Uniform stability of linear evolution equations, with applications to parallel transports

TL;DR

This paper establishes the stability of certain linear evolution equations in Banach spaces and applies these results to geometric problems like parallel transport and vector bundle sections.

Contribution

It introduces a new stability result for linear evolution equations with operator functions of a specific form and applies it to problems in differential geometry.

Findings

Bounded solutions for the class of linear evolution equations are explicitly characterized.

Parallel transport along curves is bounded by the length of their projection.

An extendability result for parallel sections in vector bundles is proved.

Abstract

I prove the bistability of linear evolution equations $x' = A(t)x$ in a Banach space $E$, where the operator-valued function $A$ is of the form $A(t) = f'(t)G(t,f(t))$ for a binary operator-valued function $G$ and a scalar function $f$. The constant that bounds the solutions of the equation is computed explicitly; it is independent of $f$, in a sense. Two geometric applications of the stability result are presented. Firstly, I show that the parallel transport along a curve $\gamma$ in a manifold, with respect to some linear connection, is bounded in terms of the length of the projection of $\gamma$ to a manifold of one dimension lower. Secondly, I prove an extendability result for parallel sections in vector bundles, thereby answering a question by Antonio J. Di Scala.