Lie Algebra of Killing Vector Fields and its Stationary Subalgebra - Popov, V. A. (Journal of mathematical sciences (New York, N.Y.), 2022)
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.