^\ast \gamma}^\beta=\hat{\Gamma}_{\alpha\gamma^\ast }^\beta,\quad \hat{\Gamma}_{\alpha\gamma}^{\beta^\ast }=-\hat{\Gamma}_{\alpha^\ast \gamma^\ast }^{\beta^\ast },\quad \hat{\Gamma}_{\alpha^\ast \gamma}^{\beta^\ast} =\hat{\Gamma}_{\alpha\gamma^\ast }^{\beta^\ast }.\label{rakxucnl} \end{align}\tag{35}\] On the other hand, \[\begin{align} \Gamma_{\alpha\gamma}^\beta\frac{\partial}{\partial v^\beta}&=&D_{\frac{\partial}{\partial v^\gamma}}\frac{\partial}{\partial v^\alpha}=\nabla_{\frac{\partial}{\partial v^\gamma}}\frac{\partial}{\partial v^\alpha}\\ &=&\frac{1}{4}(\hat{\Gamma}_{\alpha\gamma}^\beta-i\hat{\Gamma}_{\alpha^\ast \gamma}^\beta-i\hat{\Gamma}_{\;\alpha\gamma^\ast }^\beta-\hat{\Gamma}_{\alpha^\ast \gamma^\ast }^\beta)\frac{\partial}{\partial y^\beta}\\ && +\frac{1}{4}(\hat{\Gamma}_{\alpha\gamma}^{\beta^\ast }-i\hat{\Gamma}_{\alpha^\ast \gamma}^{\beta^\ast }-i\hat{\Gamma}_{\alpha\gamma^\ast }^{\beta^\ast }-\hat{\Gamma}_{\alpha^\ast \gamma^\ast }^{\beta^\ast })\frac{\partial}{\partial y^{\beta^\ast }}, \end{align}\] which together with 30 implies that \[\text{Re}\Gamma_{\alpha\gamma}^\beta=\hat{\Gamma}_{\alpha\gamma}^\beta= \hat{\Gamma}_{\alpha^\ast \gamma}^{\beta^\ast },\quad \text{Im}\Gamma_{\alpha\gamma}^\beta=-\hat{\Gamma}_{\alpha^\ast \gamma}^\beta=\hat{\Gamma}_{\alpha\gamma}^{\beta^\ast }.\] Thus we obtain ?? . This completes the proof. ◻
Theorem 25. Let \(F: T^{1,0}M\rightarrow [0,+\infty)\) be a strongly convex Kähler-Berwald metric on a complex manifold \(M\). Then \(\nabla_X \pmb{J}=0\) for all \(X\in\mathcal{H}_{\mathbb{R}}\).
Proof. Since \(F\) is a Kähler-Berwald metric, by Lemma 3.1 in [14], \(\mathbb{G}^\mu\) are holomorphic with respect to \(v\) and \(\mathbb{G}^\mu=\hat{\mathbb{G}}^\mu+i\hat{\mathbb{G}}^{\mu^\ast}\). Thus \[\begin{align} 0=\frac{\partial\mathbb{G}^\mu}{\partial \bar{v}^{\alpha}} &=&\frac{1}{2}\left(\frac{\partial}{\partial y^{\alpha}}+i\frac{\partial}{\partial y^{\alpha^\ast }}\right)\left(\hat{\mathbb{G}}^\mu+i\hat{\mathbb{G}}^{\mu^\ast}\right)\\ &=&\frac{1}{2}\left(\hat{\Gamma}_\alpha^\mu+i\hat{\Gamma}_\alpha^{\mu^\ast}+i\hat{\Gamma}_{\alpha^\ast}^\mu-\hat{\Gamma}_{\alpha^\ast}^{\mu^\ast}\right), \end{align}\] from which we get \[\hat{\Gamma}_\alpha^\mu=\hat{\Gamma}_{\alpha^\ast}^{\mu^\ast},\quad\hat{\Gamma}_\alpha^{\mu^\ast}=-\hat{\Gamma}_{\alpha^\ast}^\mu.\label{337}\tag{36}\] Differentiating 36 , we obtain \[\hat{\mathbb{G}}_{j\alpha}^\mu=\hat{\mathbb{G}}_{j\alpha^\ast}^{\mu^\ast},\quad \hat{\mathbb{G}}_{j\alpha}^{\mu^\ast}=-\hat{\mathbb{G}}_{j\alpha^\ast}^\mu.\]
By Theorem 1.2 in [15], a strongly convex Kähler-Berwald metric is necessary a real Berwald metric, hence a real Landsberg metric [16]. So \(\dot{A}^k_{\;jl}=0\), which implies \(\hat{\mathbb{G}}^k_{jl}=\hat{\Gamma}^k_{j;l}\). That is, the equalities in ?? hold. Thus \(\nabla_X \pmb{J}=0\) for all \(X\in\mathcal{H}_{\mathbb{R}}\). ◻
Let \(\Theta:\mathcal{V}^{1,0}\rightarrow\mathcal{H}^{1,0}\) denote the complex horizontal map associated to \(\mathcal{H}^{1,0}\), and let \(\hat{\Theta}:\mathcal{V}_{\mathbb{R}}\rightarrow \mathcal{H}_{\mathbb{R}}\) denote the real horizontal map associated to \(\mathcal{H}_{\mathbb{R}}\). Since \(^o:\mathcal{V}^{1,0}\rightarrow\mathcal{V}_{\mathbb{R}}\) is a \(\mathbb{R}\)-isomorphism, we get a \(\mathbb{R}\)-isomorphism \(^{\hat{}}:\mathcal{H}^{1,0}\rightarrow\mathcal{H}_{\mathbb{R}}\) given by \[\hat{H}=\hat{\Theta}((\Theta^{-1}(H))^o),\quad \forall H\in \mathcal{H}^{1,0}.\] Note that \(\widehat{\delta_\alpha}=\frac{\delta}{\delta x^\alpha}\) and \(\widehat{i\delta_\alpha}=\frac{\delta}{\delta x^{\alpha^\ast}}\).
