##
Critical Points and Curvature for Embedded Polyhedra

*Journal of Differential Geometry *

Vol. 1, No. 3, Sept. 1967, pp. 245-256

This paper establishes a connection between the total curvature of
embedded polyhedra and critical poiont theory for non-degenerate height
functions. A new interpretation of the relationship between intrinsic
and extrinsic curvature is also formulated.

A generalization of the classical critical point theorem for non-
degenerate functions on smooth manifolds is proven, and this is
used to show that the curvature of an even-dimensional differentiable
manifold depends only on intrinsic quantities.

An intrinsic form of the Guass-Bonnet theorem is proven, and these results
are generalized to other complexes and maps.