General Connections, Exponential Maps, and Second-order Differential Equations

TL;DR

This paper develops a unified geometric framework for all connections, extending classical concepts like geodesics and exponential maps to nonlinear settings, and introduces vertically homogeneous connections to include Finsler spaces.

Contribution

It introduces a comprehensive theory of connections via second-order differential equations, extending classical results and including Finsler spaces through vertically homogeneous connections.

Findings

Extended Ambrose-Palais-Singer correspondence

Locally diffeomorphic exponential maps for all connections

Full theory of nonlinear covariant derivatives

Abstract

The main purpose of this article is to introduce a comprehensive, unified theory of the geometry of all connections. We show that one can study a connection via a certain, closely associated second-order differential equation. One of the most important results is our extended Ambrose-Palais-Singer correspondence. We extend the theory of geodesic sprays to certain second-order differential equations, show that locally diffeomorphic exponential maps can be defined for all, and give a full theory of (possibly nonlinear) covariant derivatives for (possibly nonlinear) connections. In the process, we introduce vertically homogeneous connections. Unlike homogeneous connections, these complete our theory and allow us to include Finsler spaces in a completely consistent manner. This is an expanded version of the article published in Differ. Geom. Dyn. Syst. 13 (2011) 72--90. Included are the proof published in Nonlinear Anal. 63 (2005) e501--e510 (for the reader's convenience) and some new material on homogeneity.