Let \[\hat{\chi}=y^j\frac{\delta}{\delta x^j}=y^j\left(\frac{\partial}{\partial x^j}-\hat{\Gamma}_j^k\frac{\partial}{\partial y^k}\right)\quadand\quad\chi=v^\alpha\left(\frac{\partial}{\partial z^\alpha}-\Gamma_{;\alpha}^\beta\frac{\partial}{\partial v^\beta}\right)\] denote the real and complex radial horizontal vector fields associated to \(F\), respectively. Let \(\hat{\Omega}\) and \(\Omega\) denote the curvature operators of the Cartan connection \(\nabla\) and the Chern-Finsler connection \(D\) associated to \(F\), respectively.
Using some known results established by Abate and Patrizio in section 2.6 in [8], we are now able to obtain the following theorem which shows the specialities of strongly convex Kähler-Berwald metrics.
Theorem 26. Let \(F: T^{1,0}M\rightarrow[0,+\infty)\) be a strongly convex Kähler-Berwald metric on a complex manifold \(M\). Then for all \(H\in\mathcal{H}^{1,0}\) and \(V\in\mathcal{V}^{1,0}\), we have
(1) \(\chi^o=\hat{\chi}\);
(2) \(\hat{\Theta}\) commutes with \(\pmb{J}\);
(3) \((\pmb{J}H)^o= \pmb{J}H^o\);
(4) \(\mathcal{H}_{\mathbb{R}}\) is \(\pmb{J}\)-invariant;
(5) \(H=(H^o)_o\);
(6) \(\Gamma_{;\alpha}^\beta=\hat{\Gamma}_{\alpha}^\beta+i\hat{\Gamma}_{\alpha}^{\beta^\ast }\);
(7) \((\frac{\delta}{\delta x^\alpha})_o=\frac{\delta}{\delta z^\alpha}\) and \((\frac{\delta}{\delta x^{\alpha^\ast }})_o=i\frac{\delta}{\delta z^\alpha}\);
(8) \[\hat{\Gamma}_a^b=\left\{ \begin{array}{ll} \text{Re}\,\Gamma_{;\alpha}^\beta, & if\;\;1\leq a,b\leq n, \\ \text{Im}\,\Gamma_{;\alpha}^\beta, & if\;\;1\leq a\leq n\;\;\text{and}\;\;n+1\leq b\leq 2n, \\ -\text{Im}\,\Gamma_{;\alpha}^\beta, & if\;\;n+1\leq a\leq 2n\;\;\text{and}\;\;1\leq b\leq n, \\ \text{Re}\,\Gamma_{;\alpha}^\beta, & if\;\;n+1\leq a,b\leq 2n. \end{array} \right.\]
(9) \(\nabla_{\hat{\chi}}V^o=(\nabla_{\chi^o}V)^o\);
(10) \(\nabla_{\hat{H}}V^o=(\nabla_{H^o}V)^o\);
(11) \(\langle\hat{\Omega}(\hat{\chi},\hat{H})\hat{H}\vert \hat{\chi}\rangle=\langle(\Omega(H,\overline{\chi})\chi)^o\vert H^o\rangle-\langle(\Omega(\chi,\overline{H})\chi)^o\vert H^o\rangle\);
(12) \(\Gamma_{\gamma;\alpha}^\beta=\hat{\Gamma}_{\gamma;\alpha}^\beta+i\hat{\Gamma}_{\gamma;\alpha}^{\beta^\ast }\);
(13) \(\nabla_X \pmb{J}=0\) for any \(X\in\mathcal{H}_{\mathbb{R}}\);
(14) \(\langle\hat{\Omega}(\hat{\chi},\widehat{\pmb{J}\chi})\widehat{\pmb{J}\chi}\vert \hat{\chi}\rangle=2\langle\Omega(\chi,\overline{\chi})\chi,\chi\rangle\).
Proof. The assertion (1) follows from Proposition 2.6.2 in [8].
The assertion (5) follows from Theorem 2.6.4 in [8], which together with Proposition 2.6.3 in [8] implies the assertions (2)-(4) and (6)-(8).
The assertion (9) follows from Theorem 2.6.6 in [8].
The assertion (10) follows from Theorem 2.6.8 in [8] since by assertion (5), we have \(\hat{H}=H^o\).
The assertion (11) follows from Theorem 2.6.9 in [8]. Since for a Kähler-Berwald metric we have \(\frac{\partial\Gamma_{;\mu}^\alpha}{\partial \bar{v}^\gamma}=0\), hence \(\left(\nabla_\chi\tau^{\mathcal{H}}\right)(\chi,\overline{W})=0\). Thus \(\left(\left(\nabla_\chi\tau^{\mathcal{H}}\right)(\chi,\overline{\Theta^{-1}(H)})\right)^o=0\) for any \(H\in\mathcal{H}^{1,0}\).
The assertion (12) follows from Theorem 14.
The assertion (13) follows from Theorem 25.
Substituting \(H=\pmb{J}\chi=i\chi\) into assertion (11), we get \[\langle\hat{\Omega}(\hat{\chi},\widehat{\pmb{J}\chi})\widehat{\pmb{J}\chi}\vert \hat{\chi}\rangle=\langle(\Omega(i\chi,\overline{\chi})\chi)^o\vert (i\chi)^o\rangle-\langle(\Omega(\chi,\overline{i\chi})\chi)^o\vert (i\chi)^o\rangle.\] This together with ?? implies \[\begin{align} \langle\hat{\Omega}(\hat{\chi},\widehat{\pmb{J}\chi})\widehat{\pmb{J}\chi}\vert \hat{\chi}\rangle&=&\text{Re}[\langle\Omega(i\chi,\overline{\chi})\chi,i\chi\rangle-\langle \Omega(\chi,\overline{i\chi})\chi,i\chi\rangle]\\ &=&\text{Re}[\langle\Omega(\chi,\overline{\chi})\chi,\chi\rangle+\langle \Omega(\chi,\overline{\chi})\chi,\chi\rangle]\\ &=&2\langle\Omega(\chi,\overline{\chi})\chi,\chi\rangle \end{align}\] where we use the fact that \(\langle\!\langle \cdot,\chi\rangle\!\rangle=0\) and \(\langle\Omega(\chi,\overline{\chi})\chi,\chi\rangle\) is real-valued. ◻
Proposition 27. Let \(\Phi=-iG_{\alpha\bar{\beta}}dz^{\alpha}\wedge d\bar{z}^{\beta}\) be the fundamental form of a strongly convex complex Finsler metric \(F:T^{1,0}M\rightarrow[0,+\infty)\) on a complex manifold \(M\). Then \(\Phi\) is a real horizontal \((1,1)\)-form on \(\tilde{M}\). If moreover, \(\nabla_X\pmb{J}=0\) for any \(X\in\mathcal{H}_{\mathbb{R}}\), then \[\Phi(X,Y)=\langle X\vert \pmb{J}Y\rangle -\langle \pmb{J}X\vert Y\rangle\] for any \(X,Y\in\mathcal{H}_{\mathbb{R}}\).
