Riemannian Means as Solutions of Variational Problems

Abstract

We formulate a variational problem on a Riemannian manifold M whose solutions are piecewise smooth geodesies that best fit a given data set of time labelled points in M . By a limiting process, these solutions converge to a single point in M . which we prove to be the Riemannian mean of the given points for some particular Riemannian manifolds such as Euclidean spaces, connected and compact Lie groups, and spheres.