Time-dependent metrics and connections

TL;DR

This paper explores the mathematical framework of time-dependent structures in differential geometry, focusing on covariant derivatives, connections, and geodesics on product manifolds with applications to non-autonomous differential equations.

Contribution

It introduces a general approach to time-dependent covariant derivatives, parallel transport, and geodesics, extending classical concepts to time-varying geometric settings.

Findings

Defined derivatives of one-parameter families of connections

Analyzed properties of time-dependent geodesics and torsion

Extended the notion of parallel transport to time-dependent contexts

Abstract

Time-dependent structures often appear in differential geometry, particularly in the study of non-autonomous differential equations on manifolds. One may study the geodesics associated with a time-dependent Riemannian metric by extremizing the corresponding energy functional, but also through the introduction of a more general concept of time-dependent covariant derivative operator. This relies on the examination of connections on the product manifold $\mathbb{R}\times M$. For these time-dependent covariant derivatives we explore the notions of parallel transport, geodesics and torsion. We also define the derivative of a one-parameter family of connections.