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In this article we investigate a type of totally geodesic map which has itsimage being a geodesic in an anisotropic Riemannian manifold. We consider itsnonlinear stability among the family of wave maps. We first establish thefactorization property and then formulate the stability problem into a PDEsystem in a specially constructed chart of geodesic normal coordinates. With ageneralization of the hyperboloidal foliation, we establish the globalexistence result associate to small initial data for this PDE system, whichleads to the geometric stability.

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