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 formfor 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 γ in a manifold, with respect to some linear connection, is bounded in terms of the length of the projection of γ to a manifold of one dimension lower. Secondly, I prove an extendability result for parallel sections in vector bundles, thereby answering a question byis a null set since J \ J 0 is a null set; see [2, Proposition 1.2.2 a)]. Therefore {t ∈ A : f ′ (t) = 0} is a null set by means of Lemma 2.3. Since f ∈ AC(I), the set of points of I at which f is not differentiable is a null set [8, Proposition 3.8]. Taking into account that