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Pseudo-Riemannian weakly symmetric manifolds. (English) Zbl 1237.53071
The authors develop a theory of weakly symmetric pseudo-Riemannian manifolds and show that many results in the Riemannian case are generalized to the pseudo-Riemannian one.
A pseudo-Riemannian manifold \(M\) is called weakly symmetric if for any tangent vector \(X \in T_xM\), there is an isometry which preserves the point \(x\) and transforms \(X\) into \(-X\).
The authors prove that a weakly symmetric pseudo-Riemannian manifold is a homogeneous geodesically complete pseudo-Riemannian manifold and give a group-theoretic characterization of such manifolds. They study geodesics in a pseudo-Riemannian weakly symmetric space and prove that, like in the Riemannian case, they are orbits of one-parameter groups of isometries. The structure of the nilradical of the isometry group of a weakly symmetric pseudo-Riemannian space is discussed.
Several examples of pseudo-Riemannian weakly symmetric spaces are given. Some of them are similar to the Riemannian case and others are different and indicate the problems arising when extending Riemannian results to weakly symmetric pseudo-Riemannian spaces.

MSC:
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53C35 Differential geometry of symmetric spaces
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