Proof. It is clear that \(\Phi\) is real \((1,1)\)-form on \(\tilde{M}\). If \(\nabla_X \pmb{J}\equiv 0\) for all \(X\in\mathcal{H}_{\mathbb{R}}\), then \(\mathcal{H}_{\mathbb{R}}\) is \(\pmb{J}\)-invariant and by assertion (5) in Theorem 26, we have \(H=(H_o)^o\) for any \(H\in\mathcal{H}_{\mathbb{R}}\), namely \(\mathcal{H}_{\mathbb{R}}\cong\mathcal{H}^{1,0}\). Thus \[\begin{align} \Phi(X,Y) &=&-iG_{\alpha\bar{\beta}}dz^\alpha\wedge d\bar{z}^\beta(V+\overline{V},W+\overline{W})\\ &=&-i\left(G _{\alpha\bar{\beta}}V^\alpha\overline{W}^\beta-G_{\alpha\bar{\beta}}W^\alpha\overline{V}^\beta\right)\\ &=&-i\left[\langle V,W\rangle-\overline{\langle V,W}\rangle\right]\\ &=&2\text{Im}\langle V,W\rangle. \end{align}\] for \(V=\frac{1}{2}(X-i\pmb{J}X)=X_o\) and \(W=\frac{1}{2}(Y-i\pmb{J}Y)=Y_o\in \mathcal{H}^{1,0}\) with \(X,Y\in \mathcal{H}_{\mathbb{R}}\).
Since by Lemma 1, we also have \[\begin{align} \langle V,W\rangle=2\langle V\vert \overline{W}\rangle=\frac{1}{2}\left\{\langle X\vert Y\rangle+\langle JX\vert JY\rangle+i\left[\langle X\vert \pmb{J}Y\rangle -\langle \pmb{J}X\vert Y\rangle\right]\right\}, \end{align}\] this completes the proof. ◻
For a strongly convex complex Finsler manifold \((M,F)\), denote \(d:\wedge^k\tilde{M}\rightarrow\wedge^{k+1}\tilde{M}\) the exterior derivative operator on \(\tilde{M}\). Then \[\begin{align} d=d_H+d_V,\quad d_H=dx^j\wedge\nabla_{\frac{\delta}{\delta x^j}},\quad d_V=\delta y^j\wedge\nabla_{\frac{\partial}{\partial y^j}},\label{dh} \end{align}\tag{37}\] where \(\nabla\) is the Cartan connection of \(F\). If we denote \[\partial_H=dz^\alpha\wedge D_{\delta_\alpha},\; \bar{\partial}_H=d\bar{z}^\alpha\wedge D_{\delta_{\bar{\alpha}}},\; \partial_V=\psi^\gamma\wedge D_{\frac{\partial}{\partial v^\gamma}},\; \bar{\partial}_V=\bar{\psi}^\gamma\wedge D_{\frac{\partial}{\partial\bar{v}^\gamma}},\label{dv}\tag{38}\] where \(D\) is the Chern-Finsler connection of \(F\), then in general \(d_H\neq \partial_H+\bar{\partial}_H\) since in general the Cartan connection \(\nabla\) may not coincides with the Chern-Finsler connection \(D\).
Proposition 28. Let \(d_H,d_V,\partial_H,\bar{\partial}_H,\partial_V,\bar{\partial}_V\) be defined by 37 and 38 , respectively. If \(\nabla_X\pmb{J}=0\) for all \(X\in\mathcal{H}_{\mathbb{R}}\), then \[d_H=\partial_H+\bar{\partial}_H,\quad d_V=\partial_V+\bar{\partial}_V.\]
Proof. By assumption, \(F\) must be a Kähler-Berwald metric, hence \(\mathcal{H}_{\mathbb{R}}\) is \(\pmb{J}\)-invariant and \(\widetilde{\mathcal{H}}^{1,0}=\mathcal{H}^{1,0}\). ◻
Theorem 29. Let \(F: T^{1,0}M\rightarrow [0,+\infty)\) be a strongly convex complex Finsler metric on a complex manifold \(M\). If \(\nabla_X \pmb{J}=0\) for all \(X\in\mathcal{H}_{\mathbb{R}}\), then
\(\nabla_X\varPhi= 0\) for all \(X\in\mathcal{H}_{\mathbb{R}}\);
\(d_H\varPhi= 0\), i.e., \(\varPhi\) is \(d_H\)-closed.
Proof. (1) Let \(H,X,Y\in\mathcal{H}_{\mathbb{R}}\). Then by Proposition 27, \[\begin{align} (\nabla_H\Phi)(X,Y)&=&H\Phi(X,Y)-\Phi(\nabla_HX,Y)-\Phi(X,\nabla_HY)\\ &=&H\langle X\vert \pmb{J}Y\rangle-H\langle \pmb{J}X\vert Y\rangle-\langle \nabla_HX\vert \pmb{J}Y\rangle+\langle \pmb{J}(\nabla_HX)\vert Y\rangle\\ &&-\langle X\vert \pmb{J}(\nabla_HY)\rangle+\langle \pmb{J}X\vert \nabla_HY\rangle. \end{align}\] Since the Cartan connection is horizontal metric, namely \(H\langle X\vert Y\rangle=\langle \nabla_HX\vert Y\rangle+\langle X\vert \nabla_HY\rangle\), the above equality becomes \[(\nabla_H\Phi)(X,Y)=\langle X\vert \nabla_H( \pmb{J}Y)\rangle-\langle \nabla_H( \pmb{J}X)\vert Y\rangle+\langle \pmb{J}(\nabla_HX)\vert Y\rangle-\langle X\vert \pmb{J}(\nabla_HY)\rangle.\] Since \(\nabla_H \pmb{J}=0\), we have \[\begin{align} &&\langle X\vert \nabla_H( \pmb{J}Y)\rangle-\langle \nabla_H( \pmb{J}X\vert Y\rangle+\langle \pmb{J}(\nabla_HX)\vert Y\rangle-\langle X\vert \pmb{J}(\nabla_HY)\rangle\\ &=&\langle X\vert \pmb{J}(\nabla_HY)\rangle-\langle \pmb{J}(\nabla_HX)\vert Y\rangle+\langle \pmb{J}(\nabla_HX)\vert Y\rangle-\langle X\vert \pmb{J}(\nabla_HY)\rangle\\ &=&0. \end{align}\] Therefore \(\nabla_H\Phi=0\) for any \(H\in\mathcal{H}_{\mathbb{R}}\).
