October 02, 2025
We use differential forms on loop spaces to prove that the fundamental group of certain geometric transformation groups is infinite. Examples include both finite and infinite dimensional Lie groups. The finite dimensional examples are the conformal group of \(S^{4k+1}\) for a family of nonstandard metrics, and the group of pseudo-Hermitian transformations of a compact CR manifold. Infinite dimensional examples include the group of strict contact diffeomorphisms of a regular contact manifold, and other groups coming from symplectic and contact geometry.
In memory of Yuji Ito
In a series of papers [1]–[5], we developed the geometry of loop spaces and a theory of characteristic classes on loop spaces. In particular, the secondary Wodzicki-Chern-Simons (WCS) form was used to prove that \(\pi_1({\rm Isom}(M,g))\) is infinite for certain closed manifolds \(M\) with specific Riemannian metrics \(g\). In this paper, we prove similar results using alternatives to the WCS form, and with the finite dimensional Lie group \({\rm Isom}(M,g)\) replaced by certain infinite dimensional Lie groups of geometric transformations.
In our setup, \(M\) is a closed connected oriented finite dimensional manifold. Let \(\mathcal{G}(M)\) be a closed finite or infinite dimensional subgroup of \({\rm Diff}(M)\). We want examples where \(\pi_1(\mathcal{G}(M))\) is infinite. In general, this seems difficult to prove, even in the explicit case where \(\mathcal{G}(M)= {\rm Isom}(M,g).\) In all our examples, \(M\) has an \(S^1\) action via diffeomorphisms, and we prove that the associated element of \(\pi_1(\mathcal{G}(M))\) has infinite order.
Our techniques assume a smooth structure on \({\rm Diff}(M)\) and on the loop space \(LM.\) These manifolds have various smooth structures, depending on whether the model space is a Hilbert space [6], a Banach space [7], a Fréchet space or a locally convex space [8] (of if we take ILH structures [9], [10]). As long as we take the same type of structure on both \({\rm Diff}(M)\) and \(LM\), the results we use from our earlier papers are valid. If we give \(\mathcal{G}(M)\) the induced structure from \({\rm Diff}(M)\), the results on \(\pi_1(\mathcal{G}(M))\) are independent of the choice of structure.
Our previous work focused on contact and Sasakian manifolds. Recall that \((M^{2k+1}, \eta)\) is a contact manifold if \(\eta\) is a one-form on \(M\) satisfying \(\eta \wedge (d\eta)^k \neq 0.\) The characteristic vector field (or Reeb vector field) of \((M,\eta)\) is defined by \(d\eta (\xi , \cdot )=0, \eta(\xi) =1.\) A contact manifold \((M, \eta)\) is regular if the flow of the characteristic vector field \(\xi\) through any point \(m \in M\) is periodic. Such manifolds arise as the total space of circle bundles over symplectic manifolds. The extra conditions which make a contact manifold Sasakian are detailed in [5].
We proved the following results about the isometry group \({\rm Isom}(M,g)\) and the strict contactomorphism group \({\rm Diff}_{\eta, {\rm str}}(M) = \{ \phi \in {\rm Diff}(M), \phi^* \eta = \eta \},\) of contact and Sasakian manifolds \((M,g)\).
Theorem 1. (i) [2], [3] Let \((M,\omega)\) be an integral symplectic manifold of dimension \(4k\), and let \(\overline{M}_p\) be the total space of the circle bundle with first Chern class \(p\omega\). Then \(\overline{M}_p\) admits a Riemannian metric \(g_p\) such that for \(p\gg 0\), \[|\pi_1({\rm Isom}(\overline{M}_p, g_p) | = \infty.\] Equivalently, if \(\overline{M}\) is a regular contact manifold, then \(\overline{M}\) covers infinitely many strictly regular contact manifolds \((\overline{M}_p, g_p)\) with \(|\pi_1({\rm Isom}(M_p, g_p) | = \infty.\)
(ii) [5] Let \((M, g, \phi, \xi, \eta)\) be a connected, closed, regular \((4k+1)\)-dimensional Sasakian manifold. Deform the Sasakian metric \(g\) to the family \(g_\rho= g + \rho^2 \eta \otimes \eta\), \(\rho>0\), where \(\eta\) is the contact \(1\)-form on \(M\). Then \[|\pi_1({\rm Isom}(M, g_\rho) | = \infty.\]
(iii) [5] \[|\pi_1 ({\rm Diff}_{\eta, {\rm str}}(S^{4k+1}))| = \infty.\]
Part (ii) is particularly interesting when \(M= S^{4k+1}\) with the standard metric \(g= g_0\) and contact structure, since \(\pi_1({\rm Isom}(S^{4k+1}, g)) = \mathbb{Z}_2\). Therefore, \(\pi_1\) of the isometry group is discontinuous in \(\rho.\) Note that the isometry groups of closed manifolds are finite dimensional Lie groups, while the contactomorphism group in (iii) is infinite dimensional.
The main technique in this paper is to replace the WCS \(n\)-form on the loop space of an \(n\)-manifold with other \(n\)-forms \(\hat{\mathcal{K}}\) as in (2 ). We give a general condition under which \(\hat{\mathcal{K}}\) detects an element of infinite order in \(\pi_1(\mathcal{G}(M))\):
Let \(M\) be a closed oriented \(n\)-dimensional smooth manifold, and let \(\mathcal{G}(M)\) be a subgroup of \({\rm Diff}(M).\) Let \(\hat{\mathcal{K}}\in \Lambda^n(LM)\) be defined as in (2 ) with kernel \(\hat{k}\) as in (1 ). If there is a smooth action \(a:S^1\times M\to M\) such that (i) \(a(\theta,\cdot) \in \mathcal{G}(M)\) for all \(\theta\in S^1\), (ii) \(a(\theta,\cdot)^*\hat{k} = \hat{k}\) for all \(\theta\in S^1\), (iii) \(\int_M a^{L,*}\mathcal{K}\neq 0,\) then \[|\pi_1 (\mathcal{G}(M) )| = \infty.\]
For example, for one choice of \(\hat{\mathcal{K}},\) we obtain a strengthening of Thm. 1(iii).
Let \((M, \eta)\) be a closed connected regular contact manifold. Then \[|\pi_1 ({\rm Diff}_{\eta, str} (M))| = \infty.\]
This result was previously obtained in [11] using algebraic topology techniques, which gave new information on the cohomology of the classifying space \(B{\rm Diff}_{\eta, str} (M).\) Our proof is more analytic, and allows us to generalize Theorem 5.1 (see §5.2). Our idea of replacing WCS forms by more general forms on loop spaces is motivated by [11].
