  In this article we investigate a type of totally geodesic map which has its
image being a geodesic in an anisotropic Riemannian manifold. We consider its
nonlinear stability among the family of wave maps. We first establish the
factorization property and then formulate the stability problem into a PDE
system in a specially constructed chart of geodesic normal coordinates. With a
generalization of the hyperboloidal foliation, we establish the global
existence result associate to small initial data for this PDE system, which
leads to the geometric stability.
