The purpose of this paper is to initiate the submanifold theory of the probability simplex, and we start by classifying doubly totally-umbilical submanifolds. Classical information geometry developed by S. Amari and H. Nagaoka studies the differential
geometry of statistical models as submanifolds in the probability simplex [1]. The probability simplex \((\Delta^n,g^F,\nabla^{(\mathrm{e})})\) consists of the Fisher metric \(g^F\) and the exponential connection \(\nabla^{(\mathrm{e})}\) on the set of positive
probability distributions \(\Delta^n=\{p:\{1,\ldots,n+1\}\to(0,1)\mid \sum_{\omega=1}^{n+1}p(\omega)=1\}\) which is a manifold. The tools \(g^F,\nabla^{(\mathrm{e})}\) are induced by
Chentsov’s theorem under the invariance of Markov embeddings. Moreover, the Riemannian manifold \((\Delta^n,g^F)\) is isometric to an open set of the Euclidean sphere of radius \(2\), thus
its sectional curvature is constant equal to \(\frac{1}{4}\). On \(\Delta^n\), another affine connection called the mixture connection \(\nabla^{(\mathrm{m})}\) is defined by the following equation\(:\)\[\begin{align} \label{conjeq} Xg^F(Y,Z) =
g^F\left(\nabla^{(\mathrm{e})}_X Y,Z\right) + g^F\left(Y,\nabla^{(\mathrm{m})}_X Z\right),\quad X,Y,Z\in\Gamma(TM).
\end{align}\tag{1}\] A submanifold \(M\subset\Delta^n\) is called a statistical model, or a family of probability distributions. In particular, it is known that \(M\) is an exponential family if and only if it is \(\nabla^{(\mathrm{e})}\)-autoparallel, and \(M\) is a mixture family if and only if it is
\(\nabla^{(\mathrm{m})}\)-autoparallel. While these two classes have been extensively studied, much less is known about more general submanifolds in the probability simplex [1].
On the other hand, the geometry of statistical manifolds studies the differential-geometric structure arising in information geometry. A pair \((g,\nabla)\) of a Riemannian metric \(g\) and
a torsion-free affine connection \(\nabla\) on a manifold \(M\) is called a statistical structure if \(\nabla g\) is a totally symmetric tensor
field, and here the triplet \((M,g,\nabla)\) is called a statistical manifold. Another affine connection called the conjugate connection\(\nabla^*\) is defined by the
equation analogous to 1 . H. Furuhata introduced doubly totally-umbilical submanifolds as the natural analogues of totally-umbilical submanifolds in Riemannian geometry [2]. Totally-umbilical submanifolds have been extensively studied and continue to attract considerable attention in Riemannian geometry [3]–[7], and in pseudo-Riemannian geometry [8], [9].
The definition of doubly totally-umbilical submanifolds is given in Defition 8. We show that the doubly totally-umbilical submanifolds in the probability simplex are described as
follows: Theorem 26. A submanifold \(M^m\) of the probability simplex \((\Delta^n,g^F,\nabla^{\mathrm{(e)}})\)
is doubly totally-umbilical if and only if it is contained in the following mixture family\(\mathrm{:}\)\[\begin{align}
\Xi_{a_{i_1},\ldots,a_{i_{\mathcal{A}}},b_{j_1},\ldots,b_{j_{\mathcal{B}}},\phi} = \left\{p\in\Delta^n\mid p(\omega_{i_k}) = a_{i_k}p(\omega_{\phi(i_k)}),\,p(\omega_{j_l})=b_{j_l}\right\},
\end{align}\] where \(i_1,\ldots,i_{\mathcal{A}},j_1,\ldots,j_{\mathcal{B}}\in\{1,\ldots,n+1\}\) are \(\mathcal{A}+\mathcal{B}=n-m\) distinct integers, \(\phi:\{i_1,\ldots,i_{\mathcal{A}}\}\to(\{1,\ldots,n+1\}\setminus\{i_1,\ldots,i_{\mathcal{A}},j_1,\ldots,j_{\mathcal{B}}\})\), and \(a_{i_1},\ldots,a_{i_{\mathcal{A}}}\) are positive real numbers
and \(b_{j_1},\ldots,b_{j_{\mathcal{B}}}\in(0,1)\).
In particular, if \(n=m+1\), then either one of the following hold\(:\)
The submanifold \(M\) is doubly totally-umbilical if and only if there exist an \(\omega\in\{1,\ldots,m+1\}\) and \(0<b<1\) such that \(M\) is contained in \[\begin{align} \{p\in\Delta^{m+1}\mid p(\omega)
= b\}.
\end{align}\]
The submanifold \(M\) is doubly autoparallel if and only if there exist two distinct \(\omega_1,\omega_2\in\{1,\ldots,m+1\}\) and an \(a>0\) such
that \(M\) is contained in \[\begin{align} \{p\in\Delta^{m+1}\mid p(\omega_1) = ap(\omega_2)\}.
\end{align}\]
It is important to note that every doubly autoparallel submanifolds are doubly totally-umbilical submanifolds.
We review the submanifold theory of statistical manifolds in Section 3. The proof relies on the fact that the statistical manifold \((\Delta^n,g^F,\nabla^{(\mathrm{e})})\) is a Hessian
manifold, namely, a statistical manifold whose affine connection is flat. Since Hessian manifolds carry the Hessian curvature tensor, we derive formulas for submanifolds in terms of the Hessian curvature of the ambient space. In particular, it turns out
that doubly totally-umbilical submanifolds in the probability simplex are also Hessian manifolds.
In Theorem 25, we also classify the doubly totally-umbilical submanifolds in the denormalized state space \((\mathbb{R}^+)^{n+1}\) with the
Hessian structure \((g_0,D)\) from [1] or [10]. This is because the probability simplex \((\Delta^n,g^F,\nabla^{(\mathrm{e})})\) is embedded into \(((\mathbb{R}^+)^{n+1},g_0,D)\) as a doubly
totally-umbilical statistical submanifold. Therefore, any doubly totally-umbilical submanifold \(M\) of the probability simplex can be embedded into \(((\mathbb{R}^+)^{n+1},g_0,D)\) as a
doubly totally-umbilical submanifold. Consequently, the submanifold \(M\) can be realized as the intersection of \(\Delta^n\) and a doubly totally-umbilical submanifold of the denormalized
state space.
All objects in this paper are assumed to be smooth, and \(M=M^m\) denotes a connected manifold of dimension \(m\geq2\). We denote by \(\Gamma(E)\) the set
of sections of a vector bundle \(E\) over \(M\). In particular, we denote by \(T^{(p,q)}M\) the tensor bundle of type \((p,q)\).
Let \(g\) be a Riemannian metric on \(M\), and \(\nabla\) a torsion-free affine connection on \(M\). The triplet \((M,g,\nabla)\) is called a statistical manifold if \[(\nabla_X g)(Y,Z) = (\nabla_Y g)(X,Z)\] holds for any vector fields \(X,Y,Z\in\Gamma(TM)\). Here,
the pair \((g,\nabla)\) is called the statistical structure on \(M\). The Levi-Civita connection of \(g\) will be denoted by \(\nabla^g\). The equation \(\nabla g = 0\) is equivalent to \(\nabla=\nabla^g\).
Definition 1. The conjugate connection\(\nabla^{*}\) of a statistical manifold \((M,g,\nabla)\) is a torsion-free affine connection on \(M\) defined by \[Xg(Y,Z)=g(\nabla_X Y,Z)+g(Y,\nabla^{*}_X Z),\quad X,Y,Z\in\Gamma(TM).\]
Remark 1. Conjugate connections are often called dual connections and are sometimes denoted by \(\overline{\nabla}\) instead of \(\nabla^{*}\).
Remark 2. For a statistical manifold \((M,g,\nabla)\), the triplet \((M,g,\nabla^{*})\) is also a statistical manifold. In fact, the equality \(\nabla g=-\nabla^{*}g\) holds.
The statistical manifold in the next example is the main object of this paper and is called the probability simplex. See [1], [11], [12] for its role in information geometry. For differential geometric aspects of the probability simplex, see [10], [13]–[15] for
details.
Example 1. Let \(\Omega_{n+1}=\{1,\ldots,n+1\}\) be a finite set. A positive probability distribution\(p\) on \(\Omega_{n+1}\)
is a map \(p:\Omega_{n+1}\to(0,1)\) such that \[\sum_{\omega=1}^{n+1}p(\omega) = 1.\] Denote by \(\Delta^n\) the set of all the positive probability
distributions on \(\Omega_{n+1}\). The set \(\Delta^n\) is a smooth manifold with an atlas consisting of a single coordinate system \((\eta^1,\ldots,\eta^n)\) defined by \((\eta^1(p),\ldots,\eta^n(p))=(p(1),\ldots,p(n))\). The Fisher metric\(g^F\) and the exponential
connection\(\nabla^{(\mathrm{e})}\) are defined by the following equations\(:\)\[\begin{align}
g^F\left(\frac{\partial}{\partial \eta^i},\frac{\partial}{\partial \eta^j}\right) &= \sum_{\omega = 1}^{n+1}\left(\frac{\partial}{\partial \eta^i}\log (p(\omega))\cdot\frac{\partial}{\partial \eta^j}\log (p(\omega))\right)p(\omega)\\ &=
\frac{\delta_{ij}}{\eta^i} + \frac{1}{1-\sum_{l=1}^{n}\eta^l},\label{fishelem}\\[6pt] g^F\left(\nabla^{(\mathrm{e})}_{\frac{\partial}{\partial \eta^i}}\frac{\partial}{\partial \eta^j},\frac{\partial}{\partial \eta^k}\right)&=\sum_{\omega =
1}^{n+1}\left(\frac{\partial}{\partial \eta^i}\frac{\partial}{\partial \eta^j}\log (p(\omega))\cdot\frac{\partial}{\partial \eta^k}\log (p(\omega))\right)p(\omega)\\ &= -\frac{\delta_{ij}\delta_{jk}}{\eta^i} +
\frac{1}{\left(1-\sum_{l=1}^{n}\eta^l\right)^2},
\end{align}\tag{2}\] where \(\delta_{ij}\) denotes the Kronecker delta. The triplet \((\Delta^n,g^F,\nabla^{(\mathrm{e})})\) is a statistical manifold, known as the
probability simplex. The conjugate connection of \((\Delta^n,g^F,\nabla^{(\mathrm{e})})\) is called the mixture connection, often denoted by \(\nabla^{(\mathrm{m})}\).
Definition 2. The \(\alpha\)-connections\(\nabla^{(\alpha)}\) on a statistical manifold \((M,g,\nabla)\) is a family of affine
connections defined by \[\nabla^{(\alpha)}_X Y = \frac{1+\alpha}{2}\nabla_X Y + \frac{1-\alpha}{2}\nabla^{*}_X Y,\quad X,Y\in\Gamma(TM)\] for each \(\alpha\in\mathbb{R}\). By definition,
\(\nabla^{(1)}=\nabla\) and \(\nabla^{(-1)}=\nabla^{*}\) hold. Moreover, we have \(\nabla^{(-\alpha)}=(\nabla^{(\alpha)})^{*}\).
Remark 3. In Definition 2, the triplet \((M,g,\nabla^{(\alpha)})\) is also a statistical manifold for each \(\alpha\in\mathbb{R}\).
The difference tensor \(K\in\Gamma(T^{(1,2)}M)\) of a statistical manifold \((M,g,\nabla)\) is defined by \[\begin{align}
\label{difftendef} K_X Y = K(X,Y) &= \nabla_X Y - \nabla^g_X Y,\quad X,Y\in\Gamma(TM).
\end{align}\tag{3}\] The conditions \(\nabla g = 0\) and \(K=0\) are equivalent. The difference tensor \(K^{(\alpha)}\) of \((M,g,\nabla^{(\alpha)})\) for each \(\alpha\in\mathbb{R}\) is given by \(K^{(\alpha)}=\alpha K\).
2.2 Curvature tensor fields on the statistical manifold↩︎
For an affine connection \(\nabla\) on \(M\), we define the curvature tensor field\(R^\nabla \in \Gamma(T^{(1,3)}M)\) of \(\nabla\) by \[R^\nabla (X,Y)Z=\nabla_X \nabla_Y Z-\nabla_Y \nabla_X Z-\nabla_{[X,Y]}Z,
\quad X, Y, Z \in \Gamma(TM).\] On a statistical manifold \((M,g,\nabla)\), we often denote \(R^{\nabla}\) by \(R\), \(R^{\nabla^g}\) by \(R^g\), and \(R^{\nabla^{*}}\) by \(R^{*}\), and we denote the curvature tensor field of \(\nabla^{(\alpha)}\) by \(R^{(\alpha)}\). For \(X,Y,Z,W\in\Gamma(TM)\), the curvature tensor fields are related as follows: \[\begin{align} && g(R(X,Y)Z,W) = -g(Z,R^*(X,Y)W), \tag{4}\\ && R(X,Y)Z = R^g(X,Y)Z + (\nabla^g_X K)(Y,Z) - (\nabla^g_Y K)(X,Z) + \lbrack K_X,K_Y \rbrack Z, \tag{5}\\ && R(X,Y)Z + R^*(X,Y)Z = 2R^g(X,Y)Z +
2[K_X,K_Y]Z.\tag{6}
\end{align}\]
Definition 3. A statistical manifold \((M,g,\nabla)\) is said to be conjugate symmetric if \(R=R^*\) holds.
Remark 4. The conjugate symmetry of \((M,g,\nabla)\) is equivalent to each of the following conditions\(:\)
\(g(R(X,Y)Z,W) = -g(Z,R(X,Y)W)\) for all \(X,Y,Z,W\in\Gamma(TM)\).
\(\nabla^gK\) is totally symmetric on \(M\).
Definition 4. A statistical manifold \((M,g,\nabla)\) is said to have constant curvature\(k\) if the following equation holds for some real number \(k\)\(:\)\[\label{constcurv} R(X,Y)Z=k(g(Y,Z)X-g(X,Z)Y),\quad X,Y,Z\in\Gamma(TM).\tag{7}\]
If a statistical manifold \((M,g,\nabla)\) has constant curvature, then the statistical manifold \((M,g,\nabla^{*})\) also has constant curvature. It is easy to see that statistical
manifold of constant curvature is also conjugate symmetric. In fact, the following proposition is known [16].
Proposition 5. Let \((M,g,\nabla)\) be a statistical manifold. The following conditions are equivalent\(:\)
The statistical manifold \((M,g,\nabla)\) has constant curvature.
*The statistical manifold \((M,g,\nabla)\) is conjugate symmetric and \(\nabla\) is projectively flat.*
If there is a \(k^{(\alpha)}\in\mathbb{R}\) such that \(R^{(\alpha)}\) satisfies the equation 7 for each \(\alpha\in\mathbb{R}\), then we say that \((M,g,\nabla^{(\alpha)})\)has constant curvature for each\(\alpha\). In this case, \(k^{(\alpha)}\) can be determined by \[\begin{align} k^{(\alpha)} = \alpha^2k^{(1)} + (1-\alpha^2)k^{(0)}.
\end{align}\] This equation follows from the conjugate symmetry of \((M,g,\nabla)\) and the identity \[\begin{align} R^{(\alpha)}(X,Y)Z &= R^g(X,Y)Z + \alpha^2[K_X,K_Y]Z\\ &=
\alpha^2R(X,Y)Z + (1-\alpha^2)R^g(X,Y)Z,\quad X,Y,Z\in\Gamma(TM).
