Some properties of the Riemannian distance function and the position vector X, with applications to the construction of Lyapunov functions

The quadratic distance function on a Riemannian manifold can be expressed in terms of the position vector, which in turn can be constructed using geodesic normal coordinates through consideration of the exponential map. The formulas for the derivative of the distance are useful to study Lyapunov stability of dynamical systems, and to build cost functions for optimal control and estimation.