Locally Homogeneous Manifolds Defined by Lie Algebra of Infinitesimal Affine Transformations

Abstract

This article deals with Lie algebra G of all infinitesimal affine transformations of the manifold M with an affine connection, its stationary subalgebra ℌ⊂G, the Lie group G corresponding to the algebra G, and its subgroup H⊂G corresponding to the subalgebra ℌ⊂G. We consider the center ℨ⊂G and the commutant [G,G] of algebra G. The following condition for the closedness of the subgroup H in the group G is proved. If ℌ∩ℨ+G;G=ℌ∩[G;G], then H is closed in G. To prove it, an arbitrary group G is considered as a group of transformations of the set of left cosets G/H, where H is an arbitrary subgroup that does not contain normal subgroups of the group G. Among these transformations, we consider right multiplications. The group of right multiplications coincides with the center of the group G. However, it can contain the right multiplication by element 𝒽¯, belonging to normalizator of subgroup H and not belonging to the center of a group G. In the case when G is in the Lie group, corresponding to the algebra G of all infinitesimal affine transformations of the affine space M and its subgroup H corresponding to its stationary subalgebra ℌ⊂G, we prove that such element 𝒽¯ exists if subgroup H is not closed in G. Moreover 𝒽¯ belongs to the closures H¯ of subgroup H in G and does not belong to commutant G,G of group G. It is also proved that H is closed in G if P+ℨ∩ℌ=P∩ℌ for any semisimple algebra P∈G for which P+ℜ=G.