Infinitesimally homogeneous manifolds with prescribed structure groups

TL;DR

This paper characterizes classes of infinitesimally homogeneous manifolds with specific structure groups, detailing their curvature, torsion, and inner torsion properties for various Lie groups.

Contribution

It provides a classification of infinitesimally homogeneous manifolds with specific structure groups, expanding understanding of their geometric properties.

Findings

Characterization of manifolds with identity structure group

Classification for finite and diagonal groups

Analysis of special linear, orthogonal, and unitary groups

Abstract

We explore the class of triples (M, nabla, P) where M is a manifold, nabla is an affine connection in M and P is a G-structure in M. Inside this class there are infinitesimally homogeneous manifolds, characterized by having G-constant curvature, torsion and inner torsion. For each matrix Lie group G subgroup of GL(Rn) there is a class of infinitesimally homogeneous manifolds with structure group G. In this paper we characterize the classes of infinitesimally homogeneous manifolds for some specific values of the structure group G including: identity group, finite groups, diagonal group, special linear group, orthogonal group and unitary group.