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1984 (Publication #45)

Normal Curvatures and Euler Classes for Polyhedral Surfaces in 4-Space

Proc. A.M.S.,92 (1984), 593-596.

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The total curvature of an immersed surface in n-space can be related to singularities of projections into lines. This allows for a unified treatment of curvature for both smooth surfaces and polyhedral surfaces. We show that 2-surfaces in 4-space admit an analogous treatment of normal curvature in terms of singularities of projections into oriented 3-space. This leads to a new curvature quantitiy for smooth immersions. We then give a function that assigns a real number to each vertex such that the normal Euler class of the immersion equals the sum of the normal curvatures at the vertices.