A general formulation for geodesic distance propagation of surfaces is presented. Starting from a surface lying on a 3-manifold in IR4, we set up a partial differential equation governing the propagation of surfaces at equal geodesic distance (on the 3-manifold) from the given original surface. This propagation scheme generalizes a result of Kimmel et al. [11] and provides a way to compute distance maps on manifolds. Moreover, the propagation equation is generalized to any number of dimensions. Using an eulerian formulation with level-sets, it gives stable numerical algorithms for computing distance maps. This theory is used to present a new method for surface matching which generalizes a curve matching method [5]. Matching paths are obtained as the orbits of the vector field defined as the sum of two distance maps' gradient values. This surface matching technique applies to the case of large deformation and topological changes.