\end{align}\] Computations of these formulas can be found in many papers, such as [16], [17].
Example 2. The probability simplex \((\Delta^n,g^F,\nabla^{(\mathrm{e})})\) in Example 1 is a statistical manifold which \((\Delta^n,g^F,\nabla^{(\alpha)})\) is constant curvature for each \(\alpha\in\mathbb{R}\). In fact, the curvature tensor field of \(\nabla^{(\mathrm{e})}\) and
\(\nabla^{(\mathrm{m})}\) are zero. This can be proved since the coordinate system \((\eta^1,\ldots,\eta^n)\) is an affine coordinate system of \(\nabla^{(\mathrm{m})}\), that is, \[\begin{align} \nabla^{(\mathrm{m})}_{\frac{\partial}{\partial \eta^i}}\frac{\partial}{\partial \eta^j} = 0
\end{align}\] holds. On the other hand, the immersion \(\Delta^n\ni p\to(2\sqrt{p(1)},\ldots,2\sqrt{p(n+1)})\in S^{n}(2)\) is an isometric embedding of \((\Delta^n,g^F)\) into the
Euclidean sphere \((S^n(2),g_0)\) of radius \(2\). Thus, \((\Delta^n,g^F)\) can be regarded as an open Riemannian submanifold of \((S^n(2),g_0)\), for details, see [13]. Therefore, the Riemannian manifold \((\Delta^n,g^F)\) has constant curvature \(\frac{1}{4}\), and \[\begin{align} R^{(\alpha)}(X,Y)Z = \frac{1-\alpha^2}{4}\left(g^F(Y,Z)X-g^F(X,Z)Y\right),\quad
X,Y,Z\in\Gamma(T\Delta^n)
\end{align}\] holds for each \(\alpha\in\mathbb{R}\).
A statistical manifold \((M,g,\nabla)\) with flat connection \(\nabla\) is called a Hessian manifold.
Definition 5. Let \((M,g,\nabla)\) be a Hessian manifold. If there exists a \(c\in\mathbb{R}\) such that \[\begin{align} (\nabla_X K)(Y,Z) = -\frac{c}{2}\left(g(X,Y)Z+g(X,Z)Y\right),\quad X,Y,Z\in\Gamma(TM),\label{chc}
\end{align}\tag{8}\] then \((M,g,\nabla)\) is said to have constant Hessian curvature of \(c\). We abbreviate this by CHC \(c\).
As seen in Example 2, the probability simplex is a Hessian manifold. In fact, it has CHC \(-1\), see [10]. The proof of the following proposition can be found in [14] for example.
Proposition 6. *Let \((M,g,\nabla)\) be a Hessian manifold that has CHC \(c\). Then the Riemannian manifold \((M,g)\) has constant
curvature \(-\frac{c}{4}\).*
Example 3. Let \(\mathbb{R}^+=\{y\in\mathbb{R}\mid y>0\}\), denote by \(g_0\) the Euclidean metric restricted to \((\mathbb{R}^+)^n\),
and let s\(D\) be a torsion-free affine connection on \((\mathbb{R}^+)^n\) defined by \[\begin{align} D_{\frac{\partial}{\partial y^i}}\frac{\partial}{\partial
y^j} &= -\frac{\delta_{ij}}{y^i}\frac{\partial}{\partial y^i}.
\end{align}\] The triplet \(((\mathbb{R}^+)^n,g_0,D)\) is a Hessian manifold of CHC \(0\)[10].
3 Doubly totally-umbilical submanifolds of statistical manifolds↩︎
We fix the notation and terminology used throughout this paper, most of which are borrowed from [2]. Denote by \((\widetilde{M},\widetilde{g},\widetilde{\nabla})\) a statistical manifold of dimension \(n=m+p\). Given an immersion \(\iota:M\to\widetilde{M}\), it induces a
statistical structure \((g,\nabla)\) on \(M\) by the following: \[\begin{align} g=\iota^*\widetilde{g},\quad g\left(\nabla_X
Y,Z\right)=\widetilde{g}\left(\widetilde{\nabla}_X \iota_*Y,\iota_*Z\right),\quad X,Y,Z\in\Gamma(TM).\label{induce}
\end{align}\tag{9}\] Here, the connection on \(\iota^{*}T\widetilde{M}\) induced by \(\widetilde{\nabla}\) is also denoted by \(\widetilde{\nabla}\).
Definition 6. Let \((M,g,\nabla),(\widetilde{M},\widetilde{g},\widetilde{\nabla})\) be statistical manifolds. If there exists an immersion \(\iota:M\to\widetilde{M}\)
such that the equation 9 holds, then \(\iota:(M,g,\nabla)\to(\widetilde{M},\widetilde{g},\widetilde{\nabla})\) is called a statistical immersion, and \((M,g,\nabla)\) is called a statistical submanifold of \((\widetilde{M},\widetilde{g},\widetilde{\nabla})\).
Example 4. Consider the Hessian manifold \((((\mathbb{R}^+)^{n+1},g_0,D))\) of CHC 0 in Example 3. The probability simplex is embedded
into \((\mathbb{R}^+)^{n+1}\) by \(\iota:\Delta^n\ni p\to(2\sqrt{p(1)},\ldots,2\sqrt{p(n+1)})\in(\mathbb{R}^+)^{n+1}\). Define a coordinate system \((\eta^1,\ldots,\eta^{n+1})\) on \((\mathbb{R}^+)^{n+1}\) by \(\eta^i=\frac{(y^i)^2}{4},\,i\in\{1,\ldots,n+1\}\). If we denote by \(D^*\) the conjugate connection of \(((\mathbb{R}^+)^{n+1},g_0,D)\), then \((\eta^1,\ldots,\eta^{n+1})\) is an affine coordinate system of \(D^*\), and we have \[\begin{align} \label{denorprobsimp} g_0\left(\frac{\partial}{\partial \eta^i},\frac{\partial}{\partial \eta^j}\right) &=
\frac{\delta_{ij}}{\eta^i},\\ D_{\frac{\partial}{\partial \eta^i}}\frac{\partial}{\partial \eta^j} =& -\frac{\delta_{ij}}{\eta^i}\frac{\partial}{\partial \eta^i}.
\end{align}\tag{10}\] With this coordinate system and the coordinate system \((\eta^1,\ldots,\eta^n)\) on \(\Delta^n\) defined in Example 1, the embedding \(\iota:\Delta^n\to(\mathbb{R}^+)^{n+1}\) is expressed by \(\iota(\eta^1,\ldots,\eta^n)=(\eta^1,\ldots,\eta^n,1-\sum_{i=1}^n\eta^i)\). It is easy to see that the statistical structure \((g^F,\nabla^{(\mathrm{e})})\) on \(\Delta^n\) is the one induced by \((g_0,D)\), thus the probability simplex \((\Delta^n,g^F,\nabla^{(\mathrm{e})})\) is a statistical submanifold of \(((\mathbb{R}^+)^{n+1},g_0,D)\). Following [1], [18], the statistical manifold \(((\mathbb{R}^+)^{n+1},g_0,D)\) is called the denormalized state space.
Example 5. For an embedding \(\iota:\Delta^m\to\Delta^n\) where \(m\leq n\), suppose there is a family of non-empty subsets \(\{C_1,\ldots,C_{m+1}\}\subset\Omega_{n+1}\) such that \[\begin{align} \Omega_{m+1} = \bigsqcup_{l = 1}^{m+1} C_l
\end{align}\] is a disjoint union. The embedding \(\iota\) is called a Markov embedding if there exist functions \(Q_l:\Omega_{n+1}\to[0,\infty)\) whose support is contained
in \(C_l\) for each \(l\in\Omega_{m+1}\) such that \[\begin{align} \label{markovemb} \iota(p) = \sum_{l =
1}^{m+1}p(l)Q_l,\quad p\in\Delta^{m}.
\end{align}\tag{11}\] With the statistical structure \((g^F,\nabla^{(\mathrm{e})})\) on the probability simplex in Example 1, Markov
embeddings \(\iota:(\Delta^m,g^F,\nabla^{(\mathrm{e})})\to(\Delta^n,g^F,\nabla^{(\mathrm{e})})\) are statistical immersions. In fact, it is known that the scalar multiples of the Fisher metric, the \(\alpha\)-connections of \((g^F,\nabla^{\mathrm{(e)}})\) are the only \((0,2)\)-type tensor field, affine connections, respectively, such that it preserves any
Markov embeddings [19].
Definition 7. If a statistical immersion \(f:(M,g,\nabla)\to(\widetilde{M},\widetilde{g},\widetilde{\nabla})\) is a diffeomorphism, then \(f\) is called a statistical
diffeomorphism.
Given a statistical immersion \(\iota:(M,g,\nabla)\to(\widetilde{M},\widetilde{g},\widetilde{\nabla})\), we decompose the vector bundle \(\iota^*T\widetilde{M}\) with respect to \(\widetilde{g}\) by \[\begin{align} \iota^*T\widetilde{M} = \iota_*TM\oplus TM^{\perp}.
\end{align}\] From this decomposition, we define \(B\in\Gamma(TM^{\perp}\otimes T^{(0,2)}M)\), \(A\in\Gamma(T^{(1,1)}M\otimes(TM^{\perp})^*)\), and a connection \(\nabla^{\perp}\) on \(TM^{\perp}\) by the following formulas: \[\begin{align} \widetilde{\nabla}_X&\iota_{*} Y = \iota_*(\nabla_XY) +
B(X,Y),\tag{12}\\ \widetilde{\nabla}_X\xi &= -\iota_*A_{\xi}X + \nabla^{\perp}_X\xi,\quad X,Y\in\Gamma(TM),\,\xi\in\Gamma(TM^{\perp}).\tag{13}
\end{align}\] We call \(B\) the second fundamental form, \(A\) the shape operator, and
\(\nabla^{\perp}\) the normal connection of \(\iota\), all with respect to \(\widetilde{\nabla}\). For each \(\alpha\in\mathbb{R}\), it is easy to see that \((M,g,\nabla^{(\alpha)})\) is a statistical submanifold of \((\widetilde{M},\widetilde{g},\widetilde{\nabla}^{(\alpha)})\) since the equation 9 holds. Thus, we define \(B^{(\alpha)}\), \(A^{(\alpha)}\), and \(\nabla^{\perp(\alpha)}\) for \(\iota\) by 12 and 13 with respect to \(\widetilde{\nabla}^{(\alpha)}\) for each \(\alpha\in\mathbb{R}\).
Remark 7. Let \(\iota:(M,g,\nabla)\to(\widetilde{M},\widetilde{g},\widetilde{\nabla})\) be a statistical immersion. For each \(\alpha\in\mathbb{R}\), the following
equations hold\(:\)\[\begin{align} B^{(0)} &= \frac{B^{(\alpha)}+B^{(-\alpha)}}{2},\\ A^{(0)} &= \frac{A^{(\alpha)}+A^{(-\alpha)}}{2}.
\end{align}\]
The following Propositions 8 and 9 are obtained by simple computations (see [20] for example).
Proposition 8. *Let \(\iota:(M,g,\nabla)\to(\widetilde{M},\widetilde{g},\widetilde{\nabla})\) be a statistical immersion. For each \(\alpha\in\mathbb{R}\), we have \[\begin{align} \widetilde{g}(B^{(\alpha)}(X,Y),\xi) &= g(A^{(-\alpha)}_{\xi}X,Y),\\ X\widetilde{g}(\xi,\eta) = \widetilde{g}(\nabla^{\perp(\alpha)}_X\xi,&\eta) + \widetilde{g}(\xi,\nabla^{\perp(-\alpha)}_X\eta),
\end{align}\] where \(X,Y\in\Gamma(TM)\) and \(\xi,\eta\in\Gamma(TM^{\perp})\).*
Proposition 9. *Let \(\iota:(M,g,\nabla)\to(\widetilde{M},\widetilde{g},\widetilde{\nabla})\) be a statistical immersion, and \(\widetilde{R},R\) be the curvature tensor
fields of \(\widetilde{\nabla},\nabla\), respectively. The following equations hold for \(X,Y,Z,W\in\Gamma(TM)\) and \(\xi,\eta\in\Gamma(TM^{\perp})\)\(:\)\[\begin{align} \widetilde{g}\left(\widetilde{R}(\iota_*X,\iota_*Y)\iota_*Z,\iota_*W\right) &= g\Big(R(X,Y)Z - A_{B(Y,Z)}X + A_{B(X,Z)}Y,W\Big),\\
\widetilde{g}\left(\widetilde{R}(\iota_*X,\iota_*Y)\iota_*Z,\xi\right) &= \widetilde{g}\Big(\left(\nabla_X B\right)(Y,Z) - \left(\nabla_Y B\right)(X,Z),\xi\Big),\\ \widetilde{g}\left(\widetilde{R}(\iota_*X,\iota_*Y)\xi,\iota_*Z\right) &=
g\Big(\left(\nabla_Y A\right)(X,\xi) - \left(\nabla_X A\right)(\xi,Y),Z\Big),\\ \widetilde{g}\left(\widetilde{R}(\iota_*X,\iota_*Y)\xi,\eta\right) &= \widetilde{g}\Big(R^{\nabla^{\perp}}(X,Y)\xi-B(X,A_{\xi}Y)+B(Y,A_{\xi}X),\eta\Big).
\end{align}\] Here, \[\begin{align} (\nabla_X B)(Y,Z) &= \nabla^{\perp}_XB(Y,Z) - B(\nabla_X Y,Z) - B(Y,\nabla_X Z),\\ (\nabla_X A)(\xi,Y) &= \nabla_X A_{\xi}Y - A_{\nabla^{\perp}_X\xi}Y - A_{\xi}\nabla_X Y,
\end{align}\] where \(X,Y\in\Gamma(TM)\) and \(\xi\in\Gamma(TM^{\perp})\), and \(R^{\nabla^{\perp}}\) is the curvature tensor field of \(\nabla^{\perp}\).*
The following important classes of statistical submanifolds were introduced in [2], [21]. For a statistical immersion \(\iota:(M,g,\nabla)\to(\widetilde{M},\widetilde{g},\widetilde{\nabla})\), we define the mean curvature
tensor field\(H^{(\alpha)}\in\Gamma(TM^{\perp})\) with respect to \(\widetilde{\nabla}^{(\alpha)}\) for each \(\alpha\in\mathbb{R}\) by \[\begin{align} H^{(\alpha)} = \frac{1}{m}\sum_{i=1}^mB^{(\alpha)}(e_i,e_i),
\end{align}\] where \(\{e_1,\ldots,e_m\}\) is an orthonormal frame of \((M,g)\).
Definition 8. Let \(\iota:(M,g,\nabla)\to(\widetilde{M},\widetilde{g},\widetilde{\nabla})\) be a statistical submanifold.
The immersed submanifold \(M\) is said to be doubly autoparallel if \(B^{(1)}=B^{(-1)}=0\).
The immersed submanifold \(M\) is said to be doubly totally-umbilical if \(B^{(1)}=H^{(1)}\otimes g\) and \(B^{(-1)}=H^{(-1)}\otimes g\)
hold.
Remark 10 ([2]). The following hold in Definition 8.
If there is an \(\alpha,\beta\in\mathbb{R}\) such that \(\alpha\neq\beta\) and \(B^{(\alpha)}=B^{(\beta)}=0\) hold, then the submanifold \(M\) is doubly autoparallel.
