Characterizations of Global Transversal Exponential Stability

We study the relationship between the global exponential stability of an invariant manifold and the existence of a positive semidefinite Riemannian metric which is contracted by the flow. In particular, we investigate how the following properties are related to each other (in the global case): 1) A manifold is globally "transversally" exponentially stable; 2) the corresponding variational system admits the same property; 3) there exists a degenerate Riemannian metric which is contracted by the flow and can be used to construct a Lyapunov function. We show that the transverse contraction rate being larger than the expansion of the shadow on the manifold is a sufficient condition for the existence of such a Lyapunov function. An illustration of these tools is given in the context of global full-order observer design.