(2) Let \(X_1,X_2,X_3\in\mathcal{H}_{\mathbb{R}}\). Then \[\begin{align} (d_H\Phi)(X_1,X_2,X_3)&=&X_1(\Phi(X_2,X_3))-X_2(\Phi(X_1,X_3))+X_3(\Phi(X_1,X_2))\\ &&-\Phi([X_1,X_2],X_3)+\Phi([X_1,X_3],X_2)-\Phi([X_2,X_3],X_1). \end{align}\] Using \(\nabla_{X_j}\Phi=0\), this becomes \[\begin{align} (d_H\Phi)(X_1,X_2,X_3)&=&\Phi(\nabla_{X_1}X_2-\nabla_{X_2}X_1,X_3) -\Phi(\nabla_{X_1}X_3-\nabla_{X_3}X_1,X_2)\\ &&+\Phi(X_1,\nabla_{X_3}X_2-\nabla_{X_2}X_3)\\ &&-\Phi([X_1,X_2],X_3)+\Phi([X_1,X_3],X_2)-\Phi([X_2,X_3],X_1). \end{align}\] Since \(\theta(X_j,X_k)=\nabla_{X_j}X_k-\nabla_{X_k}X_j-[X_j,X_k]\), and \(\theta(X_j,X_k)\in\mathcal{V}_{\mathbb{R}}\), while \(\Phi\) is horizontal, we get \((d_H\Phi)(X_1,X_2,X_3)=0\). Hence, \(d_H\Phi=0\). ◻
Remark 30. The operator \(d_H\) is not the same one defined on page 95 in [8], where \(d_H\Phi=0\) is equivalent to \(F\) being a Kähler-Finsler metric. Thus converse of Theorem 29 may not be true. So far in literature, however, explicit examples of Kähler-Finsler metrics [1]–[4], [17] were all proved to be Kähler-Berwald metrics. It is still open whether there exists an example of Kähler-Finsler metric which is not a Kähler-Berwald metric.
Theorem 31. Let \(F: T^{1,0}M\rightarrow [0,+\infty)\) be a strongly convex weakly Kähler-Finsler metric on a complex manifold \(M\) and \(\sigma:[0,1]\rightarrow M\) a smooth regular curve in \(M\). Then the following assertions are equivalent:
\(F\) is a strongly convex Kähler-Berwald metric;
The types of complexified vectors in \(T_{\mathbb{R}}M\) are preserved under parallel transport along \(\sigma\) with respect to \(\nabla\);
For any parallel real vector field \(V\) along \(\sigma\) with respect to \(\nabla\), \(JV\) is also parallel along \(\sigma\) with respect to \(\nabla\);
\(\nabla_X \pmb{J}\equiv 0\) for all \(X\in\mathcal{H}_{\mathbb{R}}\);
Proof. (1)\(\Rightarrow\)(2): It suffices to show that a type \((1,0)\) complexified tangent vector is preserved under parallel transport along \(\sigma\) with respect to \(\nabla\). Let \(\xi^{1,0}(0)\) a complexified tangent vector of type \((1,0)\) at \(\sigma(0)\), namely there exists a real tangent vector \(\xi_0\) at \(\sigma(0)\) such that \(\xi^{1,0}(0)=\frac{1}{2}(\xi_0-iJ\xi_0)\). Let \(\zeta\) be the parallel transport of \(\xi^{1,0}(0)\) along \(\sigma\) with respect to \(\nabla\). Write \(\zeta=\zeta^{1,0}+\zeta^{0,1}\), where \(\zeta^{1,0}\) and \(\zeta^{0,1}\) denote the \((1,0)\) and \((0,1)\) parts of \(\zeta\), respectively. It suffices to show that \(\zeta^{1,0}=\zeta\) and \(\zeta^{1,0}(0)=\xi^{1,0}(0)\), namely \(\zeta^{0,1}\equiv 0\) along \(\sigma\).
