Geometry of polyhedron from the view point of differential geometry

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The analogous results for polyhedron of Gauss's Theorema Egregium and Weyl's volume formula were proved and written in the 2-dimensional case. The Cohn-Vossen type inequality for 2-polyhedron, the total curvature of graphs and its tightness are written.There are the related new problems, for examples, the acute troangulations, the structure of essential cut locus, the length of cycles of cut locus, etc. All these problems are very exploratory and are expected to produce greate results by continuing studies.With respect to the acute triangulations, we proved that the cubed surface admitts an acute triangulations with 24 triangles, the icosahedral surface admitts an acute triangulations with 12 triangles, and these are the least numbers. 'The dodecahedral surface does not have any acute triangulations with triangles less than 11 and admits an acute triangulation with 14 triangles. Moreover we discussed several other cases and got some fundamental ideas how to treat the general convex surfaces.Withrespect to the essential cut locus, we define it in the case of a surface as the essential part of cut locus containing all critical points of distance function, and proved that the number of end points or the degree of vertices is related with several invariants of its inner metric. Moreover, we consider its structure in the case of convex polyhedron in general dimension.With respect to the length of cycles of cut locus, we proved that there is a point p on any torus with diameter 1 such taht the length of cycles in the cut locus of p is greater than 2. It is the best possible estimate and there is no upper bound.