  This paper proposes several Converse Lyapunov Theorems for nonlinear
dynamical systems defined on smooth connected Riemannian manifolds and
characterizes properties of corresponding Lyapunov functions in a normal
neighborhood of an equilibrium. We extend the methods of constructing of
Lyapunov functions for ordinary differential equations on $\mathds{R}^{n}$ to
dynamical systems defined on Riemannian manifolds by employing the differential
geometry. By employing the derived properties of Lyapunov functions, we
obtained the stability of perturbed dynamical systems on Riemannian manifolds.
The results are obtained by employing the notions of normal neighborhoods, the
injectivity radius on Riemannian manifolds and existence of bump functions on
manifolds.