For this purpose, let \(V=V^j(t)\frac{\partial}{\partial x^j}\) be the parallel transport of \(\xi_0\) along \(\sigma\) with respect to \(\nabla\) such that \(V(0)=\xi_0\), namely \[\frac{dV^k}{dt}+V^l\hat{\Gamma}_{l;j}^k(\sigma(t);\dot{\sigma}(t))\dot{\sigma}^j=0.\label{NNN-a}\tag{39}\]
Setting \(JV=U^k(t)\frac{\partial}{\partial x^k}\). In the following we shall show that \(JV\) is also parallel along \(\sigma\) with respect to \(\nabla\). It is clear that \[U^\beta(t)=-V^{\beta^\ast}(t),\quad U^{\beta^\ast}(t)=V^\beta(t).\label{NNN-b}\tag{40}\] By assumption, \(F\) is a strongly convex Kähler-Berwald metric. Thus by Theorem 25, \(\nabla_X\pmb{J}=0\) for all \(X\in\mathcal{H}_{\mathbb{R}}\). Hence by Theorem 14, the horizontal Cartan connection coefficients \(\hat{\Gamma}_{j;k}^l\) satisfy ?? , namely \[\hat{\Gamma}_{\alpha^\ast;j}^\beta=-\hat{\Gamma}_{\alpha;j}^{\beta^\ast},\quad\hat{\Gamma}_{\alpha^\ast;j}^{\beta^\ast}=\hat{\Gamma}_{\alpha;j}^\beta.\label{NNN-c}\tag{41}\] Note that \[\frac{\pmb{D}U^k}{dt}=\frac{dU^k}{dt}+U^l\hat{\Gamma}_{l;j}^k(\sigma(t);\dot{\sigma}(t))\dot{\sigma}^j(t).\] This together with 39 41 implies that \[\begin{align} \frac{\pmb{D}U^\beta}{dt}&=&\frac{dU^\beta}{dt}+U^l\hat{\Gamma}_{l;j}^\beta(\sigma(t);\dot{\sigma}(t))\dot{\sigma}^j(t)\\ &=&-\frac{dV^{\beta^\ast}}{dt}-V^{\alpha^\ast}\hat{\Gamma}_{\alpha^\ast;j}^{\beta^\ast}(\sigma(t);\dot{\sigma}(t))\dot{\sigma}^j(t)-V^\alpha\hat{\Gamma}_{\alpha;j}^{\beta^\ast}\dot{\sigma}^j(t)\\ &=&-\frac{dV^{\beta^\ast}}{dt}-V^l\hat{\Gamma}_{l;j}^{\beta^\ast}(\sigma(t);\dot{\sigma}(t))\dot{\sigma}^j(t)=0. \end{align}\] By similar reason, we have \(\frac{\pmb{D}U^{\beta^\ast}}{dt}=0\). Thus \(JV\) is also parallel along \(\sigma\) with respect to \(\nabla\), namely \[\frac{\pmb{D}U^k}{dt}=\frac{dU^k}{dt}+U^l\hat{\Gamma}_{l;j}^k(\sigma(t);\dot{\sigma}(t))\dot{\sigma}^j(t)=0.\label{NNN-d}\tag{42}\] So that \(W=\frac{1}{2}(V-iJV)\) is parallel along \(\sigma\) with respect to \(\nabla\). Since \(W(0)=\frac{1}{2}(V(0)-iJV(0))=\frac{1}{2}(\xi_0-iJ\xi_0)=\zeta^{1,0}(0)\). Thus \(\zeta\equiv W\), namely \(\zeta^{1,0}\equiv0\) along \(\sigma\).
(2) \(\Rightarrow\) (3): Let \(V\) be a real vector field which is parallel along \(\sigma\) with respect to \(\nabla\). Write \(V=V^{1,0}+V^{0,1}\), where \(V^{1,0}=\frac{1}{2}(V-iJV)\) and \(V^{0,1}=\frac{1}{2}(V+iJV)\). Let \(W_1\) and \(W_2\) be the parallel transport of \(V^{1,0}(0)\) and \(V^{0,1}(0)\) along \(\sigma\) with respect to \(\nabla\). By assertion (2), \(W_1\) and \(W_2\) are vector fields of type \((1,0)\) and \((0,1)\) along \(\sigma\), respectively, and \(W=W_1+W_2\) is parallel along \(\sigma\) with respect to \(\nabla\). Since \(W(0)=W_1(0)+W_2(0)=V^{1,0}(0)+V^{0,1}(0)=V(0)\) so that \(W\equiv V\), \(W_1=V^{1,0}\) and \(W_2=V^{0,1}\) along \(\sigma\). Thus \(JV=J(V^{1,0}+V^{0,1})=i(V^{1,0}-V^{0,1})\) is also parallel along \(\sigma\) with respect to \(\nabla\).
(3) \(\Rightarrow\) (4): Suppose both \(V=V^k(t)\frac{\partial}{\partial x^k}\) and \(JV=U^k(t)\frac{\partial}{\partial x^k}\) are parallel along \(\sigma\) with respect to \(\nabla\). Then by 39 we have \[\begin{align} \frac{dV^\beta}{dt}+V^\alpha\hat{\Gamma}_{\alpha;j}^\beta(\sigma(t);\dot{\sigma}(t))\dot{\sigma}^j+V^{\alpha^\ast}\hat{\Gamma}_{\alpha^{\ast};j}^\beta(\sigma(t);\dot{\sigma}(t))\dot{\sigma}^j&=&0,\tag{43}\\ \frac{dV^{\beta^\ast}}{dt}+V^\alpha\hat{\Gamma}_{\alpha;j}^{\beta^\ast}(\sigma(t);\dot{\sigma}(t))\dot{\sigma}^j+V^{\alpha^\ast}\hat{\Gamma}_{\alpha^{\ast};j}^{\beta^\ast}(\sigma(t);\dot{\sigma}(t))\dot{\sigma}^j&=&0.\tag{44} \end{align}\] Similarly by 40 and 42 , we have \[\begin{align} -\frac{dV^{\beta^\ast}}{dt}-V^{\alpha^\ast}\hat{\Gamma}_{\alpha;j}^\beta(\sigma(t);\dot{\sigma}(t))\dot{\sigma}^j+V^{\alpha}\hat{\Gamma}_{\alpha^{\ast};j}^\beta(\sigma(t);\dot{\sigma}(t))\dot{\sigma}^j&=&0,\tag{45}\\ \frac{dV^{\beta}}{dt}-V^{\alpha^\ast}\hat{\Gamma}_{\alpha;j}^{\beta^\ast}(\sigma(t);\dot{\sigma}(t))\dot{\sigma}^j+V^{\alpha}\hat{\Gamma}_{\alpha^{\ast};j}^{\beta^\ast}(\sigma(t);\dot{\sigma}(t))\dot{\sigma}^j&=&0.