Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 39, No. 2, pp. 379-393 (1998)

The Angle Defect for Arbitrary Polyhedra

Ethan D. Bloch

Bard College, Annandale-on-Hudson, NY 12504, e-mail:

Abstract: The classical polyhedral Gauss-Bonnet Theorem for surfaces uses the angle defect to measure curvature. Using a natural stratification for all polyhedra (not necessarily manifolds), the angle defect is generalized to arbitrary polyhedra in all dimensions. A Gauss-Bonnet type theorem is then proved for arbitrary polyhedra, using a modified Euler characteristic based on this stratification rather than the standard Euler characteristic.

Keywords: polyhedra, Gauss-Bonnet Theorem, curvature, stratification, Euler characteristic

Classification (MSC91): 52A25

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