On essential conformal groups and a conformal invariant
The group \(\text{Diff}(M)\) of \(C^{\infty}\) diffeomorphisms of a manifold \(M\) acts on conformal classes of tensor fields \(\gamma\) on \(M\), since if \(\tau, \tau'\in \gamma\), i.e., \(\tau' =\lambda \tau\), then \(\varphi^{*}\tau' = (\lambda\circ\varphi)\varphi^{*}\tau\) for all \(\varphi\in \text{Diff}(M)\). Let \(\text{Diff}_{\gamma}(M)\) be the group of automorphisms of conformal class \(\gamma\). Let \(\text{Diff}_{\alpha}(M)\) be the group of diffeomorphisms preserving tensor field \(\alpha\). For any conformal structure \(\gamma\) and \(\alpha \in \gamma\), we have: \(\text{Diff}_{\alpha}(M)\subset \text{Diff}_{\gamma}(M)\). Definition. The conformal group \(\text{Diff}_{\gamma}(M)\) is said to be inessential if there exists \(\tau_{*}\in\gamma\) such that \(\text{Diff}_{\tau_{*}}(M)= \text{Diff}_{\gamma}(M)\). It is said to be essential otherwise. Theorem 1. Consider the following conformal manifolds \((M,\gamma)\): 1. \((M,\gamma)\) a contact manifold. 2. \((M,\gamma)\) any of these conformal symplectic manifolds of \(\dim(M)\geq 4\): (2.a) \(M=\mathbb{R}^{2n}\), \(\gamma=[\tau]\), \(\tau=dx_1\wedge dx_2+\dots +dx_{2n-1}\wedge dx_{2n}\). (2.b)\ \(M\) a Stein manifold with complex structure \(J\) and \(\gamma=[\omega_{\varphi}]\), where \(\omega_{\varphi}\) is the symplectic form \(d(J^{*}d\varphi)\), defined by a strictly \(J\)-convex function \(\varphi:M\to \mathbb{R}\) and more generally, any non compact symplectic manifold with a complete Liouville field. 3. \((M,\gamma)\) the Lorentz space \(H^{1.1}=\{(x,y,z)\in \mathbb{R}^3\mid x^2+y^2-z^2=r^2\}\) with the Lorentz metric \(g=dx^2+dy^2-dz^2\). Then the conformal group \(\text{Diff}_{\gamma}(M)\) is essential. Let \((M,\gamma)\) be a conformal structure and \(\tau \in \gamma\). Given \(\varphi\in \text{Diff}_{\gamma}(M)\) we let \(\lambda_{\varphi}\) be the smooth positive function such that \(\varphi^{*}\tau=\lambda_{\varphi}\tau\). The mapping \(D_{\tau}: \text{Diff}_{\gamma}(M)\rightarrow C^{\infty}(M)\), \(\varphi\to \ln(\lambda_{\varphi^{-1}})\) is a 1-cocycle on \(\text{Diff}_{\gamma}(M)\) with values in the \(\text{Diff}_{\gamma}(M)\)-module \(C^{\infty}(M)\) and its cohomology class in \(H^1(\text{Diff}_{\gamma}(M),C^{\infty}(M))\) is a conformal invariant \(\delta(\gamma,M)\). Theorem 2. The group \(\text{Diff}_{\gamma}(M)\) is inessential if and only if \(\delta(\gamma,M)=0\).
scientific article; zbMATH DE number 1507990