On transverse exponential stability and its use in incremental stability, observer and synchronization

We study the relation between the exponential stability of an invariant manifold and the existence of a Riemannian metric for which the flow is “transversally” contracting. More precisely, we investigate how the following properties are related to each other: i). A manifold is “transversally” exponentially stable; ii). The “transverse” linearization along any solution in the manifold is exponentially stable; iii). There exists a Riemannian metric for which the flow is “transversally” contracting. We show the relevance of these results in the study of incremental stability, observer design and synchronization.