On closedness of stationary subgroup of affine transformations group

Abstract

This article deals with Lie algebra g of all infinitesimal affine transformations on a manifold with affine connection and its stationary subalgebra h ⊂ g. Let G be simply connected group generated by algebra g and H ⊂ G be the subgroup generated by subalgebra h ⊂ g and let dimg/h = dimM. Then if algebra g has zero center the subgroup H is closed in G. Thus any infinitesimal affine transformation X ⊂ g on a manifold M = G/H can be extended to affine transformation f: M → M. For Riemannian manifolds the condition dimg/h = dimM can be omitted and the main result can be generalized for algebra g with non-zero center.