Structures of flat piecewise Riemannian 2-polyhedra

The object of our research is a piecewise Riemannian 2-polyhedron which is a combinatorial 2-polyhedron such that each 2-simplex is isometric to a triangle bounded by three smooth curves on some Riemannian 2-manifold. In the previous paper [4], which is a joint work with J. Itoh, we have introduced the concept of total curvature for piecewise Riemannian 2-polyhedra and proved a generalized Gauss-Bonnet theorem and a generalized Cohn-Vossen theorem. In this paper, we shall give a definition of flatness of piecewise Riemannian 2-polyhedra and characterize them.