Normal Curvatures and Euler Classes for Polyhedral Surfaces in 4-Space
by Thomas F. Banchoff

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.