Similarly, if there is an \(\alpha,\beta\in\mathbb{R}\) such that \(\alpha\neq\beta\), \(B^{(\alpha)}=H^{(\alpha)}\otimes g\) and \(B^{(\beta)}=H^{(\beta)}\otimes g\) are satisfied, then the submanifold \(M\) is doubly totally-umbilical.
For each \(\alpha\in\mathbb{R}\), the condition \(B^{(\alpha)}=H^{(\alpha)}\otimes g\) is equivalent to \[\begin{align}
\label{umcon} A^{(-\alpha)}_{\xi}X = \widetilde{g}(H^{(\alpha)},\xi)X,\quad X\in\Gamma(TM),\,\xi\in\Gamma(TM^{\perp}).
\end{align}\tag{14}\]
Remark 11. If the submanifold \(M\) of \((\widetilde{M},\widetilde{g},\widetilde{\nabla})\) satisfies \(B^{(\alpha)}=0\) for some \(\alpha\in\mathbb{R}\), then \(M\) is called a \(\widetilde{\nabla}^{(\alpha)}\)-autoparallel submanifold. It is known that \(M\) is a \(\nabla^{(\mathrm{e})}\)-autoparallel submanifold of \((\Delta^n,g^F,\nabla^{(\mathrm{e})})\) if and only if \(M\) is
an exponential family, and \(M\) is a \(\nabla^{(\mathrm{m})}\)-autoparallel submanifold of \((\Delta^n,g^F,\nabla^{(\mathrm{e})})\) if and only if
\(M\) is a mixture family[22].
Example 6. The embedding \(\iota:\Delta^n\to(\mathbb{R}^+)^{n+1}\) described in Example 4 shows that \(\Delta^n\) is a doubly totally-umbilical submanifold of \(((\mathbb{R}^+)^{n+1},g_0,D)\). See [13] for details.
3.1 New results on doubly totally-umbilical submanifolds↩︎
Even when a statistical immersion exists, properties of the ambient space such as conjugate symmetry and having constant curvature are not necessarily inherited by the submanifold. However, if the submanifold is doubly totally-umbilical, these desirable
properties are inherited. We begin by computing the equations in Proposition 9 for doubly totally-umbilical submanifolds.
Lemma 1. Let \(\iota:(M,g,\nabla)\to(\widetilde{M},\widetilde{g},\widetilde{\nabla})\) be a statistical immersion and \(\widetilde{R}^{(\alpha)},R^{(\alpha)}\) be the
curvature tensor fields of \(\widetilde{\nabla}^{(\alpha)},\nabla^{(\alpha)}\), respectively. If \(M\) is doubly totally-umbilical, then the following equations hold for \(X,Y,Z,W\in\Gamma(TM)\) and \(\xi,\eta\in\Gamma(TM^{\perp})\)\(:\)\[\begin{align}
\widetilde{g}\left(\widetilde{R}^{(\alpha)}(\iota_*X,\iota_*Y)\iota_*Z,\iota_*W\right) =& g\Big(R^{(\alpha)}(X,Y)Z,W\Big)\\ &- \widetilde{g}\left(H^{(\alpha)},H^{(-\alpha)}\right)\Big(g(Y,Z)g(X,W)-g(X,Z)g(Y,W)\Big),\tag{15}\\
\widetilde{g}\left(\widetilde{R}^{(\alpha)}(\iota_*X,\iota_*Y)\iota_*Z,\xi\right) &= \widetilde{g}\Big(g(Y,Z)\nabla^{\perp{(\alpha)}}_X H^{(\alpha)} - g(X,Z)\nabla^{\perp{(\alpha)}}_Y H^{(\alpha)},\xi\Big),\tag{16}\\
\widetilde{g}\left(\widetilde{R}^{(\alpha)}(\iota_*X,\iota_*Y)\xi,\iota_*Z\right) &= -\widetilde{g}\Big(g(Y,Z)\nabla^{\perp{(-\alpha)}}_X H^{(-\alpha)} - g(X,Z)\nabla^{\perp{(-\alpha)}}_Y H^{(-\alpha)},\xi\Big),\tag{17}\\
\widetilde{g}\left(\widetilde{R}^{(\alpha)}(\iota_*X,\iota_*Y)\xi,\eta\right) =& \widetilde{g}\Big(R^{\nabla^{\perp(\alpha)}}(X,Y)\xi,\eta\Big).\tag{18}
\end{align}\]
Proof. We prove the equations using Proposition 9 one by one. For each \(X,Y,Z,W\in\Gamma(TM)\), it holds that \[\begin{align} \widetilde{g}\left(\widetilde{R}^{(\alpha)}(\iota_*X,\iota_*Y)\iota_*Z,\iota_*W\right) =& g\Big(R^{(\alpha)}(X,Y)Z - g(Y,Z)A^{(\alpha)}_{H^{(\alpha)}}X + g(X,Z)A^{(\alpha)}_{H^{(\alpha)}}Y,W\Big)\\ =&
g\Big(R^{\alpha}(X,Y)Z,W\Big)\\ &- \widetilde{g}\left(H^{(\alpha)},H^{(-\alpha)}\right)\Big(g(Y,Z)g(X,W)-g(X,Z)g(Y,W)\Big),
\end{align}\] where we used 14 in the last equality. To prove equations 16 and 17 , it suffices to compute \(\nabla^{(\alpha)} B^{(\alpha)}\)
and \(\nabla^{(\alpha)} A^{(\alpha)}\). For each \(X,Y,Z\in\Gamma(TM)\) and \(\xi\in\Gamma(TM^{\perp})\), we have \[\begin{align}
\left(\nabla^{(\alpha)}_X B^{(\alpha)}\right)(Y,Z) =& \nabla^{\perp{(\alpha)}}_XB^{(\alpha)}(Y,Z) - B^{(\alpha)}\left(\nabla^{(\alpha)}_X Y,Z\right) - B^{(\alpha)}\left(Y,\nabla^{(\alpha)}_Y Z\right),\\ =&Xg(Y,Z)H^{(\alpha)} +
g(Y,Z)\nabla^{\perp{(\alpha)}}H^{(\alpha)}\\ &-g\left(\nabla^{(\alpha)}_X Y,Z\right)H^{(\alpha)} - g\left(Y,\nabla^{(\alpha)}_X Z\right)H^{(\alpha)}\\ =&\left(\nabla^{(\alpha)}_X g\right)(Y,Z)H^{(\alpha)} + g(Y,Z)\nabla^{\perp(\alpha)}_X
H^{(\alpha)},
\end{align}\] and from 14 , we have \[\begin{align} (\nabla^{(\alpha)}_X A^{(\alpha)})(\xi,Y) &= \nabla^{(\alpha)}_X A^{(\alpha)}_{\xi}Y - A^{(\alpha)}_{\nabla^{\perp^{(\alpha)}}_X\xi}Y -
A^{(\alpha)}_{\xi}\nabla^{(\alpha)}_X Y\\ &= \left(\nabla^{(\alpha)}_X\widetilde{g}\left(H^{-(\alpha)},\xi\right)Y\right) - \widetilde{g}\left(H^{(-\alpha)},\nabla^{\perp(\alpha)}_X \xi\right)Y -
\widetilde{g}\left(H^{(-\alpha)},\xi\right)\nabla^{(\alpha)}_X Y\\ &=X\widetilde{g}\left(H^{(-\alpha)},\xi\right)Y - \widetilde{g}\left(H^{(-\alpha)},\nabla^{\perp(\alpha)}_X \xi\right)Y\\ &=\widetilde{g}\left(\nabla^{\perp(\alpha)}_X
H^{(-\alpha)},\xi\right)Y.
\end{align}\] Equation 16 is obtained since \(\nabla g\) is symmetric. Lastly, the equation 18 is obtained by 14 and the symmetry of \(B^{(\alpha)}\). ◻
Proposition 12. Let \(\iota:(M,g,\nabla)\to(\widetilde{M},\widetilde{g},\widetilde{\nabla})\) be a statistical immersion, where \(M\) is a doubly totally-umbilical
submanifold.
If \((\widetilde{M},\widetilde{g},\widetilde{\nabla})\) is conjugate symmetric, then so is \((M,g,\nabla)\).
*If \((\widetilde{M},\widetilde{g},\widetilde{\nabla})\) has constant curvature \(\widetilde{k}\), then \(\widetilde{g}\left(H^{(1)},H^{(-1)}\right)\)
is a constant function and \((M,g,\nabla)\) has constant curvature of \(k=\widetilde{k}+\widetilde{g}\left(H^{(1)},H^{(-1)}\right)\).*
Proof. Claim \((1)\) immediately follows from equation 15 . Suppose that \((\widetilde{M},\widetilde{g},\widetilde{\nabla})\) has constant curvature
\(\widetilde{k}\). We prove that \[\begin{align} \label{parmean} \nabla^{\perp(1)}_X H^{(1)}=0,\quad X\in\Gamma(TM)
\end{align}\tag{19}\] holds. The statistical manifold \((\widetilde{M},\widetilde{g},\widetilde{\nabla})\) is also conjugate symmetric from Proposition 5. For any orthonormal pair \(\{X,Y\}\) on \((M,g)\) and \(\xi\in\Gamma(TM^{\perp})\), from 16 we have
\[\begin{align} \widetilde{g}\left(\widetilde{R}^{(1)}(\iota_*X,\iota_*Y)\iota_*Y,\xi\right) = \widetilde{g}\Big(\nabla^{\perp{(1)}}_X H^{(1)},\xi\Big).
\end{align}\] Since the left-hand side of 16 vanishes if \((\widetilde{M},\widetilde{g},\widetilde{\nabla})\) has constant curvature, we obtain 19 . The equation
\(\nabla^{\perp(-1)}_X H^{(-1)}=0\) can be proved in the same manner. Thus, for any \(X\in\Gamma(TM)\) we have \[\begin{align}
X\widetilde{g}\left(H^{(1)},H^{(-1)}\right) = \widetilde{g}\left(\nabla^{\perp(1)}_X H^{(1)},H^{(-1)}\right) + \widetilde{g}\left(H^{(1)},\nabla^{\perp(-1)}_X H^{(-1)}\right) = 0,
\end{align}\] so we conclude that \(\widetilde{g}\left(H^{(1)},H^{(-1)}\right)\) is a constant function. For any orthonormal pair \(\{X,Y\}\) on \((M,g)\), from 15 we have \[\begin{align} g\Big(R^{(1)}(X,Y)Y,X\Big) &= \widetilde{g}\left(\widetilde{R}^{(1)}(\iota_*X,\iota_*Y)\iota_*Y,\iota_*X\right) +
\widetilde{g}\left(H^{(1)},H^{(-1)}\right)\\ &=\widetilde{k} + \widetilde{g}\left(H^{(1)},H^{(-1)}\right),
\end{align}\] therefore, the statistical manifold \((M,g,\nabla)\) has constant curvature.
◻
Remark 13. Let \(\iota:(M,g,\nabla)\to(\widetilde{M},\widetilde{g},\widetilde{\nabla})\) be a statistical immersion, where \(M\) is a doubly totally-umbilical submanifold
and \((\widetilde{M},\widetilde{g},\widetilde{\nabla})\) has constant curvature. In [23], it is stated that if the function
\(\widetilde{g}\left(H^{(1)},H^{(-1)}\right)\) is constant, then \((M,g,\nabla)\) also has constant curvature, however, by Proposition 12 we see that this function is always constant under these assumptions.
Example 7. Fix \(\omega\in\{1,\ldots,n+1\}\) and \(b\in(0,1)\). Then \[\begin{align} M = \{p\in\Delta^n\mid p(\omega)=b\}
\end{align}\] is a doubly totally-umbilical submanifold of the probability simplex \((\Delta^n,g^F,\nabla^{(\mathrm{e})})\). Indeed, we have \(B^{(0)} = H^{(0)}\otimes g\), since the
image of \(\Delta^n\ni p\to(2\sqrt{p(1)},\ldots,2\sqrt{p(n+1)})\in S^{n}(2)\) restricted to \(M\) is an open Riemannian submanifold of \(S^{n-1}(2\sqrt{1-b})\subset
S^n(2)\), which is a well-known totally-umbilical submanifold in Riemannian geometry (see [7] for example). In order to prove that \(M\) is a doubly totally-umbilical submanifold, we prove \(B^{(-1)}=0\). Consider the global affine coordinate system \((\eta^1,\ldots,\eta^n)\) of \(\nabla^{(\mathrm{m})}\) defined in Example 1. Since \(M\) is a hyperplane of \(\Delta^n\) with respect to the coordinate system \((\eta^1,\ldots,\eta^n)\), it is clear that \(M\) is a \(\nabla^{(\mathrm{m})}\)-autoparallel submanifold, thus we have \(B^{(-1)}=0\).
Proposition 14. Let \(\iota:(M,g,\nabla)\to(\widetilde{M},\widetilde{g},\widetilde{\nabla})\) be a statistical immersion, where \(M\) is a doubly totally-umbilical
submanifold. If the following inequality holds for each orthonormal pair of tangent vectors \(\{X,Y\}\) on \((\widetilde{M},\widetilde{g})\), then the difference tensor field \(K\) of \((M,g,\nabla)\) does not vanish\(\mathrm{:}\)\[\label{ineq1}
\widetilde{g}\left(\widetilde{K}_X\widetilde{K}_YY-\widetilde{K}_Y\widetilde{K}_XY,X\right)<0.\qquad{(1)}\] Here, \(\widetilde{K}\) is the difference tensor of \((\widetilde{g},\widetilde{\nabla})\).
Proof. Assume that \(K=0\) holds. We have \[\begin{align} \widetilde{K}_{\iota_*X}{\iota_*Y} = g(X,Y)\left(H^{(1)}-H^{(0)}\right)
\end{align}\] for each \(X,Y\in\Gamma(TM)\). Thus, for each orthonormal pair \(\{X,Y\}\) on \((M,g)\), we have \[\begin{align}
\widetilde{g}\left(\widetilde{K}_{\iota_*X}\widetilde{K}_{\iota_*Y}\iota_*Y-\widetilde{K}_{\iota_*Y}\widetilde{K}_{\iota_*X}\iota_*Y,\iota_*X\right)&=\widetilde{g}\left(\widetilde{K}_{\iota_*X}\iota_*X,\widetilde{K}_{\iota_*Y}\iota_*Y\right)\\
&=\|H^{(1)}-H^{(0)}\|_{\widetilde{g}}^2\geq 0,
\end{align}\] which contradicts the inequality ?? .
◻
Let \(\iota:(M^m,g,\nabla)\to(\widetilde{M}^{m+1},\widetilde{g},\widetilde{\nabla})\) be a statistical immersion. If \(n=m+1\), the statistical submanifold \(M\) is called a statistical hypersurface of \(\widetilde{M}\). We also assume that \(M\) is orientable in this subsection.
Since \(M\) is orientable, let \(\boldsymbol{n}\) be a unit normal vector field along \(\iota\). For each \(\alpha\in\mathbb{R}\), there exists a symmetric \(h^{(\alpha)}\in\Gamma(T^{(0,2)}M)\) such that \(B^{(\alpha)} = \boldsymbol{n}\otimes h^{(\alpha)}\). There also
exists an 1-form \(\tau^{(\alpha)}\in\Gamma(T^*M)\) for each \(\alpha\in\mathbb{R}\) such that \[\begin{align} \nabla^{\perp{(\alpha)}}_X \boldsymbol{n} =
\tau^{(\alpha)}(X)\boldsymbol{n},\quad X\in\Gamma(TM).
