Time-dependent metrics and connections

TL;DR

The paper develops a geometric framework for time-dependent metrics and connections on a manifold by working on the product manifold $\mathbb{R}\times M$ and employing the suspension construction. It introduces a time-dependent covariant derivation operator $\mathcal{D}=(\nabla,C,A,B)$, defines parallel transport and geodesics in this setting, and introduces the time-dependent torsion operator $\mathcal{T}^{\mathcal{D}}$, linking these notions to a generalized covariant calculus. Euler–Lagrange analysis yields geodesic equations for $g_t$ and connects them to the Levi-Civita connection of the suspension, while an explicit double pendulum example with variable masses demonstrates how time dependence enters dynamics. The work lays groundwork for future work in curvature, second variation, and broader physical applications, aiming to extend classical Riemannian concepts to genuinely time-dependent geometric structures.

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.

Theorems & Definitions (29)

• Proposition 2.1
• proof
• Definition 2.2
• Proposition 2.3
• Definition 3.1
• Example 3.2
• Proposition 4.1
• Definition 4.2
• Proposition 4.3
• Proposition 4.4