In this paper, we report several new geometric and Lyapunov characterizations of incremental stable systems on Finsler and Riemannian manifolds. First, a new and intrinsic proof of an important theorem in contraction analysis is given via the complete lift of a vector field... Based on this, two Lyapunov characterizations of incremental stable systems are derived. The first one is a converse contraction theorem, and the second one reveals a connection between incremental stability and stability of an equilibrium point. This result recovers and extends the classical Krasovskii's method. At the end, we show how to extend the results to discrete time systems. (read more)