As an outline of the paper, in §2 we review some calculations for differential forms on loop spaces, and introduce the forms \(\hat{\mathcal{K}}\) (2 ). We use these forms to give a criterion for proving \(|\pi_1(\mathcal{G}(M))| = \infty\) (Thm. 2). In §3, we discuss the conformal transformation group of \(S^{4k+1}, k>0.\) We obtain
For \(\rho\neq 0\), \[|\pi_1({\rm Conf}(S^{4k+1}, g_\rho) )| = \infty.\] This conformal group is finite dimensional [12]. As above, this theorem fails for the standard metric \(g_0.\)
§4 is devoted to applications of Thm. 2 to finite and infinite dimensional groups of transformations that preserve a contact or canonical one-form. In §4.1, we discuss the strict contactomorphism group, and a generalization in §4.2. In §4.3, we also prove (Thm. 7) \[|\pi_1({\rm Psh}(M))| = \infty,\] where \({\rm Psh}(M)\) is the finite dimensional group of pseudo-Hermitian transformations of a pseudo-Hermitian (or CR) manifold, the odd dimensional analogue of a symplectic manifold with a compatible almost complex structure. In §4.4, we consider a subgroup of the contact transformations \(\mathbb{R}^{2k}\), and in §4.5 we generalize this to the cotangent bundle of a closed manifold. Finally, in §4.6 we consider Hamiltonian transformations of symplectic manifolds. In all cases, we prove that the fundamental group of these transformation groups is infinite by determining the appropriate form on \(LM\) to use in Thm. 2.
We dedicate this paper to the late Professor Yuji Ito of Keio University. The first author had the privilege of working alongside Professor Ito as a colleague at Keio University for many years. Despite the author’s lack of prior expertise in ergodic theory and operator algebras, Professor Ito generously and patiently shared his deep knowledge and insights. Professor Ito’s intellectual guidance and encouragement remain a lasting and invaluable asset to the author.
In this section, we study differential forms on the loop space and some basic properties, as in [4], [5]. In §3, we use this material to give the general method to prove \(|\pi_1(\mathcal{G}(M))|=\infty\) for a geometric transformation group \(\mathcal{G}(M)\).
Let \(M\) be a \(n\)-dimensional manifold. We consider tensor fields \({\hat{k}} \in \Omega^1(M) \otimes \Omega^n (M)\). In local coordinates \((x^\lambda)= (x^1, \cdots , x^n)\), we have \[\label{eq:k} {\hat{k}} = {\hat{k}}_{\nu [\lambda_1 \cdots , \lambda_n]} dx^\nu \otimes dx^{\lambda_1} \wedge \cdots \wedge dx^{\lambda_n}.\tag{1}\] The square brackets denote that the indices are skew-symmetric in \((\lambda_1, \ldots, \lambda_n)\), and may be omitted if the context is clear.
Let \(LM = LM =\{ \gamma : S^1 \to M: \gamma \in C^\infty(S^1,M)\}\) be the loop space of \(M\). We fix some \(C^\ell\) topology on \(LM\) for \(\ell\gg 0.\) Given such a \({\hat{k}}\), we define \(\hat{\mathcal{K}}\in \Omega^n(LM)\) by \[\label{eq:hatk} {\hat{\mathcal{K}}} (\gamma ) (X_{\gamma ,1}, \cdots , X_{\gamma, n}) = \int_{S^1} {\hat{k}}_{\nu [\lambda_1 \cdots \lambda_n]} (\gamma (\theta)) {\dot{{\gamma}}}^\nu (\theta) X_{\gamma, 1}^{\lambda_1}(\theta) \cdots X_{\gamma, n}^{\lambda_n}(\theta) d\theta,\tag{2}\] where \(X_{\gamma, 1} = X_{\gamma,1}^{\lambda_1}\frac{\partial}{\partial x^{\lambda_1}}, \cdots , X_{\gamma, n} =X_{\gamma,n}^{\lambda_n}\frac{\partial}{\partial x^{\lambda_n}} \in T_\gamma LM\) are tangent vectors at \(\gamma\), i.e., vector fields along \(\gamma \in LM.\) We call \(\hat{k}\) the kernel of \(\hat{\mathcal{K}}.\)
On an infinite dimensional smooth Banach manifold \(N\), the exterior derivative of \(\omega\in \Lambda^{s+1}(N)\) is defined by the Cartan formula \[\begin{align} d_N\omega(X_{0},\cdots X_{s})_p &= \sum_{i=0}^s (-1)^i X_i(\omega(X_{0},\cdots,\widehat{X_i},\cdots, X_{s})\\ &\qquad + \sum_{0\leq i<j\leq s} (-1)^{i+j}\omega([X_i,X_j], X_{0},\cdots, \widehat{X_i},\cdots, \widehat{X_j},\cdots,X_{s}), \end{align}\] where \(X_i\in T_pN\) are extended to vector fields near \(p\) using a chart map (see e.g., [13]).
We recall the key formulas to show the infinite order of the fundamental group of geometric transformation group \({\mathcal{G}}(M)\).
As in the finite-dimensional case, we have
Lemma 1. [4] Let \(f:W\to L\) be a smooth map between smooth Banach manifolds, and let \(\omega\in \Omega^*(L).\) Then \(d_W f^*\omega = f^*d_L \omega\).
In fact, the proof carries over to more general settings like ILH manifolds. The following is a consequence of a direct computation of the exterior derivatives:
Lemma 2. [4] \[\begin{align} \label{dLM-formula} &(d_{LM}{\hat{\mathcal{K}} )(X_{\gamma ,0}, X_{\gamma ,1}, \cdots , X_{\gamma ,2k-1} ) }\\ & \qquad =\sum_{a=0}^n (-1)^a \int_0^{2\pi}\hat{k}_{\nu \lambda_0 \cdots \hat{\lambda_a} \cdots \lambda_n } (\gamma(\theta)) \dot{X}_{\gamma,0}^{\lambda_0} (\theta) \cdots \hat{X}_{\gamma, a}^{\lambda_a}(\theta)\cdots X_{\gamma, n}^{\lambda_n} (\theta) d\theta \nonumber \end{align}\qquad{(1)}\]
We will consider smooth actions \(a:S^1\times M\to M\), with associated maps \(a^D:S^1\to{\rm Diff}(M)\), \(a^L:M\to LM\), given by \(a^D(\theta)(x) = a^L(x)(\theta) := a(\theta, x).\) Since we want to study \(\pi_1(\mathcal{G}(M))\) for \(\mathcal{G}(M)\subset {\rm Diff}(M)\), we need to consider smooth functions \(F:[0,1]\times S^1\times M \to M\) and the associated homotopies \(F^D:[0,1]\times S^1\to {\rm Diff}(M), F^L: [0,1]\times M\to LM\), given by \(F^D(x^0,\theta)(x) = F^L(x^0,x)(\theta) := F(x^0,\theta,x).\)
Here we are assuming that \(F(x^0,\theta,\cdot)\in {\rm Diff}(M)\) for all \((x^0,\theta)\in [0,1]\times S^1.\) Then \(\{F_*(\partial/\partial x^i)\}_{i=1}^{2k-1}\) is a basis of \(T_{F(x^0,\theta,x)}M\) for all \((x^0,\theta,x).\) Therefore, there exist functions \(\alpha^i =\alpha^i(x^0,\theta,x)\), \(i = 1,\ldots,{\rm dim}(M)\), such that \[\label{eq:five} F_*\left(\frac{\partial}{\partial x^0}\right) = \alpha^iF_*\left(\frac{\partial}{\partial x^i}\right).\tag{3}\]
Using (?? ) and replacing of the WCS form \(CS^W\) by \(\mathcal{K}\) in [4], we have :
Lemma 3. We have \[F^{L,*} d_{L{\bar M}} \hat{\mathcal{K}}(\partial_{x^0}, \partial_{x^1}, \cdots, \partial_{x^{n}}) = \int_0^{2\pi} \hat{k}_{\lambda_0 [\lambda_1\ldots\lambda_{n}]} \frac{\partial \alpha^i}{\partial\theta} \frac{\partial F^{\lambda_0}}{\partial x^i} \frac{\partial F^{\lambda_1}}{\partial x^1} \cdots \frac{\partial F^{\lambda_{n}}}{\partial x^{2k-1}} d\theta.\]
Now we discuss our method for proving \(|\pi_1(\mathcal{G}(M))|=\infty,\) for any subgroup \(\mathcal{G}(M)\) of \({\rm Diff}(M).\) Here \(M\) may have nonempty boundary.