\end{align}\] For each \(\alpha\in\mathbb{R}\), the relations \[\begin{align} h^{(0)} =& \frac{h^{(\alpha)}+h^{(-\alpha)}}{2},\\ \tau^{(\alpha)} &= -\tau^{(-\alpha)}
\end{align}\] hold, since \(\tau^{(0)} = 0\).
Proposition 15. Let \(\iota:(M^m,g,\nabla)\to(\widetilde{M}^{m+1},\widetilde{g},\widetilde{\nabla})\) be statistical hypersurface immersion, where \(M\) is a doubly
totally-umbilical submanifold. For each \(\alpha\in\mathbb{R}\), we have \[\begin{align} h^{(\alpha)} &= \frac{1}{m}(\mathrm{tr}_g h^{(\alpha)})g,\\ H^{(\alpha)} &=
\frac{1}{m}\mathrm{tr}_g h^{(\alpha)}\boldsymbol{n},\\ A^{(\alpha)}_{\boldsymbol{n}}X &= \frac{1}{m}\mathrm{tr}_g h^{(-\alpha)}X,\quad X\in\Gamma(TM).
\end{align}\]
Proof. The proof is obtained by a straightforward computation using 14 .
◻
Proposition 16. Let \(\iota:(M^m,g,\nabla)\to(\widetilde{M}^{m+1},\widetilde{g},\widetilde{\nabla})\) be statistical hypersurface immersion, where \(M\) is a doubly
totally-umbilical submanifold. If \((\widetilde{M},\widetilde{g},\widetilde{\nabla})\) is conjugate symmetric, then for each \(\alpha\in\mathbb{R}\), we have \[\begin{align} \label{umhyp1} d(\mathrm{tr}_g h^{(\alpha)} - \mathrm{tr}_g h^{(-\alpha)}) = -\left(\mathrm{tr}_g h^{(\alpha)}\tau^{(\alpha)}-\mathrm{tr}_g h^{(-\alpha)}\tau^{(-\alpha)}\right).
\end{align}\qquad{(2)}\] Moreover, if \((\widetilde{M},\widetilde{g},\widetilde{\nabla})\) has constant curvature, then \[\begin{align} \label{umhyp2}
d(\mathrm{tr}_gh^{(1)}) = -\mathrm{tr}_gh^{(1)}\tau^{(1)}.
\end{align}\qquad{(3)}\]
Proof. If \((\widetilde{M},\widetilde{g},\widetilde{\nabla})\) is conjugate symmetric, then \(\widetilde{R}^{(-\alpha)}=\widetilde{R}^{(\alpha)}\). Hence, by 16 and 17 , we obtain \[\begin{align} \label{dermean} \nabla^{\perp(\alpha)}_X H^{(\alpha)} = \nabla^{\perp(-\alpha)}_X H^{(-\alpha)}
\end{align}\tag{20}\] for each \(X\in\Gamma(TM)\). Since \[\begin{align} \nabla^{\perp(\alpha)}_X H^{(\alpha)} = \frac{1}{m}X\mathrm{tr}_g h^{(\alpha)}\boldsymbol{n} +
\frac{1}{m}\mathrm{tr}_g h^{(\alpha)}\tau^{(\alpha)}(X)\boldsymbol{n}
\end{align}\] for each \(X\in\Gamma(TM)\), comparing the coefficients of \(\boldsymbol{n}\) yields ?? . If \((\widetilde{M},\widetilde{g},\widetilde{\nabla})\) has constant curvature, then 19 holds. Thus, ?? follows from the same computation with \(\alpha=1\).
◻
Proposition 17. *Let \(\iota:(M^m,g,\nabla)\to(\widetilde{M}^{m+1},\widetilde{g},\widetilde{\nabla})\) be statistical hypersurface immersion, where \(M\) is a doubly
totally-umbilical submanifold. If \((\widetilde{M},\widetilde{g},\widetilde{\nabla}^{(\alpha)})\) has constant curvature for each \(\alpha\in\mathbb{R}\), then \(\mathrm{tr}_gh^{(\alpha)}\) is a constant function for each \(\alpha\in\mathbb{R}\).*
Proof. By Proposition 12\((2)\), the function \(\widetilde{g}(H^{(0)},H^{(0)})\) is constant,
hence \(\mathrm{tr}_gh^{(0)}\) is also constant. Since \(\widetilde{g}(H^{(1)},H^{(-1)})\) is also constant, we have \[\begin{align} 0 =
m^2X\widetilde{g}(H^{(1)},H^{(-1)}) &= X\left(\mathrm{tr}_gh^{(1)}\cdot\mathrm{tr}_gh^{(-1)}\right)\\ &= X\left(\mathrm{tr}_gh^{(1)}(2\mathrm{tr}_gh^{(0)}-\mathrm{tr}_gh^{(1)})\right)\\
&=2\left(\mathrm{tr}_gh^{(0)}-\mathrm{tr}_gh^{(1)}\right)X\mathrm{tr}_gh^{(1)}.
\end{align}\] Since the manifiold \(M\) is connected, the function \(\mathrm{tr}_gh^{(1)}\) is constant, and consequently so is \(\mathrm{tr}_gh^{(-1)}\). The function \(\mathrm{tr}_gh^{(\alpha)}\) is also constant for each \(\alpha\in\mathbb{R}\), since \[\begin{align} \mathrm{tr}_gh^{(\alpha)} = \frac{1+\alpha}{2}\mathrm{tr}_gh^{(1)} + \frac{1-\alpha}{2}\mathrm{tr}_gh^{(-1)}
\end{align}\] holds.
◻
The following corollary is obtained immediately from equation ?? and Proposition 17.
Corollary 18. *Let \(\iota:(M^m,g,\nabla)\to(\widetilde{M}^{m+1},\widetilde{g},\widetilde{\nabla})\) be a statistical hypersurface immersion, where \(M\) is a doubly
totally-umbilical submanifold. If \((\widetilde{M},\widetilde{g},\widetilde{\nabla}^{(\alpha)})\) has constant curvature for each \(\alpha\in\mathbb{R}\) and \(M\) is not doubly autoparallel, then \(\tau^{(\alpha)}=0\) holds for each \(\alpha\in\mathbb{R}\).*
Laslty, we obtain the following theorem.
Theorem 19. Let \(\iota:(M^m,g,\nabla)\to(\widetilde{M}^{m+1},\widetilde{g},\widetilde{\nabla})\) be a statistical hypersurface immersion, where \(M\) is a doubly
totally-umbilical submanifold. If \((\widetilde{M},\widetilde{g},\widetilde{\nabla}^{(\alpha)})\) has constant curvature for each \(\alpha\in\mathbb{R}\) and \(\widetilde{k}^{(1)}\neq\widetilde{k}^{(0)}\), then there exists a \(\beta\in\mathbb{R}\) such that \(M\) is a \(\nabla^{(\beta)}\)-autoparallel submanifold of \(\widetilde{M}\).
Proof. If \(M\) is doubly autoparallel, the statement is clear. Suppose that \(M\) is not doubly autoparallel. We first show that \(\mathrm{tr}_gh^{(1)}\neq\mathrm{tr}_gh^{(0)}\) holds, where they are constant functions by Proposition 17. If we assume that \(\mathrm{tr}_gh^{(1)}=\mathrm{tr}_gh^{(0)}\) holds, then by Corollary 18, for any \(X\in\Gamma(TM)\) we have \[\begin{align} \widetilde{K}_{\iota_*X} \boldsymbol{n} = 0.
\end{align}\] Here, \[\begin{align} \widetilde{K}_{\boldsymbol{n}}\boldsymbol{n} &= \sum_{i=1}^m\widetilde{g}\left(\widetilde{K}_{\boldsymbol{n}}\boldsymbol{n},\iota_*e_i\right)\iota_*e_i +
\widetilde{g}\left(\widetilde{K}_{\boldsymbol{n}}\boldsymbol{n},\boldsymbol{n}\right)\boldsymbol{n}\\ &= \widetilde{g}\left(\widetilde{K}_{\boldsymbol{n}}\boldsymbol{n},\boldsymbol{n}\right)\boldsymbol{n}
\end{align}\] where \(\{e_1,\ldots,e_m\}\) is an orthonormal frame of \((M,g)\). Consequently, \[\begin{align} 0 &=
\widetilde{g}\left(\left[\widetilde{K}_{\iota_*X},\widetilde{K}_{\boldsymbol{n}}\right]\boldsymbol{n},\iota_*X\right)\\ &= \widetilde{g}\left(\widetilde{R}^{(1)}(\iota_*X,\boldsymbol{n})\boldsymbol{n} -
\widetilde{R}^{(0)}(\iota_*X,\boldsymbol{n})\boldsymbol{n},\iota_*X\right)\\ &=\widetilde{k}^{(1)}-\widetilde{k}^{(0)},
\end{align}\] where \(X\) is a unit vector on \((M,g)\). This contradicts \(\widetilde{k}^{(1)}\neq\widetilde{k}^{(0)}\). Therefore, since \(\mathrm{tr}_gh^{(1)}\neq\mathrm{tr}_gh^{(0)}\) holds, we define \(\beta\in\mathbb{R}\) by \[\begin{align} \beta =
\frac{\mathrm{tr}_gh^{(0)}}{\mathrm{tr}_gh^{(0)}-\mathrm{tr}_gh^{(1)}}.
\end{align}\] Then we obtain \[\begin{align} \mathrm{tr}_g h^{(\beta)} &= \frac{1+\beta}{2}\mathrm{tr}_g h^{(1)} + \frac{1-\beta}{2}\mathrm{tr}_g h^{(-1)}\\ &= \frac{1+\beta}{2}\mathrm{tr}_g h^{(1)} +
\frac{1-\beta}{2} \left(2\mathrm{tr}_gh^{(0)}-\mathrm{tr}_gh^{(1)}\right)\\ &= \beta \mathrm{tr}_g h^{(1)} + (1-\beta)\mathrm{tr}_gh^{(0)} = 0,
\end{align}\] which implies \(h^{(\beta)}=0\) by Proposition 15, and hence we have \(B^{(\beta)}=0\).
Therefore, \(M\) is a \(\nabla^{(\beta)}\)-autoparallel submanifold.
◻
3.3 Doubly totally-umbilical submanifolds of Hessian manifolds↩︎
Since Hessian manifolds are equipped with the Hessian curvature, it is worthwhile to prepare formulas for this curvature on doubly totally-umbilical submanifolds. Hereafter, for a statistical immersion \(\iota:(M,g,\nabla)\to(\widetilde{M},\widetilde{g},\widetilde{\nabla})\) we write \(B^{*}=B^{(-1)}\), \(\widehat{B}=B^{(0)}\), \(H^{*}=H^{(-1)}\), \(\widehat{H}=H^{(0)}\), \(A^{*}=A^{(-1)}\), \(\widehat{A}=A^{(0)}\), \(\nabla^{\perp*}=\nabla^{\perp(-1)}\), and \(\widehat{\nabla}^{\perp}=\nabla^{\perp(0)}\).
Proposition 20. *Let \(\iota:(M,g,\nabla)\to(\widetilde{M},\widetilde{g},\widetilde{\nabla})\) be a statistical immersion, where \(M\) is a doubly totally-umbilical
submanifold. The following formulas hold. \[\begin{align} \widetilde{g}\left(\left(\widetilde{\nabla}_{\iota_*X}\widetilde{K}\right)(\iota_*Y,\iota_*Z),\iota_*W\right) =& g\left(\left(\nabla_X
K\right)(Y,Z),W\right) + g(X,W)g(Y,Z)\widetilde{g}\left(H^{*},\widehat{H}-H\right)\\ + g(X,Y)&g(Z,W)\widetilde{g}\left(H,H^{*}-\widehat{H}\right) + g(X,Z)g(Y,W)\widetilde{g}\left(H,H^{*}-\widehat{H}\right),\tag{21}\\
\widetilde{g}\left(\left(\widetilde{\nabla}_{\iota_*X}\widetilde{K}\right)(\iota_*Y,\iota_*Z),\xi\right) &= g(K_X Y,Z)\widetilde{g}(H^{*},\xi) + g(Y,Z)\widetilde{g}\left(\nabla^{\perp}_X H-\nabla^{\perp}_X\widehat{H},\xi\right)\\
+g(X,Y)\widetilde{g}&\left(\widehat{\nabla}^{\perp}_Z H-\nabla^{\perp}_Z H,\xi\right) + g(X,Z)\widetilde{g}\left(\widehat{\nabla}^{\perp}_Y H-\nabla^{\perp}_Y H,\xi\right),\tag{22}\\
\widetilde{g}\left(\left(\widetilde{\nabla}_{\iota_*X}\widetilde{K}\right)(\iota_*Y,\xi),\iota_*Z\right) &= g(K_X Y,Z)\widetilde{g}(H^{*},\xi) + g(Y,Z)\widetilde{g}\left(\nabla^{\perp*}_X \widehat{H}-\nabla^{\perp*}_XH^{*},\xi\right)\\
+g(X,Z)\widetilde{g}&\left(\nabla^{\perp*}_Y H^{*}-\widehat{\nabla}^{\perp}_YH^{*},\xi\right) + g(X,Y)\widetilde{g}\left(\widehat{\nabla}^{\perp}_Z H-\nabla^{\perp}_Z H,\xi\right),\tag{23}\\
\widetilde{g}\left(\left(\widetilde{\nabla}_{\iota_*X}\widetilde{K}\right)(\iota_*Y,\xi),\eta\right) &= \widetilde{g}\left(\left(\nabla_X K^{\perp}\right)(Y,\xi),\eta\right) + g(X,Y)\widetilde{g}\left(\widetilde{K}_{\xi}\eta,H\right)\\ +
g(X&,Y)\left(\widetilde{g}\left(\widehat{H},\xi\right)\widetilde{g}\Big(H^{*},\eta\Big) - \widetilde{g}\left(\widehat{H},\eta\right)\widetilde{g}\Big(H^{*},\xi\Big)\right)\tag{24},
\end{align}\] where \(X,Y,Z,W\in\Gamma(TM)\), \(\xi,\eta\in\Gamma(TM^{\perp})\), \(K^{\perp} = \nabla^{\perp} - \widehat{\nabla}^{\perp}\), and \[\begin{align} \left(\nabla_X K^{\perp}\right)(Y,\xi) = \nabla^{\perp}_X K^{\perp}_Y\xi - K^{\perp}_{\nabla_X Y}\xi - K^{\perp}_Y \nabla^{\perp}_X \xi,
\end{align}\] for \(X,Y\in\Gamma(TM)\) and \(\xi\in\Gamma(TM^{\perp})\).*
Proof. We prove the equations one by one. For \(X,Y,Z,W\in\Gamma(TM)\), we have \[\begin{align}
\widetilde{g}\left(\left(\widetilde{\nabla}_{\iota_*X}\widetilde{K}\right)(\iota_*Y,\iota_*Z),\iota_*W\right) =& \widetilde{g}\left(\widetilde{\nabla}_X \widetilde{K}_{\iota_*Y}\iota_*Z - \widetilde{K}_{\widetilde{\nabla}_X \iota_* Y}\iota_*Z -
\widetilde{K}_{\iota_*Y} \widetilde{\nabla}_X\iota_*Z, \iota_*W \right)\\ =&\widetilde{g}\left(\widetilde{\nabla}_X\left(\iota_* K_Y Z + B(Y,Z) - \widehat{B}(Y,Z)\right),\iota_*W\right)\\ -\widetilde{g}&\left(\widetilde{K}_{\iota_* Z}\left(\iota_*
\nabla_X Y + B(X,Y)\right),\iota_*W\right)\\ -\widetilde{g}&\left(\widetilde{K}_{\iota_*Y}\left(\iota_*\nabla_X Z + B(X,Z)\right),\iota_*W\right)\\ =&\widetilde{g}\left(\iota_*\nabla_X K_Y Z - \iota_*A_{B(Y,Z)}X +
\iota_*A_{\widehat{B}(Y,Z)}X,\iota_*W\right)\\ &-\widetilde{g}\left(\iota_*K_{Z}\nabla_X Y-\iota_*A_{B(X,Y)}Z+\iota_*\widehat{A}_{B(X,Y)}Z,\iota_*W\right)\\ &-\widetilde{g}\left(\iota_*K_{Y}\nabla_X
Z-\iota_*A_{B(X,Z)}Y+\iota_*\widehat{A}_{B(X,Z)}Y,\iota_*W\right)\\ =&g((\nabla_{X}K),(Y,Z)) + g(X,W)g(Y,Z)\widetilde{g}\left(H^{*},\widehat{H}-H\right)\\ &+ g(X,Y)g(Z,W)\widetilde{g}\left(H,H^{*}-\widehat{H}\right)\\ &+
g(X,Z)g(Y,W)\widetilde{g}\left(H,H^{*}-\widehat{H}\right),
\end{align}\] where we used 14 to obtain the last row.
