Generalization of the notion of completeness of a Riemannian analytic manifold

V. A. Popov

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference "Classical and Modern Geometry" Dedicated to the 100th Anniversary of the Birth of Professor Vyacheslav Timofeevich Bazylev. Moscow, April 22-25, 2019. Part 3, Tome 181 (2020), pp. 84-101

Résumé

In this paper, we discuss the concept of an analytic continuation of a local Riemannian metric. We propose a generalization of the notion of completeness realized as an analytic continuation of an arbitrary Riemannian metric. Various Riemannian metrics are studied, primarily those related to the structure of the Lie algebra $\mathfrak{g}$ of all Killing vector fields for a local metric. We introduce the notion of a quasicomplete manifold, which possesses the property of extendability of all local isometries to isometries of the whole manifold. A classification of pseudocomplete manifolds of small dimensions is obtained. We present conditions for the Lie algebra of all Killing vector fields $\mathfrak{g}$ and its stationary subalgebra $\mathfrak{h}$ of a locally homogeneous pseudo-Riemannian manifold under which a locally homogeneous manifold can be analytically extended to a homogeneous manifold.

Détail

Keywords: Riemannian manifold, pseudo-Riemannian manifold, Lie algebra, analytic continuation, vector field, closed subgroup.
Mots-clés : Lie group