Abstract
Let 𝔤 be the Lie algebra of all Killing vector fields on a locally homogeneous, analytic Riemannian manifold M, 𝔥 be a stationary subalgebra of 𝔤, G be the simply connected group generated by the algebra 𝔤, H be the subgroup of G generated by the subalgebra 𝔥, 𝔷 be the center of the algebra g, r be its radical, and [𝔤; 𝔤] be its commutator subgroup. If dim (𝔥 ∩ (𝔷 + [𝔤, 𝔤])) = dim (𝔥 ∩ [𝔤, 𝔤]), then H is closed in G. If for any semisimple subalgebra 𝔭 ⊂ 𝔤 satisfying the condition 𝔭 + 𝔯 = 𝔤, the relation (𝔭 + 𝔷) ∩ 𝔥 = 𝔭 ∩ 𝔥 holds, then H is closed in G. We also examine the analytic continuation of the given local, analytic Riemannian manifold.
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 169, Proceedings of the International Conference “Geometric Methods in the Control Theory and Mathematical Physics” Dedicated to the 70th Anniversary of Prof. S. L. Atanasyan, 70th Anniversary of Prof. I. S. Krasil’shchik, 70th Anniversary of Prof. A. V. Samokhin, and 80th Anniversary of Prof. V. T. Fomenko. Ryazan State University named for S. Yesenin, Ryazan, September 25–28, 2018. Part II, 2019.
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Popov, V.A. Lie Algebra of Killing Vector Fields and its Stationary Subalgebra. J Math Sci 263, 404–414 (2022). https://doi.org/10.1007/s10958-022-05937-2
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DOI: https://doi.org/10.1007/s10958-022-05937-2