Research on the structure of ideal boundaries of piecewise Riemannian polyhedra from the viewpoint of total curvature

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The field of our study is piecewise Riemannian 2-polyhedra, which is a combinatorial 2-polyhedra. such that each 2-simplex is isometric to a triangle bounded by three smooth curves on some Riemannian 2-manifold. The purpose in this research project is to investigate some properties of noncompact piecewise Riemannian 2-polyhedra from the viewpoint of total curvature.Main results in this project is the following.1."Total excess on length surfaces"by Y.Machigashira and F.Ohtsuka : On length spaces satisfying some suitable conditions, which may contain some discrete singular points, we generalize the notion of total curvature (we call it total excess in this paper) and prove some generalized theorems.2."Total curvature of noncompact piecewise Riemannian 2-polyhedra"by J.Itoh and F.Ohtsuka : We introduce two kind of definitions of total curvature (total curvature and w-total curvature) on piecewise Riemannian 2-polyhedra and obtain some generalized theorems for them admitting total curvature and observe the difference of total curvature and w-total curvature.3."Structure of flat piecewise Riemannian 2-polyhedra"by F.Ohtsuka : We give a definition of flatness of piecewise Riemannian 2-polyhedra and characterize them.Henceforth I would like to investigate the structure of 2-polyhedra of constant curvature and the relationship between polyhedra and its ideal boundaries and so on.