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Isometric actions on pseudo-Riemannian nilmanifolds. (English) Zbl 1295.53081
Authors’ abstract: This work deals with the structure of the isometry group of pseudo-Riemannian \(2\)-step nilmanifolds. We study the action by isometries of several groups and we construct examples showing substantial differences with the Riemannian situation; for instance, the action of the nilradical of the isometry group does not need to be transitive. For a nilpotent Lie group endowed with a left-invariant pseudo-Riemannian metric, we study conditions for which the subgroup of isometries fixing the identity element equals the subgroup of isometric automorphisms. This set equality holds for pseudo-\(H\)-type Lie groups.

MSC:
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53C30 Differential geometry of homogeneous manifolds
22E25 Nilpotent and solvable Lie groups
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
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