For \(X,Y,Z\in\Gamma(TM)\) and \(\xi\in\Gamma(TM^{\perp})\), we have \[\begin{align}
\widetilde{g}\left(\left(\widetilde{\nabla}_{\iota_*X}\widetilde{K}\right)(\iota_*Y,\iota_*Z),\xi\right) =& \widetilde{g}\left(\widetilde{\nabla}_X\left(\iota_* K_Y Z + B(Y,Z) - \widehat{B}(Y,Z)\right),\xi\right)\\
-\widetilde{g}&\left(\widetilde{K}_{\iota_* Z}\left(\iota_* \nabla_X Y + B(X,Y)\right) + \widetilde{K}_{\iota_*Y}\left(\iota_*\nabla_X Z + B(X,Z)\right),\xi\right)\\ =& \widetilde{g}\left(B(X,K_Y Z) + \nabla^{\perp}_XB(Y,Z) -
\nabla^{\perp}_X\widehat{B}(Y,Z),\xi\right)\\ &-\widetilde{g}\left(B(Z,\nabla_X Y) - \widehat{B}(Z,\nabla_X Y)+\nabla^{\perp}_Z B(X,Y) - \widehat{\nabla}^{\perp}_Z B(X,Y),\xi\right)\\ &-\widetilde{g}\left(B(Y,\nabla_X Z) - \widehat{B}(Y,\nabla_X
Z)+\nabla^{\perp}_Y B(X,Z) - \widehat{\nabla}^{\perp}_Y B(X,Z),\xi\right)\\ =& g(X,K_Y Z)\widetilde{g}\left(H,\xi\right)\\ &-(Xg(Y,Z)-g(\nabla_X Y,Z)-g(Y,\nabla_X Z))\widetilde{g}\left(\widehat{H}-H,\xi\right)\\
&+g(Y,Z)\widetilde{g}\left(\nabla^{\perp}_X H - \nabla^{\perp}_X \widehat{H},\xi\right) - g(X,Y)\widetilde{g}\left(\nabla_Z^{\perp}H - \widehat{\nabla}_Z^{\perp}H,\xi\right)\\ &-g(X,Z)\widetilde{g}\left(\nabla_Y^{\perp}H -
\widehat{\nabla}_Y^{\perp}H,\xi\right)\\ =& g(K_X Y,Z)\widetilde{g}\left(H,\xi\right)\\ &+ 2g(K_X Y,Z)\widetilde{g}\left(\widehat{H}-H,\xi\right) + g(Y,Z)\widetilde{g}\left(\nabla^{\perp}_X H - \nabla^{\perp}_X \widehat{H},\xi\right)\\
&+g(X,Y)\widetilde{g}\left(\widehat{\nabla}_Z^{\perp}H - \nabla_Z^{\perp}H,\xi\right) + g(X,Z)\widetilde{g}\left(\widehat{\nabla}_Y^{\perp}H - \nabla_Y^{\perp}H,\xi\right),
\end{align}\] and since we have \(H^{*} = 2\widehat{H} - H\), we obtain 22 . For \(X,Y,Z\in\Gamma(TM)\) and \(\xi\in\Gamma(TM^{\perp})\), we have \[\begin{align} \widetilde{g}\left(\left(\widetilde{\nabla}_{\iota_*X}\widetilde{K}\right)(\iota_*Y,\xi),\iota_*Z\right) =&
\widetilde{g}\left(\widetilde{\nabla}_X \widetilde{K}_{\iota_*Y}\xi - \widetilde{K}_{\widetilde{\nabla}_X \iota_* Y}\xi - \widetilde{K}_{\iota_*Y} \widetilde{\nabla}_X\xi, \iota_*Z \right)\\
=&\widetilde{g}\left(\widetilde{\nabla}_X\left(-\iota_*A_{\xi}Y+\iota_*\widehat{A}_{\xi}Y + \nabla^{\perp}_Y\xi-\widehat{\nabla}^{\perp}_Y\xi\right),\iota_* Z\right)\\ -\widetilde{g}&\left(\widetilde{K}_{\xi}\left(\iota_*\nabla_X
Y+B(X,Y)\right)+\widetilde{K}_{\iota_*Y}\left(-\iota_*A_{\xi}X+\nabla^{\perp}_X\xi\right),\iota_*Z\right)\\ =&\widetilde{g}\left(\widetilde{\nabla}_X\left(-\widetilde{g}\left(H^{*}-\widehat{H},\xi\right)\iota_*Y\right) -
\iota_*A_{\nabla^{\perp}_Y\xi}X+\iota_*A_{\widehat{\nabla}^{\perp}_Y\xi}X,\iota_*Z\right)\\ +\widetilde{g}&\left(\iota_*A_\xi\nabla_X Y - \iota_*\widehat{A}_\xi\nabla_X Y - \widetilde{K}_{\xi}B(X,Y),\iota_*Z\right)\\ +
\widetilde{g}&\left(\widetilde{g}\left(H^{*},\xi\right)\widetilde{K}_{\iota_*Y}\iota_*X + \iota_*A_{\nabla^{\perp}_X\xi}Y - \iota_*\widehat{A}_{\nabla^{\perp}_X\xi}Y,\iota_*Z\right)\\
=\widetilde{g}&\left(-X\widetilde{g}\left(H^{*}-\widehat{H},\xi\right)\iota_*Y - \widetilde{g}\left(H^{*},\nabla^{\perp}_Y\xi-\widehat{\nabla}^{\perp}_Y\xi\right)\iota_*X,\iota_*Z\right)\\
&-\widetilde{g}\left(\widetilde{K}_{\xi}B(X,Y),\iota_*Z\right)\\ &+\widetilde{g}\left(\widetilde{g}\left(H^{*},\xi\right)\iota_*K_X Y + \widetilde{g}\left(H^{*}-\widehat{H},\nabla^{\perp}_X\xi\right)\iota_*Y,\iota_*Z\right).
\end{align}\] Here, it holds that \[\begin{align} \widetilde{g}\left(H^{*},\nabla^{\perp}_Y\xi-\widehat{\nabla}^{\perp}_Y\xi\right) &= Y\widetilde{g}\left(H^{*},\xi\right) -
\widetilde{g}\left(\nabla^{\perp*}_YH^{*},\xi\right)\\ &-Y\widetilde{g}\left(H^{*},\xi\right) + \widetilde{g}\left(\nabla^{\perp*}_YH^{*},\xi\right)\\ &= \widetilde{g}\left(\widehat{\nabla}^{\perp}_YH^{*}-\nabla^{\perp*}_YH^{*},\xi\right),
\end{align}\] and \[\begin{align} \widetilde{g}\left(\widetilde{K}_{\xi}B(X,Y),\iota_*Z\right) =& g(X,Y)\widetilde{g}\left(\widetilde{K}_{\iota_*Z}H,\xi\right)\\ =& g(X,Y)\widetilde{g}\left(\nabla^{\perp}_Z H -
\widehat{\nabla}^{\perp}_Z H,\xi\right),
\end{align}\] thus we obtain 23 . Lastly, to prove 24 , for \(X,Y\in\Gamma(TM)\) and \(\xi,\eta\in\Gamma(TM^{\perp})\) we have
\[\begin{align} \widetilde{g}\left(\left(\widetilde{\nabla}_{\iota_*X}\widetilde{K}\right)(\iota_*Y,\xi),\eta\right) =& \widetilde{g}\left(\widetilde{\nabla}_X\left(\widetilde{g}\left(\widehat{H}-H^{*},\xi\right)Y +
K^{\perp}_Y\xi\right),\eta\right)\\ -\widetilde{g}&\left(\widetilde{K}_{\xi}\left(\iota_*\nabla_X Y+B(X,Y)\right)+\widetilde{K}_{\iota_*Y}\left(-\widetilde{g}\left(H^{*},\xi\right)X+\nabla^{\perp}_X\xi\right),\eta\right)\\ =&
\widetilde{g}\left(\widehat{H}-H^{*},\xi\right)\widetilde{g}\left(B(X,Y),\eta\right) + \widetilde{g}\left(\nabla_X^{\perp}K^{\perp}_Y\xi,\eta\right)\\ &-\widetilde{g}\left(\nabla^{\perp}_{\nabla_X Y}\xi-\widehat{\nabla}^{\perp}_{\nabla_X
Y}\xi,\eta\right) + g(X,Y)\widetilde{g}\left(\widetilde{K}_{\xi}\eta,H\right)\\ &+g(X,Y)\widetilde{g}\left(H^{*},\xi\right)\widetilde{g}\left(H-\widehat{H},\eta\right) - \widetilde{g}\left(K^{\perp}_Y\nabla^{\perp}_X\xi,\eta\right)\\ =&
g(X,Y)\left(\widetilde{g}\left(\widehat{H},\xi\right)\widetilde{g}\Big(H,\eta\Big) - \widetilde{g}\Big(H^{*},\xi\Big)\widetilde{g}\left(\widehat{H},\eta\right)\right) \\ &+ \widetilde{g}\left(\left(\nabla_{X}K^{\perp}\right)(Y,\xi),\eta\right) +
g(X,Y)\widetilde{g}\left(\widetilde{K}_{\xi}\eta,H\right).
\end{align}\] ◻
Corollary 21. If \(\iota:(M,g,\nabla)\to(\widetilde{M},\widetilde{g},\widetilde{\nabla})\) is a statistical immersion, where \(M\) is a doubly totally-umbilical
submanifold and \((\widetilde{M},\widetilde{g},\widetilde{\nabla})\) is a Hessian manifold of CHC \(\widetilde{c}\), then the following conditions are equivalent\(:\)
The statistical manifold \((M,g,\nabla)\) is a Hessian manifold of CHC \(c=\widetilde{c} - 4\widetilde{g}(\widehat{H},\widehat{H})\).
The submanifold \(M\) is \(\nabla^{*}\)-autoparallel.
Proof. Assume that \((M,g,\nabla)\) is a Hessian manifold of CHC \(c=\widetilde{c} - 4\widetilde{g}(\widehat{H},\widehat{H})\). Since equation 8 holds for
\(\nabla K\) and \(\widetilde{\nabla}\widetilde{K}\), for an orthonormal pair \(\{X,Y\}\) on \((M,g)\) we have
\[\begin{align} \label{ceq1} 0 = g((\nabla_X K)(Y,Y),X) = \widetilde{g}\left(H^{*},H-\widehat{H}\right),
\end{align}\tag{25}\] by taking \(\xi=H-\widehat{H}\) in 21 . Consequently, \[\begin{align} \label{sizestar}
\|H^*\|_{\widetilde{g}}^2 = \widetilde{g}\left(H^*,H\right) - 2\widetilde{g}\left(H^*,H-\widehat{H}\right)
\end{align}\tag{26}\] is a constant function since \(\widetilde{g}\left(H^*,H\right)\) is also constant. Thus, for each orthonormal pair \(\{X,Y\}\) on \((M,g)\), from equation 23 we have \[\begin{align} 0 = g(K_Y Y,X)\widetilde{g}(H^*,H^*) - \widetilde{g}\left(\widehat{\nabla}_X^{\perp}H^*,H^*\right) = \widetilde{g}(K_Y
Y,X)\|H^*\|_{\widetilde{g}}^2
\end{align}\] by taking \(\xi = H^*\), which implies that \(H^*=0\) or \(K=0\). If \(H^*=0\), then \(M\) is a \(\nabla^{*}\)-autoparallel submanifold. If \(K=0\), then \((M,g,\nabla)\) is a Hessian manifold of CHC \(0\), which implies \(\widetilde{c}=4\widetilde{g}(\widehat{H},\widehat{H})\). Here, equation 21 yields \[\begin{align}
-4\widetilde{g}(\widehat{H},\widehat{H}) &= \widetilde{g}\left(\left(\widetilde{\nabla}_{\iota_*X}\widetilde{K}\right)\left(\iota_*X,\iota_*X\right),\iota_*X\right)\\ &= 2\widetilde{g}\left(H,H^*-\widehat{H}\right)
\end{align}\] for any unit vectors \(X\) on \((M,g)\). We also have \(\widetilde{g}(H^*,H)=\widetilde{g}(H^*,\widehat{H})\) from 25 , hence \[\begin{align} 0 &= 2\widetilde{g}(\widehat{H},\widehat{H}) + \widetilde{g}\left(H,H^*-\widehat{H}\right)\\ &= \widetilde{g}(H,H^*) + \widetilde{g}(H^*,\widehat{H})\\ &=
2\widetilde{g}(H,H^*),
\end{align}\] which implies \(H^*=0\) from 26 , and again, \(M\) is \(\nabla^{*}\)-autoparallel. The converse is follows
immediately from 21 . ◻
Example 8. Consider the doubly totally-umbilical submanifold \(M = \{p\in\Delta^n\mid p(\omega)=b\}\) of the probability simplex \((\Delta^n,g^F,\nabla^{(\mathrm{e})})\)
in Example 7. The submanifold \(M\) is also a \(\nabla^{(\mathrm{m})}\)-autoparallel submanifold. Since the
probability simplex \((\Delta^n,g^F,\nabla^{(\mathrm{e})})\) has CHC \(-1\), Corollary 21 implies that \((M,g,\nabla)\) also a Hessian manifold of CHC. The Hessian curvature of \((M,g,\nabla)\) is \(-(1-b)^{-1}\).
4 Complete classification of doubly totally-umbilical submanifolds in the probability
simplex↩︎
We give a complete classification of doubly totally-umbilical submanifolds in the probability simplex. From here on, we only consider submanifolds \(M\subset\widetilde{M}\) of the ambient space and not immersed
ones.
Lemma 2. Let \(M\) be a submanifold of \(\Delta^n\). Then, the submanifold \(M\) is a doubly totally-umbilical one of \((\Delta^n,g^F,\nabla^{\mathrm{(e)}})\) if and only if \(\iota(M)\) is a doubly totally-umbilical submanifold of \(((\mathbb{R}^+)^{n+1},g_0,D)\) where \(\iota:\Delta^n\to(\mathbb{R}^+)^{n+1}\) is the embedding map in Example \(\mathrm{\ref{probsimpemb}}\).
