Riemannian geometry as a curved pre-homogeneous geometry

Abstract

We define a Riemannian structure as a pre-homogeneous geometric structure with curvature R. We show that R = 0 if and only if the underlying metric has constant curvature. We define pre-homogeneous geometric structures and pose some problems.

Curvature without connection

“…The purpose of this paper is to give a detailed analysis of an affine and Riemannian PHG showing that the above mentioned technical assumption is redundant, a fact stated also in [Or1] without proof. This paper can be regarded also as a revision and extension of [Or2].…”