Riemannian geometry as a curved pre-homogeneous geometry

TL;DR

This paper explores Riemannian geometry through the lens of pre-homogeneous geometric structures, establishing a link between curvature and constant curvature metrics.

Contribution

It introduces a novel perspective by defining Riemannian structures as pre-homogeneous geometries and characterizes flatness via curvature.

Findings

R=0 iff the metric has constant curvature

Pre-homogeneous structures generalize Riemannian geometry

Poses open problems in the field

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.