Proof. The probability simplex \((\Delta^n,g^F,\nabla^{(\mathrm{e})})\) is a doubly totally-umbilical statistical submanifold of \(((\mathbb{R}^+)^{n+1},g_0,D)\). If we let \(\iota_0:M\to\Delta^n\) the inclusion map, \((g,\nabla)\) the induced statistical structure on \(M\), \(B\) the second
fundamental form of \(\iota_0\) with respect to \(\nabla^{(\mathrm{e})}\), and \(\widetilde{H}\) the mean curvature of \(\iota\) with respect to \(D\), we have \[\begin{align} D_X(\iota\circ\iota_0)_*Y &= \iota_*(\nabla^{(\mathrm{e})}_X\iota_{0*}Y) +
g^F(\iota_{0*}X,\iota_{0*}Y)\widetilde{H}\\ &= (\iota\circ\iota_0)_*(\nabla_X Y) + \iota_*B(X,Y) + g(X,Y)\widetilde{H},
\end{align}\] thus the equivalence holds.
◻
From Lemma 2, it is sufficient to classify the doubly totally-umbilical submanifolds in the denormalized state space first.
4.1 Doubly totally-umbilical submanifolds in the denormalized state space↩︎
We first classify doubly autoparallel submanifolds in the denormalized state space.
Proposition 22. A submanifold \(M^m\) of the denormalized state space \(((\mathbb{R}^+)^n,g_0,D)\) is doubly autoparallel if and only if \(M\) is contained in \[\label{dapde} \mathcal{P}^{m} = \left\{(\eta^1,\ldots,\eta^n)\in(\mathbb{R}^+)^n \mid \eta^{i_k} = a_{i_k}\eta^{\phi(i_k)},\,\eta^{j_l} =
b_{j_l}\right\}\qquad{(4)}\] where \(i_1,\ldots,i_{\mathcal{A}},j_1,\ldots,j_{\mathcal{B}}\in\{1,\ldots,n\}\) are \(\mathcal{A}+\mathcal{B}=n-m\) distinct integers, \(\phi:\{i_1,\ldots,i_{\mathcal{A}}\}\to(\{1,\ldots,n\}\setminus\{i_1,\ldots,i_{\mathcal{A}},j_1,\ldots,j_{\mathcal{B}}\})\), and \(a_{i_1},\ldots,a_{i_{\mathcal{A}}},b_{j_1},\ldots,b_{j_{\mathcal{B}}}\in\mathbb{R}^+\).
Proof. For simplicity, suppose that \(M\) is contained in ?? , where \(i_k=m+k\), and \(j_l=m+A+l\). Let \((g,\nabla)\) be the statistical structure induced on \(M\) by \((g_0,D)\) and the inclusion map \(\iota:M\to(\mathbb{R}^+)^n\).
Since \((\eta^1,\ldots,\eta^n)\) is an affine coordinate system of \(D^*\), it is clear that \(M\) is a \(D^*\)-autoparallel
submanifold of \((\mathbb{R}^+)^n\). Define a coordinate system \((\xi^1,\ldots,\xi^m)\) on \(M\) by \((\xi^1(p),\ldots,\xi^m(p))=(\eta^1(p),\ldots,\eta^m(p)),\,p\in M\). By using these coordinate systems, we have \[\begin{align} \iota_*\frac{\partial}{\partial \xi^i} &= \frac{\partial}{\partial
\eta^i} + \sum_{k\in\phi^{-1}(i)}a_k\frac{\partial}{\partial \eta^{k}}
\end{align}\] and \[\begin{align} D_{\frac{\partial}{\partial \xi^i}}\iota_*\frac{\partial}{\partial \xi^j} &= -\frac{\delta_{ij}}{\eta^i}\frac{\partial}{\partial \eta^i} -
\sum_{k\in\phi^{-1}(i)\cap\phi^{-1}(j)}\frac{(a_k)^2}{\eta^k}\frac{\partial}{\partial \eta^k}\\ &= -\frac{\delta_{ij}}{\xi^i}\frac{\partial}{\partial \eta^i} - \delta_{ij}\sum_{k\in\phi^{-1}(i)}\frac{a_k}{\xi^i}\frac{\partial}{\partial \eta^k}\\ &=
-\frac{\delta_{ij}}{\xi^i}\iota_*\frac{\partial}{\partial \xi^i}.
\end{align}\] Since this a linear combination of \(\iota_*\frac{\partial}{\partial \xi^i}\), the second fundamental form with respect to \(D\) is \(0\), hence \(M\) is a doubly autoparallel submanifold of \(((\mathbb{R}^+)^n,g_0,D)\).
Now we assume that \(M\) is a doubly autoparallel submanifold of \(((\mathbb{R}^+)^n,g_0,D)\). Since \(M\) is a \(D^*\)-autoparallel submanifold of \((\mathbb{R}^+)^n\), it is an open subset of \(P^{m}\cap(\mathbb{R}^+)^n\) where \(P^{m}\subset\mathbb{R}^n\) is an \(m\)-dimensional plane with respect to the \((\eta^1,\ldots,\eta^n)\) coordinates. After a suitable reordering of the elements
of the \(D^*\)-affine coordinates \((\eta^1,\ldots,\eta^n)\), there exists a global coordinate system \((\xi^1,\ldots,\xi^m)\) on \(P^{m}\) such that \[\begin{align} \begin{pmatrix} \eta^1\\ \vdots \\ \vdots \\ \eta^n \end{pmatrix} = \begin{pmatrix} 1 & 0 & \dots & 0 \\ 0 & 1 & &0 \\ \vdots & &
\ddots & \vdots \\ 0 & 0 & \dots & 1\\ a_{m+1,1} & a_{m+1,2} & \dots & a_{m+1,m}\\ a_{m+2,1} & a_{m+2,2} & \dots & a_{m+2,m}\\ \vdots & &\ddots & \vdots\\ a_{n-m,1} & a_{n-m,2} & & \dots
a_{n-m,m} \end{pmatrix} \begin{pmatrix} \xi^1\\ \vdots \\ \xi^m \end{pmatrix} + \begin{pmatrix} 0\\ \vdots \\ 0 \\ b_{m+1} \\ \vdots \\ b_{n-m} \end{pmatrix}.
\end{align}\] Here, we have \[\begin{align} \label{4eq1} \iota_*\frac{\partial}{\partial \xi^i} = \frac{\partial}{\partial \eta^i} + \sum_{k=1}^{n-m}a_{k,i}\frac{\partial}{\partial \eta^k}
\end{align}\tag{27}\] and \[\begin{align} \label{4eq2} D_{\frac{\partial}{\partial \xi^i}}\iota_*\frac{\partial}{\partial \xi^j} &=
-\frac{\delta_{ij}}{\xi^i}\frac{\partial}{\partial \eta^i} - \sum_{k=1}^{n-m}\frac{a_{k,i}a_{k,j}}{\sum_{l=1}^m a_{k,l}\xi^l + b_k}\frac{\partial}{\partial \eta^k}.
\end{align}\tag{28}\] Since 28 can be expressed as a linear combination of \(\left\{\iota_*\frac{\partial}{\partial \xi^1},\ldots,\iota_*\frac{\partial}{\partial \xi^m}\right\}\), we
have \(a_{k,i}a_{k,j}=0\) for each \(i\neq j\). Thus, for each \(k\in\{1,\ldots,n-m\}\) there exists a unique \(1\leq i_k\leq
m\) such that \(a_{i,k}=0\) for each \(i\neq i_k\). Here, from equations 27 and 28 , it follows that \[\begin{align} \frac{a_{k,i_k}}{\xi^{i_k}} = \frac{(a_{k,i_k})^2}{a_{k,i_k}\xi^{i_k}+b_{k}}.
\end{align}\] If \(a_{k,i_k}\neq 0\), then \(b_k=0\) hold, and consequently we have \(a_{k,i_k}>0\). If \(a_{k,i_k}=
0\), then \(b_k=\eta^k>0\). Therefore, the manifold \(\mathcal{P}^m = P^m\cap(\mathbb{R}^+)^n\) can be expressed in terms of equation ?? .
◻
Remark 23. In Proposition 22, the induced statistical structure \((g,\nabla)\) on \(M\) is
expressed by \[\begin{align} g_0\left(\frac{\partial}{\partial \xi^i},\frac{\partial}{\partial \xi^j}\right) &= \left(1+\sum_{k\in\phi^{-1}(i)}a^k\right)\frac{\delta_{ij}}{\xi^i}\\ \nabla_{\frac{\partial}{\partial
\xi^i}}\frac{\partial}{\partial \xi^j} &= -\frac{\delta_{ij}}{\xi^i}.
\end{align}\] If we define a coordinate system \((\sigma^1,\ldots,\sigma^m)\) on \(M\) by \(\sigma^i=(1+\sum_{k\in\phi^{-1}(i)}a^k)\xi^i\), then the
expressions of \((g,\nabla)\) with respect to \((\sigma^1,\ldots,\sigma^m)\) coincide with the expression of \((g_0,D)\) given by 10 . Thus, we can regard \(M\) as an open statistical submanifold of \(((\mathbb{R}^+)^m,g_0,D)\).
The doubly totally-umbilical submanifolds in the denormalized state space of codimension one that are not doubly autoparallel is already classified in [2], [13]. We provide a different proof to this classfication.
Proposition 24. A submanifold \(M^m\) of the denormalized state space \(((\mathbb{R}^+)^{m+1},g_0,D)\) is doubly totally-umbilical if and only if it is contained in
one of the following sets\(\mathrm{:}\)
The set \(\mathcal{P}^m\) in \(\eqref{dapde}\).
The set \(\Delta^m(b) = \{(\eta^1,\ldots,\eta^{m+1})\in(\mathbb{R}^+)^{m+1}\mid \sum_{i=1}^{m+1}\eta^i=b\}\), where \(b\in\mathbb{R}^+\).
Proof. Case \((1)\) follows immediately from Proposition 22. If case \((2)\) holds, it can be proved
that \(\Delta^m(b)\) is a doubly totally-umbilical submanifold of \(((\mathbb{R}^+)^{m+1},g_0,D)\) in the same manner as in Example 6.
Now we let \(M^m\) be a doubly totally-umbilical submanifold of \(((\mathbb{R}^+)^{m+1},g_0,D)\). We may assume that \(M\) is not doubly autoparallel, since
otherwise case \((1)\) follows from Proposition 22. Let \((g,\nabla)\) be the induced statistical structure by
the inclusion \(\iota:M\to(\mathbb{R}^+)^{m+1}\), and let \(K,\widetilde{K}\) be the difference tensor field of \((g,\nabla)\), \((g_0,D)\), respectively. We prove \(K\neq0\) on any point of \(M\). If \(K=0\) on some point \(p\in
M\), by 21 , for each orthonormal pair \(X,Y\in T_p M\) we have \[\begin{align} 0 = g_0\left((D_{\iota_*X}\widetilde{K})(\iota_*Y,\iota_*Y),X\right) &=
g_0\left(H^*,\widehat{H}-H\right),\\ 0 = g_0\left((D_{\iota_*X}\widetilde{K})(\iota_*X,\iota_*Y),Y\right) &= g_0\left(H,H^*-\widehat{H}\right),
\end{align}\] since \(((\mathbb{R}^+)^{m+1},g_0,D)\) is a Hessian manifold of CHC \(0\). Thus it must hold that \(H_p=H_p^*=\widehat{H}_p\). From this
result, on \(p\) we have \[\begin{align} 0 = \iota_*K_X Y &=\iota_*\nabla_X Y - \iota_*\nabla^g_X Y\\ &= D_X\iota_* Y - D^{g_0}_X\iota_* Y - g(X,Y)(H-\widehat{H}) = \widetilde{K}_{\iota_*
X}\iota_* Y
\end{align}\] for \(X,Y\in\Gamma(TM)\). Thus, if \(X\in T_{\iota(p)}(\mathbb{R}^{+})^{m+1}\) is tangent to \(M\), then it holds that \(\widetilde{K}_X X=0\). However, if we express \(X\) with respect to \((\eta^1,\ldots,\eta^{m+1})\) by \(X =
\sum_{i=1}^{m+1}X^i\frac{\partial}{\partial \eta^i}\), we have \[\begin{align} 0 = \widetilde{K}_X X = -\sum_{i=1}^{m+1} \left(X_i\right)^2\frac{1}{\eta^i}\frac{\partial}{\partial \eta^i},
\end{align}\] which implies that \(X^i=0\) for each \(i\) and consequently \(X=0\). This contradicts that the dimension of \(M\) is \(m\geq 2\), therefore we have \(K\neq0\) on any point of \(M\).
For each \(X,Y,Z\in\Gamma(TM)\), from 22 we have \[\begin{align} 0 =& g(K_X Y,Z)g_0(H^*,\boldsymbol{n}) - g(Y,Z)g_0\left(D^{\perp}_X
\widehat{H},\boldsymbol{n}\right)\\ & + g(X,Y)g_0\left(\widehat{D}^{\perp}_ZH,\boldsymbol{n}\right) + g(X,Z)\left(\widehat{D}^{\perp}_Y H,\boldsymbol{n}\right)
\end{align}\] since \(D^{\perp}H = D^{\perp*}H^* = \widehat{D}^{\perp}\widehat{H} = 0\) holds from 19 , where \(\boldsymbol{n}\) is the normal vector field
of \(\iota:M^m\to(\mathbb{R}^+)^{m+1}\). Here, from Proposition 17 and Corollary 18, we have \(D^{\perp} \widehat{H}=\widehat{D}^{\perp} H=0\). Thus \(H^*=0\), therefore \(M\) is a \(D^*\)-autoparallel submanifold of \((\mathbb{R}^+)^{m+1}\), which is contained in a hyperplane with respect to the coordinate system \((\eta^1,\ldots,\eta^{m+1})\). There exists \(a_1,\ldots,a_{m+1},b\in\mathbb{R}\) such that \((a_1,\ldots,a_{m+1})\neq(0,\ldots,0)\) and \(M\) is contained in \[\begin{align} \label{planeform} \left\{(\eta^1,\ldots,\eta^{m+1})\mid \sum_{i=1}^{m+1}a_i\eta^i = b\right\}.
