Uniform Stability of Linear Evolution Equations with Applications to Parallel Transports

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, thus answering a question by Antonio J. Di Scala.