Lie Algebra of Killing Vector Fields and its Stationary Subalgebra

• Published in: Journal of mathematical sciences (New York, N.Y.) Vol. 263; no. 3; pp. 404 - 414
• Main Author: Popov, V. A.
• Format: Journal Article
• Language: English
• Published: New York Springer US 04.05.2022; Springer; Springer Nature B.V
• Subjects: Algebra; Commutators; Fields (mathematics); Lie groups; Mathematical analysis; Mathematics; Mathematics and Statistics; Riemann manifold; Subgroups; analytic continuation; 53C20; vector field; Lie group; Riemannian manifold; Lie algebra; 54H15; closed subgroup
• ISSN: 1072-3374; 1573-8795; 1573-8795
• DOI: 10.1007/s10958-022-05937-2

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.