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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.