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Conformal actions of simple Lie groups on compact pseudo-Riemannian manifolds. (English) Zbl 1076.53083

By considering the group of conformal transformations on a manifold and by introducing the concept of compact manifolds with a bilinear structure, not necessarily symmetric or even nondegenerate, the authors determine first a lower bound on the dimension of isotropic subspaces by proving:
Theorem 1. Let \(G\) be a connected almost simple real Lie group with finite center. Assume \(G\) acts conformally (and nontrivially) on a compact manifold with a bilinear structure \(M\). Then there exists some point \(m\in M\), where the bilinear form on \(T_m(M)\) has an isotropic subspace of dimension at least \(rk_{\mathbb{R}}(G)- 1\).
Then, they determine the so-called standard models of conformal action on a compact pseudo-Riemannian manifold of signature \((p,q)\), for example \(\text{SO}(p,q)\) acting on \(C^{p,q}\), or \(\text{SL}(3,\mathbb{R})\) on \(\mathbb{P}^2\), and prove the following:
Theorem 2. With \(G\) and \(M\) as in the previous theorem, assume that the maximum dimension of an isotropic subspace of \(T_m(M)\) is at most \(rk_{\mathbb{R}}(G)- 1\), for all \(m\in M\). Then:
\(\bullet\) \(G\) must be locally isomorphic to either \(\text{SO}(p,q)\), or \(\text{SL}(3,\mathbb{R})\).
\(\bullet\) There exists a closed \(G\)-orbit in \(M\).
\(\bullet\) Any closed \(G\)-orbit in \(M\) is equivariantly and con formally diffeomorphic to a standard model.
Further conditions and results are also discussed and proven in the paper, like minimality of the action and the case of symplectic manifolds, with the 2-form not necessarily closed.

MSC:

53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
57S20 Noncompact Lie groups of transformations
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