We start with some notation.
Definition 1. (i) Let \(f:M\to M\) be smooth. \(\hat{k}\) is \(f\)-invariant if \(f^*\hat{k} = \hat{k}.\)
(ii) For a smooth map \(F:[0,1]\times S^1\times M\to M\), and for \((x^0,\theta)\in [0,1]\times S^1\), set \(a(x^0,\theta) = F(x^0,\theta,\cdot): M\to M.\) Then \(\hat{k}\) is \(F\)-invariant if \[\label{eq:kinv} a(x^0,\theta)^* {\hat{k}} = {\hat{k}},\qquad{(2)}\] for all (\(x^0,\theta)\in [0,1]\times S^1.\)
(iii) \(\hat{k}\) is \(\mathcal{G}(M)\)-invariant if \(f^*\hat{k} = \hat{k}\) for all \(f\in \mathcal{G}(M).\)
For example, the WCS form on a \((4k+1)\)-dimensional Riemannian manifold \((M,g)\) is built from \[\begin{align} \label{K95tensor} \MoveEqLeft{ \hat{k}_{\nu [\lambda_1 \cdots \lambda_{4k+1}]} } \\ &= \sum_{\sigma } {\rm sgn}({\sigma}) R_{\lambda_{\sigma (1) e_1 \nu}}{}^{e_{2}} R_{\lambda_{\sigma (2) } \lambda_{\sigma (3)} e_{3}}{}^{e_1} R_{\lambda_{\sigma (4) } \lambda_{\sigma (5)} e_{1}}{}^{e_{3}} \cdots R_{\lambda_{\sigma (4k) } \lambda_{\sigma (4k+1)} e_{2}}{}^{e_{k-1}}, \notag \end{align}\tag{4}\] for \(\sigma\) a permutation of \(\{1,\ldots,4k+1\}\), and where \(R_{ijk}^{\;\;\;\ell}\) are the components of the curvature tensor of \(g\) [4]. Then \(\hat{k}\) is \({\rm Isom}(M,g)\)-invariant.
Proposition 1. If \(\hat{k}\) is \(F\)-invariant, then \[d_{[0,1]\times M}F^{L,*}{\hat{\mathcal{K}}}=0.\]
Proof. We have \[\label{eq:5}(a_{(x^0,\theta)}^* {\hat{k}})_{j [i_1, \cdots, i_{n}] }(x) ={\hat{k}}_{\nu [\lambda_1 \cdots \lambda_n]} (a_{(x^0,\theta)}(x)) \frac{\partial a_{(x^0,\theta)}^\nu}{\partial x^j} \frac{\partial F^{\lambda_1}}{\partial x^{i_1}} \cdots \frac{\partial F^{\lambda_n}}{\partial x^{i_n}},\tag{5}\] where \(\partial a_{(x^0,\theta)}^\nu/\partial x^j\) is evaluated at \(x\), and the other partial derivatives are evaluated at \(a_{(x^0,\theta)}(x) = F(x^0,\theta,x).\) By Lem. 3, we have \[\begin{align} F^{L,*} d_{LM} \hat{\mathcal{K}}(\partial_{x^0} , \partial_{x^1}, \cdots,\partial_{x^n}) &= \int_0^{2\pi} \frac{\partial \alpha^i}{\partial \theta} \hat{k}_{\lambda_0 [\lambda_1 \cdots \lambda_n]} \frac{\partial F^{\lambda_0}}{\partial x^i} \frac{\partial F^{\lambda_1}}{\partial x^1} \cdots \frac{\partial F^{\lambda_n}}{\partial x^n} d \theta \\ &= \left(\int_0^{2\pi} \frac{\partial \alpha^i}{\partial \theta} d\theta\right) \cdot \hat{k}_{i [\lambda_1 \cdots \lambda_n]}(x) \notag \\ &=0. \notag \end{align}\] Applying Lem. 1 to \(F^L:[0,1]\times M\to LM\) gives \(d_{[0,1]\times M}F^{L,*}{\hat{\mathcal{K}}}=0.\) ◻
We now give a general formulation of [2].
Proposition 2. Let \(a_0, a_1:S^1\times M\to M\) be smooth maps such that \(a_1^D(\theta), a_2^D(\theta) \in \mathcal{G}(M)\), for all \(\theta\in S^1\).
(i) Let \(F:[0,1]\times S^1\times M\to M\) be a smooth homotopy from \(a_0\) to \(a_1\) with \(a(x^0,\theta)\in \mathcal{G}(M)\) and \(a(x^0,\theta)(\partial M) \subset \partial M\) for all \((x^0,\theta)\in [0,1]\times S^1\). If \(\hat{k}\) is \(F\)-invariant, then \(\int_{M} a_0^{L,*}{\hat{\mathcal{K}}} = \int_{M} a_1^{L,*}{\hat{\mathcal{K}}}\).
(ii) Let \(a :S^1 \times M \to M\) be a smooth action with \(a^D(\theta)\in \mathcal{G}(M)\) for all \(\theta\in S^1\). If \(\int_M a^{L,*} {\hat{\mathcal{K}}} \neq 0,\) then \(\pi_1(\mathcal{G}(M))\) is infinite.
Proof. (i) We apply Stokes’ Theorem, which is valid for \([0,1]\times M\), which may be a manifold with corners [14]. For \(i_{x^0}:M\to [0,1]\times M, i_{x^0}(m) = (x^0,m),\) we have \[\begin{align} \int_{M} a_1^{L,*}{\hat{\mathcal{K}}} - \int_{M} a_0^{L,*}{\hat{\mathcal{K}}} &= \int_{M} i_1^*F^{L,*}{\hat{\mathcal{K}}} - \int_{M} i_0^* F^{L,*}{\hat{\mathcal{K}}}\\ & = \int_{[0,1]\times M} d_{[0,1]\times M} F^{L,*}{\hat{\mathcal{K}}} =0, \end{align}\] by Prop. 1.
