Uniform stability of linear evolution equations with applications to parallel transports (Q2281309)

The author considers the linear evolution equation in the Banach space \(E\) \[ x'=A(t) x. \tag{1} \] The author proves bistability of this equation for the following operator-valued function \(A\): \[ A : I \to \mathcal{L}(E), \ A(t) = f'(t).G(t)(f(t)), \] where \(I \subset \mathbb{R}\), \(\mathcal{L}(E)\) is the Banach space of all bounded linear operators acting in \(E\), \(G : I \to \mathcal{L}(E)\) is a Bochner integrable and essentially bounded Bochner measurable function and \(f\) is a locally absolutely continuous scalar function. Two geometric applications of this stability result are presented. First, it is shown that the parallel transport along a curve in a manifold, with respect to some linear connection, is bounded by terms of the length of the projection of the curve to a manifold of one dimension lower. Finally, an extendability result for parallel sections in vector bundles is proved.