Nonlinear stability of the totally geodesic wave maps in non-isotropic manifolds

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.