(ii) Let \(a_n\) be the \(n^{\rm th}\) iterate of \(a\), i.e. \(a_n(\theta,m) = a(n\theta,m).\) We claim that \(\int_{M}a_n^{L,*}{\hat{\mathcal{K}}} = n\int_{M} a^{L,*}{\hat{\mathcal{K}}}\). By (2 ), every term in \({\hat{\mathcal{K}}}\) is of the form \(\int_{0}^{2\pi} \dot{\gamma}(\theta) f(\theta)\), where \(f\) is a periodic function on the circle. Each loop \(\gamma\in a^L_1(M)\) corresponds to the loop \(\gamma(n\cdot)\in a^L_n(M).\) Therefore the term \(\int_{0}^{2\pi} \dot{\gamma}(\theta) f(\theta)\) is replaced by \[\int_{0}^{2\pi} \frac{d}{d\theta}\gamma(n\theta) f(n\theta)d\theta = n\int_0^{2\pi} \dot{\gamma}(\theta)f(\theta)d\theta.\] Thus \(\int_{M}a_n^{L,*}{\hat{\mathcal{K}}} = n\int_{M}a^{L,*}{\hat{\mathcal{K}}}.\) By (i), \(a_n\) and \(a_m\) are not homotopic in \(\mathcal{G}(M).\) By a straightforward modification of [2], the \([a^L_n]\in \pi_1(\mathcal{G}(M))\) are all distinct. ◻
We now simplify the calculation of \(a^{L,*}\hat{\mathcal{K}}\) for actions. For \(\hat{k}\in \Lambda^1(M)\otimes \Lambda^n(M)\) and \(\xi\in \Gamma(TM)\), we have the contraction \(\hat{k}\cdot \xi = \hat{k}_{j [i_1, \cdots, i_n]}\xi^j\in \Lambda^n(M).\)
Lemma 4. Let \(a: S^1 \times M \to M\) be a smooth action with associated vector field \(\xi\in \Gamma(TM)\), \[\xi^\nu_m = \frac{\partial a^\nu(\theta,m)}{\partial \theta}\biggl|_{\theta = 0}.\] If \(\hat{k}\) is \(a(\theta,\cdot)\)-invariant for all \(\theta\in S^1,\) then \[a^{L,*}\hat{\mathcal{K}}= 2\pi \hat{k}\cdot\xi \in \Lambda^n(M).\]
Proof. Since \(a\) is an action, we have \(a(\theta + \theta^\prime , m) =a(\theta, a(\theta^\prime, m))\), which implies \[\xi^\nu_m = \frac{\partial}{\partial \theta} a^\nu (\theta ,m)\biggl|_{\theta=0} = \frac{\partial a^\nu}{\partial x^j}\biggl|_{(\theta,m)} \frac{\partial a^j}{\partial \theta^\prime}{\biggl|_{\theta^\prime =0}} = \frac{\partial a^\nu}{\partial x^j}\biggl|_{(\theta,m)}\xi^j_m.\] Therefore, \[\begin{align} a^{L,*} \mathcal{K} &= \int_0^{2\pi} {\hat{k}}_{\nu [\lambda_1 \cdots \lambda_n]}\xi^j \frac{\partial a^\nu}{\partial x^j} \frac{\partial a^\lambda_1}{\partial x^{i_1}} \cdots \frac{\partial a^\lambda_n}{\partial x^{i_n}} = 2\pi \hat{k}\cdot \xi, \notag \end{align}\] where we write \(a(\theta,\cdot)^*\hat{k} = \hat{k}\) in local coordinates as in (5 ) to see that the integrand is independent of \(\theta.\) ◻
Combining Prop. 2(ii) and Lem. 4 gives the main method to detect if \(|\pi_1(\mathcal{G}(M))| = \infty.\)
Theorem 2. Let \(M\) be a closed oriented \(n\)-dimensional smooth manifold, and let \(\mathcal{G}(M)\) be a subgroup of \({\rm Diff}(M).\) Let \(\hat{\mathcal{K}}\in \Lambda^n(LM)\) be defined as in (2 ) with kernel \(\hat{k}\) as in (1 ). If there is a smooth action \(a:S^1\times M\to M\) with associated vector field \(\xi\) such that (i) \(a(\theta,\cdot) \in \mathcal{G}(M)\) for all \(\theta\in S^1\), (ii) \(a(\theta,\cdot)^*\hat{k} = \hat{k}\) for all \(\theta\in S^1\), (iii) \(\int_M a^{L,*}\mathcal{K}= 2\pi \int_M \hat{k}\cdot\xi \neq 0,\) then \[|\pi_1 (\mathcal{G}(M) )| = \infty.\]
Remark 1. We will use a modified version of this result for regular contact manifolds. It is easy to check that if we replace (ii) in the Theorem with \(a(\theta,\cdot)^*\hat{k} = C\cdot \hat{k}\) for a nonzero constant \(C\), and (iii) with \(\int_M a^{L,*}\mathcal{K}= 2\pi C \int_M \hat{k}\cdot\xi \neq 0,\) then the proof carries over.
We will apply this Theorem to various groups \(\mathcal{G}(M)\) in §§3-5. The only real issue is finding a kernel \(\hat{k}\) which is \(\mathcal{G}(M)\)-invariant such that (iii) holds.
For a Riemannian manfold \((M,g)\), the group of the conformal diffeomorphisms of \((M,g)\) is \[\label{conf2} {\rm Conf}(M, g) =\{ \psi \in {\rm Diff}(M) | \psi^* g = f g \; \text{for some}\;f \in C^\infty (M), f>0 \}.\tag{6}\]
Let \(g_{st}\) be the standard metric on \(S^{4k+1}\). For the Hopf fibration \(\pi: S^{4k+1} \to \mathbb{CP}_{2k}\), the unit vector field \(\xi\) along the fiber is the Reeb vector field for the standard contact structure. The contact one-form \(\eta\) is the dual of \(\xi\): \[\eta (\xi) =1, \; d\eta (\xi , \cdot )=0.\] For a real parameter \(\rho \geq 0\), we take a new metric on \(S^{4k+1}\) \[g_\rho := g_{st} +\rho^2 \eta \otimes \eta.\]
In [5], we proved that the \(|\pi_1({\rm Isom}(S^{4k+1},g_\rho))| = \infty\) iff \(\rho>0.\) In this section, we study \(\pi_1({\rm Conf}(S^{4k+1}, g_\rho))\) by choosing an appropriate kernel in (1 ). Namely, we have
Theorem 3. \[| \pi_1 ({\rm Conf}(S^{4k+1},g_\rho))| = \infty,\] for \(\rho \neq 0.\)
This results fails if \(\rho =0\): \({\rm Conf}(S^{n-1}, g_{st})\) is diffeomorphic to \(SO(n-1,1)\), which has the homotopy type of its maximal compact subgroup \(SO(n-1).\) Thus \(\pi_1({\rm Conf}(S^{n-1}, g_{st})) \simeq \mathbb{Z}_2,\) a type of “discontinuity" in the fundamental group as \(\rho\to 0.\)
Proof. We verify the three conditions in Thm. 2 for \(\mathcal{G}(S^{4k+1}) = {\rm Conf}(S^{4k+1},g_\rho)\). For the action \(a\), we take \(a(\theta,\cdot)\) to be the rotation by angle \(\theta\) in the circle fibers of \(\pi.\) This is an action by isometries [5], so it is an action by conformal diffeomorphisms. Thus (i) holds.