\tag{46} \end{align}\] By 43 and 46 , we have \[V^\alpha(\hat{\Gamma}_{\alpha;j}^\beta-\hat{\Gamma}_{\alpha^{\ast};j}^{\beta^\ast})\dot{\sigma}^j+V^{\alpha^\ast}(\hat{\Gamma}_{\alpha^{\ast};j}^\beta+\hat{\Gamma}_{\alpha;j}^{\beta^\ast})\dot{\sigma}^j=0.\label{RR-e}\tag{47}\] By 44 and 45 , we have \[V^\alpha(\hat{\Gamma}_{\alpha;j}^{\beta^\ast}+\hat{\Gamma}_{\alpha^{\ast};j}^\beta)\dot{\sigma}^j+V^{\alpha^\ast}(\hat{\Gamma}_{\alpha^{\ast};j}^{\beta^\ast}-\hat{\Gamma}_{\alpha;j}^\beta)\dot{\sigma}^j=0.\label{RR-f}\tag{48}\] Since \(\hat{\Gamma}_{j;l}^k=\hat{\Gamma}_{l;j}^k\) and \(\hat{\Gamma}_{j;l}^k(x;y)y^j=\hat{\Gamma}_l^k\), it follows that 47 and 48 can be simplified as \[\begin{align} V^\alpha(\hat{\Gamma}_{\alpha}^\beta-\hat{\Gamma}_{\alpha^{\ast}}^{\beta^\ast})+V^{\alpha^\ast}(\hat{\Gamma}_{\alpha^{\ast}}^\beta+\hat{\Gamma}_{\alpha}^{\beta^\ast})&=&0,\tag{49}\\ V^\alpha(\hat{\Gamma}_{\alpha}^{\beta^\ast}+\hat{\Gamma}_{\alpha^{\ast}}^\beta)+V^{\alpha^\ast}(\hat{\Gamma}_{\alpha^{\ast}}^{\beta^\ast}-\hat{\Gamma}_{\alpha}^\beta)&=&0.\tag{50} \end{align}\] Since \(\hat{\Gamma}_j^k(\sigma;\dot{\sigma})\) are independent of \(V^j\) for \(j=1,\cdots,2n\), \(\sigma:[0,1]\rightarrow M\) is an arbitrary fixed smooth regular curve in \(M\) and \(V=V^j(t)\frac{\partial}{\partial x^j}\) is an arbitrary parallel vector field along \(\sigma\) with respect to \(\nabla\), it follows that 49 and 50 hold iff \[\hat{\Gamma}_{\alpha}^\beta(x;y)=\hat{\Gamma}_{\alpha^{\ast}}^{\beta^\ast}(x;y),\quad \hat{\Gamma}_{\alpha^{\ast}}^\beta(x;y)=-\hat{\Gamma}_{\alpha}^{\beta^\ast}(x;y),\quad \forall (x;y)\in\tilde{M}.\label{RR-i}\tag{51}\]
Differentiating 51 with respect to \(y^j\) yields \[\hat{\mathbb{G}}_{\alpha^\ast j }^\mu(x;y)=-\hat{\mathbb{G}}_{\alpha j}^{\mu^\ast }(x;y),\quad \hat{\mathbb{G}}_{\alpha^\ast j }^{\mu^\ast }(x;y)=\hat{\mathbb{G}}_{\alpha j}^\mu(x;y).\] This implies that \(\pmb{J}\) is horizontal parallel with respect to the real Berwald connection \(\breve{\nabla}\) associated to \(F\).
Since \(F\) is a strongly convex weakly Kähler-Finsler metric, by Lemma 3.1 in [14], the real spray coefficients \(\hat{\mathbb{G}}^k\) and the complex spray coefficients \(\mathbb{G}^\mu\) satisfy \[\mathbb{G}^\mu=\hat{\mathbb{G}}^\mu+i\hat{\mathbb{G}}^{\mathbb{\mu}^\ast}.\label{RR-j}\tag{52}\] Differentiating 52 with respect to \(\bar{v}^\alpha\) and using 51 , we obtain \[\frac{\partial\mathbb{G}^\mu}{\partial\bar{v}^\alpha}=\frac{1}{2}[(\hat{\Gamma}_\alpha^\mu-\hat{\Gamma}_{\alpha^\ast}^{\mu^\ast})+i(\hat{\Gamma}_\alpha^{\mu^\ast}+\hat{\Gamma}_{\alpha^\ast}^\mu)]=0.\] This implies that \(\mathbb{G}^\mu\) are locally holomorphic with respect to the fiber coordinates \(v\). Note that \(\mathbb{G}^\mu\) are \((2,0)\)-homogeneous with respect to the local holomorphic coordinates \(v\), namely \(\mathbb{G}^\mu(z;\lambda v)=\lambda^2\mathbb{G}^\mu(z;v)\). Thus \(\mathbb{G}^\mu\) must actually be quadratic with respect to \(v\), that is, \[\mathbb{G}^\mu=\frac{1}{2}\mathbb{G}_{\alpha\gamma}^\mu(z)v^\alpha v^\gamma,\] where \(\mathbb{G}_{\alpha\gamma}^\mu(z;v)=\mathbb{G}_{\alpha\gamma}^\mu(z)\) are the complex Berwald connection coefficients of \(F\). This shows that \(F\) is a weakly complex Berwald metric [15]. Since \(F\) is a strongly convex weakly Kähler-Finsler metric, thus by Theorem 1.1 in [15], \(F\) is also a real Berwald metric. Hence \(\hat{\mathbb{G}}^k_{jl}=\hat{\Gamma}^k_{j;l}\), namely the real Berwald connection coefficients \(\hat{\mathbb{G}}_{jl}^k\) coincide with the horizontal Cartan connection coefficients \(\hat{\Gamma}_{j;l}^k\). Thus the condition ?? in Theorem 14 are satisfied, hence \(\nabla_X \pmb{J}\equiv 0\) for all \(X\in\mathcal{H}_{\mathbb{R}}\).
(4)\(\Rightarrow\) (1): Theorem 23. ◻
In this section, we are able to use the geometric assumption \(\nabla \pmb{J}=0\) to classify all strongly convex complex Finsler metrics with constant holomorphic sectional curvatures.
Theorem 32. Let \(F: T^{1,0}M\rightarrow [0,+\infty)\) be a strongly convex complex Finsler metric on a complex manifold \(M\). If \(\nabla \pmb{J}=0\) and \(F\) has constant holomorphic sectional curvature \(c\neq 0\), then \(F\) is necessary a Kähler-Einstein metric on \(M\).
