Contraction theory on Riemannian manifolds

Contraction theory is a methodology for assessing the stability of trajectories of a dynamical system with respect to one another. In this work, we present the fundamental results of contraction theory in an intrinsic, coordinate-free setting, with the presentation highlighting the underlying geometric foundation of contraction theory and the resulting stability properties. We provide coordinate-free proofs of the main results for autonomous vector fields, and clarify the assumptions under which the results hold. We state and prove several interesting corollaries to the main result, study cascade and feedback interconnections of contracting systems, study some simple examples, and highlight how contraction theory has arisen independently in other scientific disciplines. We conclude by illustrating the developed theory for the case of gradient dynamics.