To define \(\hat{k}\) and verify (ii), we compute the Weyl tensor of \(g_\rho.\) Recall that \(S^{4k+1}\) has the standard Sasakian structure \((g_{st}, \phi, \xi,\eta)\) where \(\phi\) is the odd dimensional analogue of an almost complex structure: \(\phi_i{}^k \phi_k{}^j = -\delta_i{}^j + \eta_i \xi^j\) [5]. Let \(R_{kji}{}^h\) and \(\bar{R}_{kji}{}^h\) be the curvature tensors of \(g=g_{st}\) and \(g_{\rho},\) respectively. By [5], \[\begin{align} \label{eq:a} R_{kji}{}^h &= g_{ki} \delta_j{}^h - g_{ji}\delta_k{}^h \\ \bar{R}_{kji}{}^h &= R_{kji}{}^h -\rho^2( \phi_{ki}\phi_{j}{}^{h} -\phi_{k}{}^{h} \phi_{ji} +2\phi_{kj} \phi_{i}{}^{h} +2\eta_k \eta_i \delta_j{}^h -2 \eta_j \eta_i \delta_k{}^{h} +g_{ki}\eta_j \xi^h -g_{ji}\eta_k \xi^h)\notag \\ &\qquad -\rho^4(\eta_k \eta_i \delta_j{}^{h} -\eta_j \eta_i \delta_k{}^{h} ). \notag \end{align}\tag{7}\] This implies \[\begin{align} \label{eq:b} \bar{R}_{ji} &=R g_{ji}-\rho^2(\phi_{ki}\phi_{j}{}^{k} -\phi_{k}{}^{h} \phi_{ji}+2\phi_{kj} \phi_{i}{}^{k} +2\eta_i \eta_j -2(4k+1) \eta_i \eta_j+\eta_j \eta_i -g_{ji})\notag \\ &\qquad -\rho^4(\eta_j \eta_i -(4k+1) \eta_j \eta_i) \\ &= (R^2+4\rho^2) g_{ij} +((4k-1) \rho^2 +4k\rho^4)\eta_j \eta_i,\notag \end{align}\tag{8}\] where \(\bar{R}_{ji}= \bar{R}_{kji}{}^k, \bar{R} = h^{ji}\bar{R}_{ji}\) are the Ricci tensor and scalar curvature of \(g_\rho\), respectively, and \(R = (4k+1)(4k)\) is the scalar curvature of \(g\). The Weyl curvature tensor for \(\bar{R}\) is \[\begin{align} \label{eq:Weyl} \bar{C}_{kji}{}^h &= \bar R_{kji}{}^h +\frac{1}{4k-1} (\bar{R}_{ki} \delta_j{}^h -\bar{R}_{ji}\delta_k^h +g_{ki} \bar{R}_j{}^h -g_{ji}\bar{R}_k{}^h) \\ &\qquad - \frac{\bar{R}}{(4k)(4k-1)} (g_{ki} \delta_j^h -g_{ji}\delta_k{}^h ). \notag \end{align}\tag{9}\] Plugging (7 ) and (8 ) into (9 , we have \[\begin{align} \label{eq:ten} \bar{C}_{kji}{}^h &= -\rho^2 (\phi_{ki}\phi_{j}{}^{h} -\phi_{k}{}^{h} \phi_{ji} +2\phi_{kj} \phi_{i}{}^{h} ) \\ &+c_1(g_{ki} \delta_j{}^h - g_{ji}\delta_k{}^h) + c_2 (\eta_k \eta_i \delta_j{}^h - \eta_j \eta_i \delta_k{}^{h} ) +c_3 (g_{ki}\eta_j \xi^h -g_{ji}\eta_k \xi^h) \notag \end{align}\tag{10}\] where \(c_1, c_2, c_3\) are explicit nonzero constants depending on \(\rho\).
We set \[\hat{k}^{{\rm conf}}_{j i_1 \cdots i_{4k+1}} =\bar{C}_{i_1 \ell_1 j}{}^{\ell_0} \bar{C}_{i_2 i_3 \ell_2}{}^{\ell_1} \cdots \bar{C}_{i_{4k} i_{4k+1} \ell_0}{}^{\ell_{2k}}.\] This is the conformal version of (4 ); the similar expressions [5] were used to prove \(|\pi_1({\rm Isom}(S^{4k+1}, g_\rho))| = \infty\) in [5]. Since the Weyl tensor is invariant under conformal transformations of the metric, so is \(\hat{k}^{{\rm conf}}\). Therefore, (ii) is verified: \[a(\theta,\cdot)^* \hat{k}^{{\rm conf}} = \hat{k}^{{\rm conf}}.\]
On the loop space \(LM = \{ \gamma: S^1 \longrightarrow M\}\), we consider tangent vectors \(X_{1}, \cdots, X_{4k+1} \in T_{\gamma}(LM) = \Gamma(\gamma^*TM)\) and define \(\mathcal{K}^{\rm conf}\in \Lambda^n(LM)\) by \[{\mathcal{K}^{{\rm conf}}} (X_1,\ldots, X_n)_\gamma =\int_0^{2\pi} \hat{k}^{{\rm conf}}_{j[i_1 \cdots i_{4k+1}]}(\gamma (\theta) )\dot{\gamma}^j(\theta) X_{1}^{i_1}(\theta) \cdots X_{ 4k+1}^{i_{4k+1}} \;d\theta\] In [5], we computed that for \(M = S^{4k+1},\) we have \(\hat{k} = C_\rho \eta\wedge (d\eta)^{2k}\) for \(\hat{k}\) in (4 ) and for some nonzero constant \(C_\rho\). The Weyl tensor has the same symmetries as the Riemann curvature tensor, and the terms in (10 ) are the same as in [5], so the proof of [5] carries over to \(\hat{k}^{\rm conf}.\) Thus \(\int_M a^{L,*}\mathcal{K}^{\rm conf}\neq 0\) by Lem. 4. This verifies (iii). ◻
In this section, we discuss the the fundamental groups of geometric transformation groups which preserve certain one-forms. In §4.1, we prove that the group of strict contactomorphisms has infinite fundamental group (Thm. 4). In §4.2, we relax the conditions on the contact one-form to prove similar results for other groups of diffeomorphisms. In §4.3, we discuss pseudo-Hermitian transformations. In §4.4, we consider transformations of \(\mathbb{R}^{2k}\) which preserve a standard one-form, and in §4.5 we generalize this to the cotangent bundle of a closed manifolds. The groups in these subsections are infinite dimensional, except in §4.3.
Let \((M, \eta)\) be a \((2k+1)\)-dimensional connected closed contact manifold, where \(\eta\) is the contact one-form. We assume that \((M, \eta)\) is regular, i.e., its Reeb vector field \(\xi\), characterized by \[\label{xi} \eta(\xi) =1, \;d\eta (\xi, \cdot )=0,\tag{11}\] has closed orbits.
Let \[{\rm Diff}_{\eta, {\rm str}} (M) =\{ \phi \in {\rm Diff}(M); \phi^* \eta = \eta \}\] be the group of strict contactomorphisms.
Theorem 4. Let \((M,\eta)\) be a \((2k+1)\)-dimensional closed regular contact manifold. Then \[|\pi_1({\rm Diff}_{\eta, {\rm str}} (M))| = \infty.\] In particular, the homotopy clsss of the Reeb flow in \(\pi_1({\rm Diff}_{\eta, {\rm str}}(M))\) has infinite order.
As mentioned in the Introduction, this was first proved in [11].
