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1983 (Publication #40)

Critical points and curvature for embedded polyhedra II

Differential Geometry, Proc. Special Year, Maryland, Progress in Math 32, Birkh■user (1983), 34-55.

ABSTRACT

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.