\end{align}\tag{29}\] Here, if we set \(\varphi(\eta^1,\ldots,\eta^{m+1}) = \sum_{i=1}^{m+1}a_i\eta^i - b\), for each \(p\in M\) and \(X=\sum_{i=1}^{m+1}X^i\frac{\partial}{\partial \eta^i}\in T_{\iota(p)}(\mathbb{R}^+)^{m+1}\), the equation \(0 = X\varphi = \sum_{i=1}^{m+1}X^ia_i\) is equivalent to \(X\in\iota_*T_pM\), and we have \[\begin{align} \boldsymbol{n}_p = \frac{1}{\|(\operatorname{grad}_{g_0}\varphi)_{\iota(p)}\|_{g_0}}(\operatorname{grad}_{g_0}\varphi)_{\iota(p)},\quad p\in M,
\end{align}\] where \(\operatorname{grad}_{g_0}\varphi\) is the gradient vector field of \(\varphi\) on \(((\mathbb{R}^+)^{m+1},g_0)\). Particularly,
by \(g_0 = \sum_{i=1}^{m+1}(\eta^i)^{-1}d\eta^i\otimes d\eta^i\) on the coordinates \((\eta^1,\ldots,\eta^{m+1})\), we have \[\begin{align}
\operatorname{grad}_{g_0}\varphi &= \sum_{i=1}^{m+1}a_i\eta^i\frac{\partial}{\partial \eta^i},
\end{align}\] which implies \(X\|\operatorname{grad}_{g_0}\varphi\|_{g_0} = 0\) for every \(X\in\iota_*(T_pM)\). From Corollary 18 and equation 19 , we also have \[\begin{align} 0 &= g_0(D^*_X \boldsymbol{n},\boldsymbol{n})\\ &= \frac{1}{\|\operatorname{grad}_{g_0}\varphi\|_{g_0}}g_0\left(D^*_X
\operatorname{grad}_{g_0}\varphi, \boldsymbol{n}\right)\\ &= \sum_{i=1}^{m+1}X^ia_i^2
\end{align}\] for each \(X\in\iota_*T_pM\). This implies that there exists \(t\neq0\) such that \((ta_1,\ldots,ta_{m+1})=(a_1^2,\ldots,a_{m+1}^2)\),
thus for each \(1\leq i \leq m+1\) we have \(a_i=t\) or \(a_i=0\). We can assume \(t=1\) by rescaling \(b\in\mathbb{R}\) if necessary. Lastly, we prove that we have \(a_i\neq 0\) for every \(1\leq i \leq m+1\). If there exists an \(1\leq i
\leq m+1\) such that \(a_i = 0\), then \(\frac{\partial}{\partial \eta^i}\in\iota_*T_pM\). Here, from Proposition 15 we have \[\begin{align} \frac{1}{m}\mathrm{tr}_gh\cdot \frac{\partial}{\partial \eta^{i}} = D^*_{\frac{\partial}{\partial \eta^{i}}}\boldsymbol{n} = 0,
\end{align}\] which implies \(\mathrm{tr}_gh=0\). Since \(m\|H\|_{g_0}=|\mathrm{tr}_g h|\), this contradicts \(M\) is not doubly autoparallel. Hence
we have \(a_i\neq 0\), and the plane 29 is equal to \(\Delta^{m}(b)\).
◻
Combining Propositions 22 and 24, we obtain a complete classification of doubly
totally-umbilical submanifolds in the denormalized state space.
Theorem 25. A submanifold \(M^m\) of the denormalized state space \(((\mathbb{R}^+)^{n},g_0,D)\) is doubly totally-umbilical if and only if one of the following
conditions holds\(\mathrm{:}\)
It is contained in the manifold \(\mathcal{P}^m\) defined by \(\eqref{dapde}\).
There exists a doubly autoparallel submanifold \(M\subset \mathcal{P}^{m+1}\subset (\mathbb{R}^+)^{n}\) such that, with the induced statistical structure \((\widetilde{g},\widetilde{\nabla})\) on \(\mathcal{P}^{m+1}\), there exists a global \(\widetilde{\nabla}^{*}\)-affine chart \((\sigma^1,\ldots,\sigma^{m+1};(\mathbb{R}^+)^{m+1})\) on \(\mathcal{P}^{m+1}\) such that \(M\) is contained in \(\Delta^m(b) =
\{(\sigma^1,\ldots,\sigma^{m+1})\in(\mathbb{R}^+)^{m+1}\mid \sum_{i=1}^{m+1}\sigma^i=b\}\), where \(b\in\mathbb{R}^+\).
Proof. If \((1)\) holds, it is clear that \(M\) is a doubly autoparallel submanifold of \(((\mathbb{R}^+)^{n},g_0,D)\) by Proposition 22 and hence doubly totally-umbilical. Suppose that \((2)\) holds. By Proposition 24 the manifold \(M\) is a doubly totally-umbilical submanifold of \((\mathcal{P}^{m+1},\widetilde{g},\widetilde{\nabla})\). With the inclusion maps \(\iota_1:M\to \mathcal{P}^{m+1}\), \(\iota_2:\mathcal{P}^{m+1}\to (\mathbb{R}^+)^{n}\), we define \(\iota=\iota_2\circ\iota_1\), and let \((g,\nabla)\) be the statistical structure induced on \(M\) by \(\iota_1\). Since \(\mathcal{P}^{m+1}\) is doubly autoparallel in
\(((\mathbb{R}^+)^{n},g_0,D)\), for every \(\alpha\in\mathbb{R}\) and \(X,Y\in\Gamma(TM)\) we have \[\begin{align} D^{(\alpha)}_X
\iota_*Y &= D^{(\alpha)}_X \iota_{2*}\iota_{1*}Y\\ &=\iota_{2*}\widetilde{\nabla}^{(\alpha)}_X \iota_{1*}Y\\ &=\iota_*\nabla^{(\alpha)}_X Y + g(X,Y)H^{(\alpha)}.
\end{align}\] Hence \(M\) is a doubly totally-umbilical submanifold of \(((\mathbb{R}^+)^{n},g_0,D)\).
Conversely, let \(M\) be a doubly totally-umbilical submanifold of \(((\mathbb{R}^+)^{n},g_0,D)\). If \(M\) is doubly autoparallel, then \((1)\) follows from Proposition 22. Therefore, we may assume that \(M\) is not doubly autoparallel. Let \((g,\nabla)\) be the induced statistical structure on \(M\) by \(((\mathbb{R}^+)^{n},g_0,D)\) and the inclusion map \(\iota:M\to(\mathbb{R}^+)^{n}\). As in the proof of Proposition 24, the difference tensor \(K\) of \((g,\nabla)\) does not vanish on any point of \(M\). We first show that \(\|H-\widehat{H}\|_{g_0}\) is constant. Since \(g_0(H,H^{*})\) and \(g_0(\widehat{H},\widehat{H})\) are constant and \(2\widehat{H} = H + H^*\), we have \[\begin{align}
\|H-\widehat{H}\|^2_{g_0} &= \frac{1}{4}\|H-H^*\|^2_{g_0}\\ &=\frac{1}{4}\left(\|H\|^2_{g_0}+\|H^*\|^2_{g_0}-2g_0\left(H,H^*\right)\right)\\ &=\|\widehat{H}\|^2_{g_0} - g_0\left(H,H^*\right).
\end{align}\] Together with \(\widehat{D}^{\perp}\widehat{H} = 0\), \[\begin{align} 0 = X\|H-\widehat{H}\|^2_{g_0} = 2g_0\left(\widehat{D}_X^{\perp}H,H-\widehat{H}\right)
\end{align}\] holds for every \(X\in\Gamma(TM)\). Thus, for each orthonormal pair \(\{X,Y\}\) on \((M,g)\), from equation 22 we obtain \[\begin{align} 0 &= g(K_Y Y,X)g_0\left(H^*,H - \widehat{H}\right)
\end{align}\] by taking \(\xi=H-\widehat{H}\). Since \(K\) does not vanish at any point of \(M\), we conclude that \(\displaystyle g_0\left(H^*,H - \widehat{H}\right) = 0\). Consequently, \[\begin{align} \|H^*\|_{g_0}^2 = g_0\left(H^*,H\right) - 2g_0\left(H^*,H-\widehat{H}\right)
\end{align}\] is also constant. Thus, for each orthonormal pair \(\{X,Y\}\) on \((M,g)\), from equation 23 we have \[\begin{align} 0 = g(K_Y Y,X)g_0(H^*,H^*) - g_0\left(\widehat{D}_X^{\perp}H^*,H^*\right) = g(K_Y Y,X)\|H^*\|_{g_0}^2
\end{align}\] by taking \(\xi = H^*\). Again using \(K\neq0\) on any point, we conclude that \(H^*=0\), and hence \(M\) is a \(D^*\)-autoparallel submanifold of \((\mathbb{R}^+)^n\). Fix a point \(p\in M\). Since \(\widehat{H}_p\neq 0\), the vector space \(V=\iota_*T_p M\oplus\mathbb{R}\widehat{H}_p\) is \(m+1\)-dimensional. For every \(X,Y\in
\Gamma(TM)\), we have \((D^{g_0}_X \iota_*Y)_p,(D^{g_0}_X \widehat{H})_p\in V\). We identify \(\mathbb{R}^{n+1}\) with \(T_{0}\mathbb{R}^{n+1}\) by
the \(D^{g_0}\)-affine coordinate \((y^1,\ldots,y^{n+1})\) where \(0\in\mathbb{R}^{n+1}\) is the origin, so that we have \(V\subset\mathbb{R}^n\) The set \(\mathcal{P}^{m+1} = (\mathbb{R}^+)^{n}\cap\{p+v\mid v\in V\}\) is an affine \((m+1)\)-plane of the affine space \(((\mathbb{R}^+)^{n},D^{g_0})\) containing \(M\). On the other hand, for each \(X,Y\in \Gamma(TM)\), we have \((D_X \iota_*Y)_p,(D_X
\widehat{H})_p=2(D_X H)_p\in V\), so \(\mathcal{P}^{m+1}\) is also an affine \((m+1)\)-plane of the affine space \(((\mathbb{R}^+)^{n},D)\). Hence
\(\mathcal{P}^{m+1}\) is a doubly autoparallel submanifold of \(((\mathbb{R}^+)^{n},g_0,D)\). If we denote by the induced statistical structure \((\widetilde{g},\widetilde{\nabla})\), it follows that \(M\) is a doubly totally-umbilical submanifold of \((\mathcal{P}^{m+1},\widetilde{g},\widetilde{\nabla})\). Using the global chart \((\sigma^1,\ldots,\sigma^{m+1})\) from Remark 23, the statistical manifold \((\mathcal{P}^{m+1},\widetilde{g},\widetilde{\nabla})\) can be identified as \(((\mathbb{R}^+)^{m+1},g_0,D)\). Thus, Proposition 24 implies that there exists a \(b\in\mathbb{R}^+\) such that \(M\) is contained in \(\Delta^m(b)\) defined in the coordinates \((\sigma^1,\ldots,\sigma^{m+1})\).
◻
4.2 Doubly totally-umbilical submanifolds in the probability simplex↩︎
Using Theorem 25, we classify doubly totally-umbilical submanifolds in the probability simplex.
Theorem 26. A submanifold \(M^m\) of the probability simplex \((\Delta^n,g^F,\nabla^{\mathrm{(e)}})\) is doubly totally-umbilical if and only if it is contained in the
following \(\nabla^{\mathrm{(m)}}\)-autoparallel submanifold\(\mathrm{:}\)\[\begin{align} \label{dtusub} \Xi =
\Xi_{a_{i_1},\ldots,a_{i_{\mathcal{A}}},b_{j_1},\ldots,b_{j_{\mathcal{B}}},\phi} = \left\{p\in\Delta^n\mid p(\omega_{i_k}) = a_{i_k}p(\omega_{\phi(i_k)}),\,p(\omega_{j_l})=b_{j_l}\right\},
\end{align}\tag{30}\] where \(i_1,\ldots,i_{\mathcal{A}},j_1,\ldots,j_{\mathcal{B}}\in\{1,\ldots,n+1\}\) are \(\mathcal{A}+\mathcal{B}=n-m\) distinct integers, \(\phi:\{i_1,\ldots,i_{\mathcal{A}}\}\to(\{1,\ldots,n+1\}\setminus\{i_1,\ldots,i_{\mathcal{A}},j_1,\ldots,j_{\mathcal{B}}\})\), \(a_{i_1},\ldots,a_{i_{\mathcal{A}}}\in\mathbb{R}^+\), and \(b_{j_1},\ldots,b_{j_{\mathcal{B}}}\in(0,1)\). In particular, if we denote by \((g,\nabla)\) the induced statistical structure on \(\Xi\), the following hold
\(\mathrm{:}\)
The submanifold \(M\) is doubly autoparallel if and only if \(\mathcal{B} = 0\). If \(M\) is doubly autoparallel, then there exists a statistical
diffeomorphism \(f:(\Delta^m,g^F,\nabla^{(\mathrm{e})})\to(\Xi,g,\nabla)\) such that \(\iota\circ f\) is a Markov embedding, where \(\iota:\Xi\to\Delta^n\)
is the inclusion map.
If \(\mathcal{B}\neq 0\), there exists a Markov embedding \(f:\Delta^{m+1}\to\Delta^n\) such that \(\Xi\subset f(\Delta^{m+1})\), and if we identify
\(f(\Delta^{m+1})\) with \(\Delta^{m+1}\), then \((\Xi,g,\nabla)\) is a doubly totally-umbilical statistical submanifold of \((\Delta^{m+1},g^F,\nabla^{\mathrm{(e)}})\).
The affine connection \(\nabla|_{M}\) is complete if and only if \(M=\Xi\).
We note that the affine connection \(\nabla^{(\mathrm{e})}\) is complete on \(\Delta^n\), which will be proved in Example 9 of the appendix.
Proof. Define \(\Xi\) by 30 , and let \(\iota:\Delta^n\to(\mathbb{R}^+)^{n+1}\) be the embedding map in Example 4. Then \(\Xi\) is a submanifold of \(\mathcal{P}^{m+1}=P^{m+1}\cap(\mathbb{R}^+)^{n+1}\) where
\[\begin{align} \label{dubhypplane} P^{m+1} = \left\{(\eta^1,\ldots,\eta^{n+1})\in\mathbb{R}^{n+1}\large\mid \eta^{i_k} = a_{i_k}\eta^{\phi(i_k)},\,\eta^{j_l} = b_{j_l}\right\},
\end{align}\tag{31}\] and \((\eta^1,\ldots,\eta^{n+1})\) is the \(D^*\)-affine coordinate system defined in Example 4. If we define the global coordinate system \((\sigma^1,\ldots,\sigma^{m+1})\) in Remark 23 on \(\mathcal{P}^{m+1}\), then \[\begin{align} \label{xirep} \Xi = \left\{(\sigma^1,\ldots,\sigma^{m+1})\in\mathcal{P}^{m+1} \mid \sum_{i=1}^{m+1}\sigma^i = 1 -
\sum_{l=1}^{\mathcal{B}}b_{j_l}\right\}.
\end{align}\tag{32}\] By Theorem 25\((2)\), the manifold \(M\) is a doubly totally-umbilical
submanifold of \(((\mathbb{R}^+)^{n+1},g_0,D)\), and by Lemma 2 it is also a doubly totally-umbilical submanifold of \((\Delta^n,g^F,\nabla^{(\mathrm{e})})\).
Conversly, assume that \(M\) is a doubly totally-umbilical submanifold of \((\Delta^n,g^F,\nabla^{(\mathrm{e})})\). Let \(\iota:\Delta^n\to(\mathbb{R}^+)^{n+1}\) the embedding in Example 4, and identify \(\Delta^n\) with \(\iota(\Delta^n)\). Since \(\Delta^n\) is a doubly totally-umbilical but not doubly autoparallel submanifold of \(((\mathbb{R}^+)^{n+1},g_0,D)\), so is \(M\subset(\mathbb{R}^+)^{n+1}\) by Lemma 2. By Theorem 25, there exists an
affine plane \(P^{m+1}\subset\mathbb{R}^{n+1}\) of the Euclidean space \((\mathbb{R}^{n+1},g_0)\), such that \(M^m\subset P^{m+1}\) and \(\mathcal{P}^{m+1} = P^{m+1}\cap(\mathbb{R}^+)^{n+1}\) is a doubly autoparallel submanifold of \(((\mathbb{R}^+)^{n+1},g_0,D)\). Thus, the affine plane \(P^{m+1}\) is given by the form of 31 . If we set \(\Xi=\mathcal{P}^{m+1}\cap\Delta^n\), then it follows that \(\Xi\) is an \(m\)-dimensional doubly totally-umbilical submanifold of \((\Delta^n,g^F,\nabla^{(\mathrm{e})})\) and \(M\subset\Xi\) holds.
