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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 2polyhedra and characterize them. §1.Introduction. "Curvature" is one of the most important tools to investigate "Geometry" of manifolds. For our research object "polyhedra," the concept of "Curvature" has been introduced and remarkable results are obtained by Banchoff [3] for any dimensional compact piecewise linear polyhedra, and by Ballman-Brin [1] and Ballman-Buyalo [2] for 2-dimensional cocompact piecewise Riemannian polyhedra. My interest is based particularly on the study of noncompact case from the view point of total curvature. In our previous paper [4] with J. Itoh, we have defined two kinds of total curvature for noncompact piecewise Riemannian 2-polyhedra, total curvature and weak total curvature, which both coincide with the usual definitions for Riemannian manifolds or compact 2-polyhedra. It is naturally and easily seen that a generalized Gauss-Bonnet theorem holds under these total curvatures. Furthermore, in [4], we have shown the difference between the geometric meanings of these two kinds of total curvature, and under the assumption of admitting total curvature (not weak total curvature) we have proved a generalized Cohn-Vossen theorem. The aim of my research is to clarify the meaning of "Curvature" of polyhedra and characterize them in terms of curvature. In this paper, as a first step of this research direction, we shall define the flatness of polyhedra and classify Partially supported by Grant-in-aid for Scientific Research (C) No. 13640060, Japan Society for the Promotion of Science. Received May 11, 2004. 2000 Mathematics Subject Classification. Primary: 53C23, Secondary: 57M20.

Erratum to: “Structures of flat piecewise Riemannian 2-polyhedra”