Abstract
In this paper, we discuss the concept of an analytic prolongation of a local Riemannian metric. We propose a generalization of the notion of completeness realized as an analytic prolongation of an arbitrary Riemannian metric. Various Riemannian metrics are studied, primarily those related to the structure of the Lie algebra 𝔤 of all Killing vector fields for a local metric. We introduce the notion of a quasi-complete manifold, which possesses the property of prolongability of all local isometries to isometries of the whole manifold. A classification of pseudo-complete manifolds of small dimensions is obtained. We present conditions for the Lie algebra of all Killing vector fields 𝔤 and its stationary subalgebra 𝔥 of a locally homogeneous pseudo-Riemannian manifold under which a locally homogeneous manifold can be analytically prolonged to a homogeneous manifold.
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 181, Proceedings of the International Conference “Classical and Modern Geometry” Dedicated to the 100th Anniversary of Professor V. T. Bazylev. Moscow, April 22-25, 2019. Part 3, 2020.
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Popov, V.A. Generalization of the Notion of Completeness of a Riemannian Analytic Manifold. J Math Sci 276, 776–792 (2023). https://doi.org/10.1007/s10958-023-06801-7
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DOI: https://doi.org/10.1007/s10958-023-06801-7
Keywords and phrases
- Riemannian manifold
- pseudo-Riemannian manifold
- Lie algebra
- analytic continuation
- vector field
- Lie group
- closed subgroup
- 53C20
- 54H15