In order to prove (1) and (2), we prove that \(\Xi=\Xi_{a_{i_1},\ldots,a_{i_{\mathcal{A}}},b_{j_1},\ldots,b_{j_{\mathcal{B}}},\phi}\) is contained in the sphere \(S^m(r)\) for some \(0<r\leq2\). Indeed, on the standard coordinate system \((y^1,\ldots,y^{n+1})\) of \(\mathbb{R}^{n+1}\), we have \(\Delta^n=\{(y^1,\ldots,y^{n+1})\in(\mathbb{R}^+)^{n+1}\mid \sum_{i=1}^{n+1}(y^i)^2 = 4\}\), thus \(\Xi\subset P^{m+1}\cap S^{n}(2)\). Again, we identify \(\mathbb{R}^{n+1}\) with \(T_{0}\mathbb{R}^{n+1}\) by the \(D^{g_0}\)-affine coordinate \((y^1,\ldots,y^{n+1})\), where \(0\in\mathbb{R}^{n+1}\) is the origin. Let \(p\in P^{m+1}\) be the point minimizing \(\|p\|_{g_0}\) on \(P^{m+1}\), and let \(V=\{q-p\mid q\in P^{m+1}\}\). We have \(P^{m+1} = \{p+v\mid v\in V\}\), and for each \(q=p+v\in P^{m+1}\cap S^{n}(2)\), since \(g_0(p,v)=0\) it holds that \[\begin{align} 4=\|q\|_{g_0}^2=\|p\|_{g_0}^2+\|v\|_{g_0}^2.
\end{align}\] Hence \(P^{m+1}\cap S^{n}(2)\) is an Euclidean sphere \(S^m(r)\) where \(r=\sqrt{4-\|p\|_{g_0}^2}\).
We prove (1). The sphere \(S^m(r)\) is an autoparallel manifold of the Euclidean sphere \((S^n(2),\nabla^{g_0})\) if and only if \(r=2\) (see [7] for example). In this case, \(p\in P^{m+1}\) is the origin \(0\in\mathbb{R}^{n+1}\), which
is equivalent to \(\mathcal{B}=0\). Since \((\Delta^n,g^F)\) is an open Riemannian submanifold of \((S^n(2),g_0)\) and \(M\)
is already a \(\nabla^{\mathrm{(m)}}\)-autoparallel submanifold of \(\Delta^n\), the manifold \(M\) is a doubly autoparallel submanifold of \((\Delta^n,g^F,\nabla^{\mathrm{(e)}})\), if and only if \(\mathcal{B}=0\).
For the doubly autoparallel submanifold \(\Xi=\Xi_{a_{i_1},\ldots,a_{i_{\mathcal{A}}},\phi}\), we can assume that \((i_1,\ldots,i_{\mathcal{A}})=(m+2,\ldots,n+1)\). Define \(C_l=\{l\}\cup\phi^{-1}(l)\) and \(Q_l:\Omega_{n+1}\to[0,\infty)\) by \[\begin{align} Q_l(k) = \begin{cases} 1, &k\in\Omega_{m+1},\\ a_k, &k\in\phi^{-1}(l),\\
0, &k\notin\Omega_{m+1}\cup\phi^{-1}(l) \end{cases}
\end{align}\] for each \(l\in\Omega_{m+1}\). Then, if we define the diffeomorphism \(f:\Delta^{m}\to\Xi\) as in equation 11 by using \(\{C_l\}\) and \(\{Q_l\}\), the map \(\iota\circ f\) is a Markov embedding, and \(\iota\circ
f:(\Delta^m,g^F,\nabla^{(\mathrm{e})})\to(\Delta^n,g^F,\nabla^{(e)})\) is a statistical immersion. Thus, the diffeomorphism \(f:(\Delta^m,g^F,\nabla^{(\mathrm{e})})\to(\Xi,g,\nabla)\) is a statistical one.
Now we prove (2). Suppose that \(\mathcal{B}\neq 0\). Again, we assume that \((i_1,\ldots,i_{\mathcal{A}})=(m+2,\ldots,m+1+\mathcal{A})\) and \((j_1,\ldots,j_{\mathcal{B}})=(m+2+\mathcal{A},\ldots,n+1)\), by reordering the elements of \((\eta^1,\ldots,\eta^{n+1})\) if necessary. We identify \(\mathbb{R}^{n+1}\) with \(T_{0}\mathbb{R}^{n+1}\) by the \(D^{*}\)-affine coordinate \((\eta^1,\ldots,\eta^{n+1})\), where \(0\in\mathbb{R}^{n+1}\) is the origin. Since \(\mathcal{B}\neq0\), the point \(p\in P^{m+1}\) minimizing \(\|p\|_{g_0}\) is \(p=(0,\ldots,0,b_{j_1},\ldots,b_{j_{\mathcal{B}}})\). We define vector spaces \(V,W\) by \(V = \{q-p\mid q\in P^{m+1}\}\) and \[\begin{align} W = V\oplus\mathbb{R}p.
\end{align}\] By taking linear combinations of a basis of \(W\), we obtain \[\begin{align} W = \left\{(\eta^1,\ldots,\eta^{n+1})\in\mathbb{R}^{n+1}\large\mid \eta^{i_k} =
a_{i_k}\eta^{\phi(i_k)},\,\eta^{j_l} = \frac{b_{j_l}}{b_{j_1}}\eta^{j_1}\right\}.
\end{align}\] Hence \(\mathcal{W}=W\cap(\mathbb{R}^+)^{n+1}\) is a doubly autoparallel submanifold of \(((\mathbb{R}^+)^{n+1},g_0,D)\). By \((1)\),
the set \(\Theta = \mathcal{W}\cap\Delta^{n}\) is a doubly autoparallel submanifold of \((\Delta^{n},g^F,\nabla^{(\mathrm{e})})\), and there exists a Markov embedding \(f:\Delta^{m+1}\to\Delta^n\) such that \(f(\Delta^{m+1})=\Theta\) holds. Since \(\Xi\subset\Theta\) holds by construction, if we identify \(\Theta\) by \(\Delta^{m+1}\), then \((\Xi,g,\nabla)\) is a doubly totally-umbilical statistical submanifold of \((\Delta^{m+1},g^F,\nabla^{(\mathrm{e})})\).
Lastly, we prove \((3)\). Using the global coordinate system \((\sigma^1,\ldots,\sigma^{m+1})\) in Remark 23 on
\(\mathcal{P}^{m+1}\), we identify \((\mathcal{P}^{m+1},\widetilde{g},\widetilde{\nabla})\) with the statistical submanifold \(((\mathbb{R}^+)^{m+1},g_0,D)\). We also have that \(\Xi\) is expressed by 32 on this global coordinate, so we define a coordinate system \((\rho^1,\ldots,\rho^m)\) on \(\Xi\) by \((\rho^1(p),\ldots,\rho^m(p))=(\sigma^1(p),\ldots,\sigma^m(p)),\,p\in\Xi\). Then \((\rho^1,\ldots,\rho^m)\) is an affine coordinate system of the conjugate connection \(\nabla^{*}\) of the statistical structure \((g,\nabla)\) on \(\Xi\) induced by \((\widetilde{g},\widetilde{\nabla})\). If we define \[\begin{align} \psi(\rho^1,\ldots,\rho^m) = \sum_{i=1}^m\rho^{i}\log\rho^i +
\left(1-\sum_{l=1}^{\mathcal{B}}b_{j_l}-\sum_{i=1}^m\rho^i\right)\log\left(1-\sum_{l=1}^{\mathcal{B}}b_{j_l}-\sum_{i=1}^m\rho^i\right)
\end{align}\] on the coordinate system \((\rho^1,\ldots,\rho^m)\), we have \(g=\nabla^*d\psi\). From Proposition 29, we obtain that \(\nabla\) is complete on \(\Xi\). Consequently \(\nabla|_{M}\) is complete on \(M\) if and only if \(M=\Xi\).
◻
Remark 27. Theorem 26 also gives the complete classification of doubly autoparallel submanifolds in the probability simplex. A. Ohara and H. Ishi also gave this
classification by an algebraic characterization [24].
Remark 28. For the probability simplex \((\Delta^n,g^F,\nabla^{(\mathrm{e})})\), if we fix \(1\leq\mathcal{B}\leq n-2\) distinct integers \(j_1,\ldots,j_{\mathcal{B}}\in\{1,\ldots,n+1\}\), we obtain a foliation \[\begin{align} \left\{\Xi_{b_{j_1},\ldots,b_{j_{\mathcal{B}}}}\mid
(b_{j_1},\ldots,b_{j_{\mathcal{B}}})\in\Delta^{\mathcal{B}}\right\},
\end{align}\] of \(\Delta^n\) where each leaf is a doubly totally-umbilical submanifold. Indeed, if we assume \(n+1\notin\{j_1,\ldots,j_{\mathcal{B}}\}\) for simplicity, then we have
\(\eta^{j_l}=b_{j_l}\) on the \(\nabla^{(\mathrm{m})}\)-affine coordinate system \((\eta^1,\ldots,\eta^n)\) defined in Example 1. Foliations of statistical manifolds have been extensively studied in information geometry [25]–[27].
We assume that the manifold \(M\) is connected throughout this appendix.
Definition 9. Let \(\nabla\) be an affine connection on \(M\). A \(\nabla\)-geodesic is a curve \(\gamma:(a,b)\to M\) satisfying \[\begin{align} \nabla_{\dot{\gamma}}\dot{\gamma} = 0,
\end{align}\] where \(\dot{\gamma}\) is the velocity vector field of \(\gamma\). The affine connection is called complete, if for any \((p,v)\in
TM\) there exists a \(\nabla\)-geodesic \(\gamma:\mathbb{R}\to M\) such that \(\gamma(0)=p\) and \(\dot{\gamma}(0)=v\).
In general, completeness of the affine connection \(\nabla\) of a statistical manifold \((M,g,\nabla)\) is difficult to determine. However if \((M,g,\nabla)\) is Hessian and there is a global affine coordinate system \((x^1,\ldots,x^m)\) with respect to \(\nabla\), on \((x^1,\ldots,x^m)\) the \(\nabla\)-geodesics in this coordinate are given by \[\begin{align} x^i(\gamma(t)) = a^it + b^i,
\end{align}\] where \(a^1,\ldots,a^m,b^1,\ldots,b^m\) are constants. In particular, the completeness of \(\nabla\) is equivalent to the coordinate map \((x^1,\ldots,x^m)\) is onto \(\mathbb{R}^m\). Thus, the mixture connection \(\nabla^{(\mathrm{m})}\) in Example 1 is not complete.
The following proposition characterizes completeness of the conjugate connection.
Proposition 29. *Let \((M,g,\nabla)\) be a simply connected Hessian manifold. Assume that there exists a global affine coordinate system \((x^1,\ldots,x^m)\) on \(M\) with respect to \(\nabla\) and a \(\psi\in C^{\infty}(M)\) such that \(g=\nabla d\psi\). Then the conjugate connection \(\nabla^{*}\) is complete if and only if the map \[\begin{align} \label{diffeom} \Psi(p) = \left(\frac{\partial \psi}{\partial x^1}(p),\ldots,\frac{\partial
\psi}{\partial x^m}(p)\right), \quad p\in M
\end{align}\tag{33}\] is a diffeomorphism from \(M\) to \(\mathbb{R}^m\).*
Proof. Define functions \[\begin{align} (y^1,\ldots,y^m) = \left(\frac{\partial \psi}{\partial x^1}(p),\ldots,\frac{\partial \psi}{\partial x^m}(p)\right).
\end{align}\] Then, \((y^1,\ldots,y^n)\) is a global affine coordinate system of \(M\) with respect to \(\nabla^{*}\) (see [22] for example). Indeed, if we let \[\begin{align} g_{ij} = g\left(\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^j}\right),
\end{align}\] then the condition \(g = \nabla d\psi\) implies \[\begin{align} \frac{\partial}{\partial x^i} = \sum_{l=1}^m \frac{\partial y^l}{\partial x^i}\frac{\partial}{\partial y^l} =
\sum_{l=1}^m \frac{\partial^2 \psi}{\partial x^i \partial x^l}\frac{\partial}{\partial y^l} = \sum_{j=1}^m g_{il}\frac{\partial}{\partial y^l}.
\end{align}\] Hence, we have \[\begin{align} \frac{\partial}{\partial y^i} = \sum_{l=1}^m g^{il}\frac{\partial}{\partial x^l},
\end{align}\] where \(g^{ij}\) is the \((i,j)\)-element of the inverse matrix of \(G=(g_{ij})\). For any vector field \(X\in\Gamma(TM)\), we obtain \[\begin{align} g\left(\nabla^*_{X}\frac{\partial}{\partial y^i},\frac{\partial}{\partial x^j}\right) &= Xg\left(\frac{\partial}{\partial y^i},\frac{\partial}{\partial
x^j}\right) - g\left(\frac{\partial}{\partial y^i},\nabla_X\frac{\partial}{\partial x^j}\right)\\ &= X\left(\sum_{l=1}^mg^{il}g_{lj}\right) = X\delta_{ij} = 0.
\end{align}\] Since \(\left\{\frac{\partial}{\partial x^1},\ldots,\frac{\partial}{\partial x^m}\right\}\) is a frame of \(TM\), it follows that \(\nabla^{*}_{X}\frac{\partial}{\partial y^i}=0\) for all \(X\in\Gamma(TM)\). Therefore the coordinate system \((y^1,\ldots,y^m)\) is an affine coordinate of \(\nabla^{*}\). The claim now follows since \(\nabla\) is complete if and only if \((y^1,\ldots,y^m)\) is a coordinate map onto \(\mathbb{R}^{m}\).
◻
Example 9. Consider the probability simplex \((\Delta^n,g^F,\nabla^{(\mathrm{m})})\), in Example 1, but with the mixture
connection. The function represented by \[\begin{align} \psi(\eta^1,\ldots,\eta^n) = \sum_{i=1}^n\eta^{i}\log\eta^i + \left(1-\sum_{i=1}^n\eta^i\right)\log\left(1-\sum_{i=1}^n\eta^i\right)
\end{align}\] on the affine coordinate \((\eta^1,\ldots,\eta^n)\) with respect to \(\nabla^{(\mathrm{m})}\) satisfies \(g^F=\nabla^{(\mathrm{m})}d\psi\). Moreover, \[\begin{align} \frac{\partial \psi}{\partial \eta^i} = \log\eta^i - \log\left(1-\sum_{l=1}^n\eta^l\right).
\end{align}\] Thus the map given by 33 is a diffeomorphism between \(\{(\eta^1,\ldots,\eta^n)\in(\mathbb{R}^+)^n\mid \sum_{l=1}^n\eta^l < 1\}\) and \(\mathbb{R}^n\), hence by Proposition 29 the exponential connection \(\nabla^{(\mathrm{e})}\) is complete.
The author would like to express his gratitude to Professor Takashi Kurose and Professor Hitoshi Furuhata for their valuable support and insightful discussions during the course of this research.
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