Proof. Using Definition 2.5.2 in [8], it follows that \(F\) has constant holomorphic sectional curvature \(c\) along a nonzero tangent vector \(v \in T_z^{1,0}M\) iff \[-2G_\alpha\frac{\partial \Gamma^\alpha_{;\mu}}{\partial \bar{z}^{\nu}}v^{\mu} \overline{v^\nu}=cG^2.\label{chc-1}\tag{53}\] Setting \(A^\alpha_{\beta;\mu\bar{\nu}}:=-\frac{\partial\Gamma^{\alpha}_{\beta;\mu} }{\partial \bar{z}^\nu}\). By Theorem 23, \(F\) must be a Kähler-Berwald metric since by assumption \(\nabla\pmb{J}=0\) implies particularly \(\nabla_X\pmb{J}=0\) for any \(X\in\mathcal{H}_{\mathbb{R}}\). So that we have \(\Gamma_{\beta;\mu}^\alpha(z)=\Gamma_{\mu;\beta}^\alpha(z)\), hence \(A_{\beta;\mu\bar{\nu}}^\alpha=A_{\mu;\beta\bar{\nu}}^\alpha\) and \(\Gamma_{;\mu}^\alpha=\Gamma_{\beta;\mu}^\alpha(z)v^\beta\) are holomorphic and complex linear with respect to \(v\). Differentiating 53 with respect to \(\bar{v}^\gamma\) yields \[\label{chc-2} G_{\alpha\bar{\gamma}}A^\alpha_{\beta;\mu\bar{\nu}} v^{\beta} v^{\mu} \overline{v^\nu}+G_{\alpha}A^\alpha_{\beta;\mu\bar{\gamma}} v^{\beta} v^{\mu} =cGG_{\bar{\gamma}}.\tag{54}\]
Rewritten 54 , we obtain \[\left\langle\Omega(\chi,\bar{\chi})\chi,\frac{\delta}{\delta z^\gamma} \right\rangle_v+\left\langle\Omega\left(\chi,\frac{\delta}{\delta \bar{z}^\gamma}\right)\chi,\chi \right\rangle_v=cGG_{\bar{\gamma}}. \label{chc-22}\tag{55}\]
Using Proposition 2.5.1 in [8], we immediately have \[\begin{align} \left\langle\Omega(\chi,\bar{\chi})\chi,\frac{\delta}{\delta z^\gamma} \right\rangle_v&=&\overline{\left\langle\Omega(\chi,\bar{\chi})\frac{\delta}{\delta z^\gamma},\chi \right\rangle_v},\\ \left\langle\Omega\left(\chi,\frac{\delta}{\delta \bar{z}^\gamma}\right)\chi,\chi \right\rangle_v&=&\overline{\left\langle\Omega\left(\frac{\delta}{\delta z^\gamma},\bar{\chi}\right)\chi,\chi \right\rangle_v}. \end{align}\] Using the first equality of (2.3.21) in [8], it follows that \[\begin{align} \label{hs-1} \left\langle\Omega(\chi,\bar{\chi})\frac{\delta}{\delta z^\gamma},\chi \right\rangle_v&=&-G_\alpha\frac{\partial \Gamma^\alpha_{\gamma;\mu}}{\partial \bar{v}^\nu}v^\mu\overline{v^\nu}-G_\alpha\Gamma^{\alpha}_{\gamma\sigma}\frac{\partial\Gamma^\sigma_{;\mu}}{\partial \bar{v}^\nu}v^\mu\overline{v^\nu}\nonumber. \end{align}\tag{56}\]
By assumption \(\nabla\pmb{J}=0\), it follows from Theorem 24 that the Cartan connection of \(F\) coincides with the Chern-Finsler connection of \(F\), hence the vertical Chern-Finsler connection coefficients \(\Gamma_{\gamma\sigma}^\alpha\) and the vertical Cartan connection coefficients \(\hat{\Gamma}_{bc}^a\) satisfy the equality \(\Gamma_{\gamma\sigma}^\alpha=\hat{\Gamma}^{\alpha}_{\gamma\sigma}+i\hat{\Gamma}^{\alpha^\ast}_{\gamma\sigma}\). On the other hand, the vertical Cartan connection coefficients \(\hat{\Gamma}_{bc}^a\) of \(F\) always satisfy \[G_a\hat{\Gamma}_{bc}^a=\frac{1}{4}G_ag^{as}\frac{\partial^3F^2}{\partial y^b\partial y^c\partial y^s}=\frac{1}{2}\frac{\partial^3F^2}{\partial y^b\partial y^c\partial y^s}y^s=0,\] hence \[\begin{align} 2G_\alpha\Gamma^{\alpha}_{\gamma\sigma}&=&(G_{\alpha}-iG_{\alpha^\ast})(\hat{\Gamma}^{\alpha}_{\gamma\sigma}+i\hat{\Gamma}^{\alpha^\ast}_{\gamma\sigma})\\ &=&(G_{\alpha}\hat{\Gamma}^{\alpha}_{\gamma\sigma}+G_{\alpha^\ast}\hat{\Gamma}^{\alpha^\ast}_{\gamma\sigma})-i(G_{\alpha^\ast}\hat{\Gamma}^{\alpha^\ast}_{\gamma^\ast\sigma}+G_{\alpha}\hat{\Gamma}^{\alpha}_{\gamma^\ast\sigma})\\ &=&0, \end{align}\] Thus \[\begin{align} \left\langle\Omega(\chi,\bar{\chi})\frac{\delta}{\delta z^\gamma},\chi \right\rangle_v =-G_\alpha\frac{\partial\Gamma^\alpha_{\gamma;\mu}}{\partial \bar{v}^\nu}v^\mu\overline{v^\nu} =\left\langle\Omega\left(\frac{\delta}{\delta z^\gamma},\bar{\chi}\right)\chi,\chi \right\rangle_v \end{align}\] since \(\Gamma_{\gamma;\mu}^\alpha=\Gamma_{\mu;\gamma}^\alpha\). Therefore we actually obtain \[\left\langle\Omega(\chi,\bar{\chi})\chi,\frac{\delta}{\delta z^\gamma} \right\rangle_v=\left\langle\Omega\left(\chi,\frac{\delta}{\delta \bar{z}^\gamma}\right)\chi,\chi \right\rangle_v.\] Thus 55 can be simplified as \[\label{chc-4} 2A^\alpha_{\beta;\mu\bar{\nu}} v^{\beta} v^{\mu} \overline{v^\nu} =cGv^{\alpha}.\tag{57}\] Differentiating 57 with respect to \(\bar{v}^\nu\) yields \[2A^\alpha_{\beta;\mu\bar{\nu}} v^{\beta} v^{\mu} =cG_{\bar{\nu}}v^{\alpha}.\label{chc-aaa}\tag{58}\] Differentiating 58 with respect to \(v^\beta\) and \(v^\mu\) successively yields \[\label{chc-5} 4A^\alpha_{\beta;\mu\bar{\nu}}=c\left(G_{\beta\bar{\nu}}\delta^{\alpha}_{\mu}+cG_{\mu\bar{\nu}}\delta^{\alpha}_{\beta}+cG_{\beta\bar{\nu}\mu}v^{\alpha}\right).\tag{59}\] Taking \(\alpha=\beta\) and then summing \(\alpha\) from \(1\) to \(n\), we obtain \[\label{chc-6} 4\sum_{\alpha=1}^nA^\alpha_{\alpha;\mu\bar{\nu}}=c(n+1)G_{\mu\bar{\nu}}\quadequivalently\quad G_{\mu\bar{\nu}}=\frac{4}{c(n+1)}\text{Ric}_{\mu\bar{\nu}}.\tag{60}\] This implies that \(F\) is a Kähler-Einstein metric on \(M\) (since \(\text{Ric}_{\mu\bar{\nu}}\) are independent of \(v\), hence \(G_{\mu\bar{\nu}}\) of \(F\) are also independent of the fiber coordinates \(v\)). ◻
Now we immediately obtain the following rigidity theorem.