Proof. As in §2, we set \[\begin{align} \hat{k}^{\eta} &= \eta\otimes \left(\eta \wedge (d\eta)^k\right), \\ \hat{\mathcal{K}}^{\eta} (X_{\gamma, 1}, \cdots , X_{\gamma, 2k+1})_\gamma &= \int_0^{2\pi} \hat{k}^{\eta}(\gamma(\theta))_{j[i_1 \cdots i_{2k+1}]} X_{ 1}{}^{i_1} (\theta) \cdots X_{2k+1}{}^{i_{2k+1}}(\theta)d\theta \in \Lambda^{2k+1}(LM). \end{align}\] where \(\gamma \in LM\) and \(X_{ 1}, \cdots, X_{ 2k+1} \in T_{\gamma} (LM)\).
We now find an action \(a_{\rm rot}:S^1\times M\to M\) satisfying Conditions (i) – (iii) in Thm. 2, as modified in Rem. 1. This will complete the proof.
Let \(\psi_t (m)\) be the one-parameter group generated by \(\xi.\) After choosing a metric associated with the contact structure (see [5]), we can apply Wadsley’s theorem [15] to conclude that there exists \(N>0\) such that \(N\) is an integral multiple of the period of each Reeb orbit. Therefore, we can modify the flow to \(\bar\psi_t(m) := \psi_{(2\pi)^{-1} N t}(m)\) to get an \(S^1\) action \[a_{\rm rot}: S^1 \times M \to M,\;a_{\rm rot}(\theta , m) = \bar\psi_\theta (m).\] It follows from (11 ) and the Cartan formula for the Lie derivative that \(\mathcal{L}_{\xi} \eta =0.\) As in Rmk. 1, \(a_{\rm rot}(\theta,\cdot)^* \eta =(N/2\pi)\eta\) (Condition (i)) which implies \(a_{\rm rot}(\theta,\cdot)^*\hat{k}^\eta = C\cdot\hat{k}^\eta\) for some \(C\neq 0.\) (Condition (ii)). For Condition (iii), by Lem. 4, \[\int_M (a_{\rm rot}^L)^* {\hat{\mathcal{K}}} = 2\pi C \int_M \hat{k}^\eta\cdot\xi = 2\pi C\int_M \eta(\xi)\;\eta \wedge (d\eta)^k =2\pi C\int_M \eta \wedge (d\eta)^k \neq 0.\] ◻
The proof of Thm.4 immediately carries over to more general setups. Let \(M\) be a closed, connected, oriented smooth \((2k+1)\)-manifold, and let \(\eta\) be a one-form on \(M\). Assume there is a vector field \(\xi\) on \(M\) that satisfies the following:
The flow of the vector field \(\xi\) is periodic with period independent of the orbit.
\(L_{\xi} \eta =0.\)
\(\int_M \eta(\xi) \eta \wedge (d \eta)^{2k} \neq 0.\)
Set \({\rm Diff}_{\eta} (M) = \{\phi\in {\rm Diff}(M): \phi^*\eta = \eta\}.\)
Theorem 5. Under the assumptions (A1)-(A3), we have \[|\pi_1 ({\rm Diff}_{\eta} (M))| = \infty.\]
We give an example satisfying (A1) – (A3). Let \(T^3 = S^1 \times S^1 \times S^1\) be the 3-torus with coefficients \(u = (u^1, u^2, u^3)\). Set \[\eta (u) = \eta_1 (u^2) du^1 + \eta_3 (u^2) du^3, \; \xi (u) = \partial_{u^1} .\] Clearly, (A1) holds, and it is easily checked that \(L_{\xi} \eta =0.\) Noting that \[d\eta = \frac{\partial \eta_1}{\partial u^2} du^2 \wedge du^1 + \frac{\partial \eta_3}{\partial u^2} du^2 \wedge du^3,\] we have \[\int_{T^3} \eta (\xi) \eta \wedge d\eta = (2\pi)^2 \int_0^{2\pi} \eta_1(u^2)\left(\eta_1(u^2) \frac{\partial \eta_3}{\partial u^2} - \eta_3 (u^2)\frac{\partial \eta_1}{\partial u^2} \right)du^2.\] For \(\eta_1 (u^2) = \cos u^2 , \eta_3 (u^2) = \sin 2 u^2,\) we get \[\begin{align} \int_{T^3} \eta (\xi) \eta \wedge d\eta &= (2\pi)^2 \int_0^{2\pi} \left(2 (\cos u^2)^2 \cos 2u^2 -\cos u^2 \sin 2 u^2 \sin u^2\right)du^2= 2\pi^3\neq 0. \end{align}\]
Thus, we get
Corollary 1. Let \(T^3\) be a 3-torus with coordinates \((u^1, u^2,u^3)\), let \(\eta (u) = \eta_1(u^2) du^1 + \eta_3 (u^2) du^3\), and let \(\xi = \partial_{u^1}.\) Then \(|\pi_1 ({\rm Diff}_{\eta} (T^3))|= \infty\). Specifically, the loop of diffeomorphisms given by rotation in the \(u^1\) direction has infinite order in \(\pi_1 ({\rm Diff}_{\eta} (T^3))\).
In a second direction, we can replace \(\eta\wedge (d\eta)^k\) with a general top degree form. Let \(M\) be an oriented closed \(C^\infty\) \(n\)-manifold. We choose \(\eta\in \Lambda^1 (M)\) and \(\mu \in \Lambda^n (M)\). We note that \(\mu\) is not necessarily a volume form on \(M\).
Set \({\rm Diff}_{\mu, \eta }(M) =\{\phi \in {\rm Diff}(M): \phi^*\eta = \eta, \phi^*\mu = \mu.\}\).
If \(\mu\) is a volume form, then the Lie algebra of \({\rm Diff}_{\mu} (M) = \{\phi\in{\rm Diff}(M): \phi^*\mu = \mu\}\) is the space of divergence-free vector fields, which is infinite dimensional. We expect that \({\rm Diff}_{\eta} (M)\) and \({\rm Diff}_{\mu, \eta } (M)\) are also infinite dimensional.
We assume that there is a vector field \(\xi\) on \(M\) such
The flow of the vector field \(\xi\) is periodic with period independent of the orbit.
\(L_{\xi} \eta = L_\xi \mu = 0.\)
\(\int_M \eta (\xi ) \mu \neq 0\).
Then, we have
Theorem 6. Under the assumptions (B1) – (B3), we have \(|\pi_1 ({\rm Diff}_{\mu, \eta} (M)) | = \infty\).
Proof. We take \[\hat{k} = \eta\otimes \mu \in \Lambda^1 (M) \otimes \Lambda^n (M).\] It is clear that \(\hat{k}\) is invariant under the group \({\rm Diff}_{\mu, \eta} (M)\). Since \(\int_M \hat{k}\cdot \xi = \int_M {\hat{k}}_{j [i_1 \cdots i_n] } \xi^j \neq 0\), Lem. 4 and Thm. 2 give the result. ◻
We give a simple example that Theorem 6 holds on the torus \(T^2 = S^1 \times S^1\) with coordinates \((u^1, u^2)\). The flow of \(\xi = \partial_{u^1}\) satisfies (B1). Set \[\label{eq:mueta} \mu = du^1 \wedge du^2, \;\eta(u) = \eta_1 (u^2) du^1 + \eta_2 (u^2) du^2,\tag{12}\] where \(\eta_1(u^2) >0\) on \(T^2\). Then \(L_\xi \eta =0\), and \(L_\xi \mu =0\). Note that \[\int_{T^2} \eta (\xi ) \mu = \int_{T^2} \eta_1 (u^2) du^1 \wedge du^2 =2\pi \int_0^{2\pi} \eta_1(u^2) du^2 > 0.\] Thus, (B1) – (B3) are satisfied, and we have
Corollary 2. For the choice of \(\mu\) and \(\eta\) in (12 ), we have \(|\pi_1 ({\rm Diff}_{\mu, \eta} (T^2))| = \infty\).
