Some elementary observations regarding reductive Cartan geometries

TL;DR

This paper generalizes fundamental concepts like covariant derivatives and geodesics from Riemannian geometry to reductive Cartan geometries, proving analogous elementary results including a concise version of the Hopf-Rinow theorem.

Contribution

It introduces new generalizations of covariant derivatives and geodesics for reductive Cartan geometries and proves elementary results similar to those in Riemannian geometry.

Findings

Generalized covariant derivatives and geodesics for reductive Cartan geometries

Proved elementary results analogous to Riemannian geometry

Provided a concise proof of the Hopf-Rinow theorem in this context

Abstract

After defining generalizations of the notions of covariant derivatives and geodesics from Riemannian geometry for reductive Cartan geometries in general, various results for reductive Cartan geometries analogous to important elementary results from Riemannian geometry are proven using these generalizations. In particular, a generalization of the Hopf-Rinow theorem is given with a pleasantly concise proof.