Lie algebra of killing vector fields and its stationary subalgebra

V. A. Popov

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference "Geometric Methods in Control Theory and Mathematical Physics" dedicated to the 70th anniversary of S.L. Atanasyan, the 70th anniversary of I.S. Krasilshchik, the 70th anniversary of A.V. Samokhin, and the 80th anniversary of V.T. Fomenko. S.A. Esenin Ryazan State University, Ryazan, September 25–28, 2018. Part 2, Tome 169 (2019), pp. 56-66

Résumé

Let $\mathfrak{g}$ be the Lie algebra of all Killing vector fields on a locally homogeneous, analytic Riemannian manifold $M$, $\mathfrak{h}$ be a stationary subalgebra of $\mathfrak{g}$, $G$ be the simply connected group generated by the algebra $\mathfrak{g}$, $H$ be the subgroup of $G$ generated by the subalgebra $\mathfrak{h}$, $\mathfrak{z}$ be the center of the algebra $\mathfrak{g}$, $\mathfrak{r}$ be its radical, and $[\mathfrak{g};\mathfrak{g}]$ be its commutator subgroup. If $\dim\big(\mathfrak{h}\cap\big(\mathfrak{z} + [\mathfrak{g}, \mathfrak{g}] \big)\big) = \dim \big(\mathfrak{h} \cap [\mathfrak{g}, \mathfrak{g}]\big)$, then $H$ is closed in $G$. If for any semisimple subalgebra $\mathfrak{p}\subset\mathfrak{g}$ satisfying the condition $\mathfrak{p}+\mathfrak{r}=\mathfrak{g}$, the relation $(\mathfrak{p}+\mathfrak{z})\cap\mathfrak{h} =\mathfrak{p}\cap\mathfrak{h}$ holds, then $H$ is closed in $G$. We also examine the analytic continuation of the given local, analytic Riemannian manifold.

Détail

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