We discuss the transformation group of a psuedo-Hermitian structure (CR structure) on a closed regular contact manifold. Let \(M\) be a closed regular contact manifold with contact form \(\eta\). Assume that there exists a complex structure \(J\) on the contact bundle \(\text{Ker}\; \eta\) and that the Levi form \(d\eta \circ J\) is a positive definite Hermitian form. Then, \(( \eta, J)\) is called a psuedo-Hermitian structure on \(M\). For the Riemannian metric \(g := d\eta \circ J + \eta \otimes \eta\) on \(M\), the group of pseudo-Hermitian transformations of \(M\) is \[{\rm Psh}(M) = \{ h \in {\rm Diff}(M) \, | \, h^* \eta = \eta, h_* \circ J = J \circ h_*: \text{ on } \text{Ker}\, \eta \}.\] We note that \[\label{psh95isom} {\rm Psh}(M) \subset {\rm Isom}(M, g).\tag{13}\]
We have the following result.
Theorem 7. Let \(M\) be a \((2k+1)\)-dimentional closed regular contact manifold with a psuedo-Hermitian structure. Assume that the Reeb vector field \(\xi\) defined by \(\eta (\xi) =1\) and \(d\eta (\xi , \cdot)=0\) generates a periodic one-parameter transformation group of psuedo-Hermitian transformations. Then \(|\pi_1({\rm Psh}(M)|= \infty\).
Proof. We take as kernel function \[\label{kernel95psh} \hat{k}^{{\rm Psh}} = \eta\otimes {\rm dvol}_g,\tag{14}\] where \({\rm dvol}_g\) is the volume form for \(g\). By (13 ), the kernel function defined by (14 ) is preserved by pseudo-Hermitian transformations. Note that \[{\mathcal{K}}^{psh} = \int_{M} \hat{k}^{{\rm Psh}}\cdot\xi = {\rm vol}(M) \neq 0.\] Thus, Thm. 4 gives the result. ◻
Let \(\mathbb{R}^{2k}\) be Euclidean \(2k\)-space with the one-form \[\alpha = \frac{1}{2} \sum_{i=1}^k x^i d\xi^i -\xi^i dx^i,\] where \(z=(z^1, \cdots, z^k)\) are the coordinates on \(\mathbb{R}^{2k}= \mathbb{C}^k\) and \(z^i=(x^i, \xi^i)\). We note that \(d\alpha = \omega\) is the standard symplectic form on \(\mathbb{R}^{2k}.\)
For \(\overset{\circ}{\mathbb{R}}{}^{2k}= \mathbb{R}^{2k} - \{0\}\), \(\phi \in {\rm Diff}(\mathbb{R}^{2k})\) is of homogeneous degree one if \[\phi (rx,r\xi) = r\cdot \phi(x,\xi) \, \text{ for } r>0.\]
Let \({\rm Diff}^{(1)} (\mathbb{R}^{2k})\) be the subgroup of \({\rm Diff}(\mathbb{R}^{2k})\) consisting of homogeneous degree one diffeomorphisms. \({\rm Diff}^{(1)}(\mathbb{R}^{2k})\) contains the subgroup \[\begin{align} {\rm Diff}^{(1)}_{\alpha, {\rm str}} (\mathbb{R}^{2k}) &= \{ \phi \in {\rm Diff}^{(1)} (\mathbb{R}^{2k}) \,| \phi^* \alpha = \alpha\}. \end{align}\] This group is an infinite dimensional Lie group with good differential structures e.g., ILH-structures, Fréchet structures, etc. [10].
Let \(S^{2k-1}\) be the unit sphere of \((2k-1)\) dimensional with the origin as the center, and \(i: S^{2k-1} \longrightarrow \mathbb{R}^{2k}\) be the standard embedding of the unit sphere \(S^{2k-1}\) into \(\mathbb{R}^{2k}\). We define a map \(A:{\rm Diff}(S^{2k-1})\to {\rm Diff}^{(1)}(\mathbb{R}^{2k})\) by \((A\hat{\phi})(r\hat{x}, r\hat{\xi}) = r \hat{\phi}(\hat{x}, \hat{\xi})\), for \((\hat{x}, \hat{\xi})\in S^{2k-1}\) and \(\hat{\phi}\in {\rm Diff}(S^{2k-1}).\) \(A\) is clearly not surjective. We define \[\begin{align} {\rm Diff}^{(1)}_{\alpha, {\rm str}, A} (\mathbb{R}^{2k}) &= \{ \phi \in {\rm Diff}^{(1)}_{\alpha, {\rm str}}(\mathbb{R}^{2k}) \, | \, \phi\in {\rm Im(A)} \}. \end{align}\]
Note that \(\bar{\alpha}: = i^* \alpha\) gives a contact structure on \(S^{2k-1}\). Let \(\phi \in {\rm Diff}^{(1)}_{\alpha, {\rm str},A}(\mathbb{R}^{2k})\). We have \[\phi^*(r\alpha) = r\hat{\phi}^*(\bar{\alpha}) ,\] where \(\hat{\phi}(\hat{x}, \hat{\xi})= \phi \circ i(\hat{x}, \hat{\xi})\). Thus, \(\hat{\phi} \in {\rm Diff}_{\bar{\alpha},\text{str}}(S^{2k-1})\), which implies
Lemma 5. \[{\rm Diff}^{(1)}_{\alpha, {\rm str}, A}(\mathbb{R}^{2k}) = {\rm Diff}_{\bar{\alpha}, {\rm str}} (S^{2k-1} ).\]
Since \(S^{2k-1}\) is a regular contact manifold, Thm. 4 implies
Corollary 3. \[|\pi_1({\rm Diff}^{(1)}_{\alpha, {\rm str}, A} (\mathbb{R}^{2k} ))| = \infty.\]
Considering \(\mathbb{R}^{2k}\) as \(T^*\mathbb{R}^k\), we have a similar situation for the cotangent bundle of a symplectic manifold. Let \((M^k,\omega)\) be a closed, connected \(C^\infty\) symplectic manifold with cotangent bundle \(\pi:T^*M \longrightarrow M.\) On \(\overset{\circ}{T^*M} := T^*M - \{\text{zero-section}\}\), we define the canonical/contact one-form \(\alpha = \sum_{i=1}^k \xi^i dx^i\) in local coordinates \((x, \xi)\) on \(T^*M\).
Let \({\rm Diff}^{(1)} (\overset{\circ}{T^*M})\) be the group of diffeomorphisms of \(\overset{\circ}{T^*M}\) of homogeneous degree one in the fiber direction; i.e., if we write \(\phi \in {\rm Diff}^{(1)} (\overset{\circ}{T^*M})\) as \[\phi (x,\xi) = (\phi_1 (x,\xi ) , \phi_2 (x,\xi)),\] where \(\phi_1,\) resp. \(\phi_2\), involve only \(x\), resp. \(\xi\), coordinates, then \[\label{phi22} \phi_2(x, r\xi) = r\phi_2(x, \xi) \text{ for } r>0.\tag{15}\] Since \(\phi_2\) changes by a function of \(M\) only under a change of coordinates on \(M\), (15 ) is independent of local coordinates.