Theorem 33. Let \(F: T^{1,0}M\rightarrow[0,+\infty)\) be a complete strongly convex complex Finsler metric on a simply connected complex manifold \(M\). If \(\nabla\pmb{J}=0\) and \(F\) has constant holomorphic sectional curvature \(c\in \mathbb{R}\). Then
if \(c<0\), \((M,F)\) is a Kähler manifold which is holomorphically isometric to the open unit ball \(B_n\) in \(\mathbb{C}^n\) with a constant multiple of the Bergman metric, namely \[F^2(z;v)=-\frac{4}{c}\frac{(1-\| z\| ^2)\| v\| ^2+\vert \langle z,v\rangle\vert ^2}{(1-\| z\| ^2)^2}\]
if \(c=0\), \((M,F)\) is locally a complex Minkowski space which is locally holomorphic isometric to \(\mathbb{C}^n\) with some complex Minkowski metric \(F(z;v)=f(v)\);
if \(c>0\), \((M,F)\) is a Kähler manifold which is holomorphically isometric to the complex projective space \(\mathbb{CP}^n\) with a constant multiple of the Fubini-Study metric, namely \[F^2(z;v)=\frac{4}{c}\frac{(1+\| z\| ^2)\| v\| ^2-\vert \langle z,v\rangle\vert ^2}{(1+\| z\| ^2)^2}.\]
Proof. For \(c=0\), it follows from Theorem 2.1 in [7] that \(F\) is locally a complex Minkowski metric.
For \(c\neq 0\), it follows from Theorem 32 that \(F\) is a Kähler-Einsetin metric on \(M\). Since \(M\) is simply connected and \(F\) has constant holomorphic sectional curvature \(c\), the classical result of Kähler geometry (cf. page 171 in [18]) implies the assertions (i) and (iii). ◻
The nontrivial case of Theorem 33 is to show the case whenever \(M\) admits both Hermitian quadratic and non-Hermitian quadratic Kähler-Berwald metrics, the constant holomorphic sectional curvature property implies Kähler-Berwald metrics must actually be Kähler metrics. Indeed, there exist lots of complex manifolds which admit both Hermitian quadratic and non-Hermitian quadratic Kähler-Berwald metrics, such as the unit polydisks in \(\mathbb{C}^n\) (\(n\geq 2\)) [1],[2] and irreducible bounded symmetric domains [3],[4]. There are also complex manifolds, such as the open unit ball \(B_n\) in \(\mathbb{C}^n\) and the complex projective space \(\mathbb{CP}^n\) [17], which admit no non-Hermitian quadratic Kähler-Berwald metrics other than constant scaling of the standard Kähler metrics on \(B_n\) and \(\mathbb{CP}^n\), respectively.
Remark 34. In [7], under the assumptions that \((M,F)\) is a simply connected and complete complex manifold modeled on a complex Minkowski space \((\mathbb{C}^n,f)\), that \(F\) is a Kähler-Finsler metric and satisfies the Royden condition, and that \((M,F)\) has constant holomorphic sectional curvature \(c\), Aikou outlined a proof of the above theorem (using the nontrivial assertion that "if a Kähler-Berwald manifold \((M,F)\) is of constant holomorphic sectional curvature \(c\), then \((M,h_M)\) is also of constant holomorphic sectional curvature \(c\)" without proof). More precisely, the equality \(F(z;v)=f(v)\) in assertion (ii) is replaced by \(F(z;v)=f(v)\leq \sum_{k=1}^n\vert v^k\vert ^2\) and the "\(=\)" in assertion (iii) is replaced by "\(\leq\)" in [7]. In contrast, our proof of Theorem 33 is based on the rigidity Theorem 32 which is essentially different from that of [7]. Moreover, our hypothesis \(\nabla\pmb{J}=0\) is a simple geometric condition.
Remark 35. The complex Minkowski metric \(f(v)\) in assertion (ii) can not be replaced by the canonical Euclidean metric \(f(v)=\sum_{\alpha=1}^n\vert v^\alpha\vert^2\). Indeed, for any fixed \(t\in[0,+\infty)\) and integer \(k\geq 2\), \[f_{t,k}(v)=\vert v^1\vert^2+\cdots+\vert v^n\vert^2+t\sqrt{\vert v^1\vert^{2k}+\cdots+\vert v^n\vert^{2k}},\quad \forall v=(v^1,\cdots,v^n)\in\mathbb{C}^n\] is a strongly convex complex Minkowski metric on \(\mathbb{C}^n\) [1], and \((\mathbb{C}^n,f_{t,k})\) is a complex Minkowski space with vanishing holomorphic sectional curvature. However, \((\mathbb{C}^n,f_{t_1,k_1})\) is not holomorphic isometric to \((\mathbb{C}^n,f_{t_2,k_2})\) whenever \(t_1\neq t_2\) and \(k_1\neq k_2\).
Remark 36. The method developed in this paper can also be used to study the parallelism of the canonical complex structure \(\pmb{J}\) on the pull-back bundle \(\pi^\ast T_{\mathbb{R}}M\) with respect to the Chern-Rund connection of \(F\), this will appear elsewhere.
Acknowledgement.
Corresponding author: zcp@xmu.edu.cn↩︎