  For a bounded domain equipped with a piecewise Lipschitz continuous
Riemannian metric g, we consider harmonic map from $(\Omega, g)$ to a compact
Riemannian manifold $(N,h)\subset\mathbb R^k$ without boundary. We generalize
the notion of stationary harmonic map and prove the partial regularity. We also
discuss the global Lipschitz and piecewise $C^{1,\alpha}$-regularity of
harmonic maps from $(\Omega, g)$ manifolds that support convex distance
functions.