We set \[{\rm Diff}^{(1)}_{\alpha, {\rm str}} (\overset{\circ}{T^*M}) = \{ \phi \in {\rm Diff}^{(1)} (\overset{\circ}{T^*M}) \, | \phi^* \alpha =\alpha \}.\] This is also an infinite dimensional Lie group, since for a \(C^\infty\) diffeomorphism \(f: M \longrightarrow M\), we have (15 ) for \((df)^*\) (the adjoint of the differential \(df\)), and \((df)^{*} \alpha = \alpha\) by the cotangent bunlde lift theorem [16]. (Here we abuse notation by using \((df)^*\) instead of \((df)^{**}\) for the pullback on one-forms associated to \((df)^*\).)
A choice of metric \(g\) on \(M\) gives an inner product on each cotangent fiber and allow us to define the unit cosphere bundle \(S^*M= \{ (x, \xi) \in \overset{\circ}{T^*M}| \;|\xi|_g =1 \}\). We note that for the inclusion \(i:S^*M\longrightarrow T^*M\), \(\bar\alpha := i^*\alpha\) is a contact form on \(S^*M\). We set \[{\rm Diff}^{(1)}_{\alpha, {\rm str}, g} (\overset{\circ}{T^*M}) = \{ \phi \in {\rm Diff}^{(1)}_{\alpha, {\rm str}} (\overset{\circ}{T^*M}) | \; |(x,\xi)|_g = 1\Rightarrow| \phi_2(x,\xi) |_g=1 \}.\]
Let \(a:S^1\times M\to M\) be a smooth \(S^1\)-isometric action: i.e., \(a\) is a smooth action, and \(a^D:S^1 \to {\rm Diff}(M)\), defined by \(a^D(\theta)(x) := a(\theta,x),\) has \(a^D(\theta)\in {\rm Isom}(M,g)\) for all \(\theta\). Then we have:
Corollary 4. Let \((M^k,g,\omega)\) be a closed connected \(C^\infty\) Riemannian \(k\)-manifold with a smooth \(S^1\)-isometric action on \((M,g)\). Then \[|\pi_1 ({\rm Diff}^{(1)}_{\alpha, {\rm str},g} (\overset{\circ}{T^*M})| = \infty.\]
Proof. Since \(a^D\) is an \(S^1\) action on \({\rm Diff}(M),\) the adjoint \((da^D)^*\) of the differential \(da^D\) is an \(S^1\) action on \(T^*M\). Since \((da^D)^*\) is linear in each fiber, \((da^D)^*\) gives an action on \({\rm Diff}^{(1)}_{\alpha,A} (\overset{\circ}{T^*M})\); this uses \((da^D)^*\alpha = \alpha\) as above. As in Lem. 5, \[{\rm Diff}^{(1)}_{\alpha, {\rm str},g} (\overset{\circ}{T^*M}) = {\rm Diff}_{\bar \alpha, {\rm str}}(S^*M),\] where \(S^*M\) is the unit cotangent bundle and \(\bar\alpha = i^*\alpha\) for the inclusion \(i:S^*M \longrightarrow T^*M.\)
For each \(\theta\), \(|(x,\xi)|_g = 1\) implies \(|da^D(\theta)(x.\xi)|_g = 1\), since the action is via isometries. Therefore, \((da^D)^*\) descends to an \(S^1\) action \(\hat{a}\) on \({\rm Diff}_{\bar \alpha, {\rm str}}(S^*M)\) satisfying condition (iii) in Thm. 2: \(\int_{S~*M} \hat{a}^{L,*}\mathcal{K} \neq 0,\) where \(\mathcal{K}\) has kernel \(k= \bar{\alpha} \otimes \bar{\alpha} \wedge (d\bar{\alpha})^k.\) By Thm. 2, \(\pi_1({\rm Diff}_{\bar \alpha, {\rm str}}(S^*M))\) is infinite, which gives the result. ◻
We give an application to an interesting subgroup of the Lie algebra of the Poisson algebra of smooth functions on a symplectic manifold \(N\). The main reference for this subsection is [17].
Let \((N,\omega)\) be a closed symplectic \(2k\)-dimensional manifold with a symplectic form \(\omega\), which is is integrable, i.e., \([\omega] \in H^2(N, \mathbb{Z}).\) Then there is an \(S^1\)-bundle \(\pi: (M, \eta) \longrightarrow (N,\omega)\), where \((M, \eta)\) is a contact manifold with \(\pi^* (\omega) = d\eta\). As usual, for a smooth function \(H(x,\xi)\) on \((N,\omega)\), we define the Hamiltonian vector field \(X_H\) by \(\omega( X_H, \cdot) = dH,\) and define the Poisson bracket \(\{ \;, \;\}\) by \(\{ H , H^\prime \} = X_H H^\prime.\) It is standard that \((C^\infty (N), \{ \, , \, \})\) is an infinite-dimensional Lie algebra.
We consider a vector field \(V_f\) on \((M, \eta)\) associated with a smooth function \(f \in C^\infty (M)\) defined by \[\eta (V_f) = -f , \quad d\eta ( V_f , \cdot ) = df\] It is easily seen that \(L_{V_f} \eta = 0\), so \(V_f\) is by definition a strict contact vector field.
Let \(\mathcal{X}_{\eta , {\rm str}}(M)\) be the Lie algebra of strict contact vector fields on \((M, \eta)\). For any \(V \in \mathcal{X}_{\eta , st} (M)\), there is a smooth function \(f\) such that \(V=V_f\), so \[\mathcal{X}_{\eta, {\rm str}} (M) = \{ V_f | f \in C^\infty (M) \}.\] For \(H \in C^\infty (N)\), we denote by \(H^L \in C^\infty (M)\) the lift of \(H\), i.e., \(H^L = \pi^* H\), and set \[\mathcal{X}_0 (M) = \{ V_{H^L} | H \in C^\infty (N) \}.\] It is easily seen that \(\mathcal{X}_0 (M)\) is a closed Lie algebra of \(\mathcal{X}_{\eta ,{\rm str}}.\)
We define the contact diffeomorphism \(\phi = \phi_H = \exp( V_{H^L})\) on \((M, \eta)\). We set \(G_0(M)\) to be the Lie group which is finitely generated by \(\exp (V_{H^L})\), and let \(\bar{G}_o(M)\) be the closure of \(G_0(M)\) in \({\rm Diff}_{\alpha , {\rm str}} (M)\).
This procedure gives a Lie group \(\bar{G}_0(M)\), which we call the Lie group of Hamiltonian symplectic transformations (cf. [17]), whose Lie algebra is a subalgebra of \((C^\infty (N), \{ \cdot , \cdot \})\).
Theorem 8. Let \((N, \omega)\) be an integral closed symplectic manifold. Then \[|\pi_1 (\bar{G}_0 (M)) | = \infty.\]
Proof. We follow the proof of Theorem 4, using \(\hat{K}^\eta.\) To show (i) and (ii) in Theorem 2, it is enough to use \(\phi^* \alpha =\alpha\). To show (iii), we take \(H=1\). ◻