Vector fields on polyhedra

Author:
Michael A. Penna

Journal:
Trans. Amer. Math. Soc. **232** (1977), 1-31

MSC:
Primary 57D25; Secondary 58D99

DOI:
https://doi.org/10.1090/S0002-9947-1977-0451258-6

MathSciNet review:
0451258

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Abstract | References | Similar Articles | Additional Information

Abstract: This paper presents a bundle theory for studying vector fields and their integral flows on polyhedra and applications.

Every polyhedron has a tangent object in the category of simplicial bundles in much the same way as every smooth manifold has a tangent object in the category of smooth vector bundles. One can show that there is a correspondence between piecewise smooth flows on a polyhedron *P* and sections of the tangent object of *P* (i.e., vector fields on *P*); using this result one can prove existence results for piecewise smooth flows on polyhedra. Finally an integral formula for the Euler characteristic of a closed, oriented, even-dimensional combinatorial manifold is given; as a consequence of this result one obtains a representation of Euler classes of such combinatorial manifolds in terms of piecewise smooth forms.

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DOI:
https://doi.org/10.1090/S0002-9947-1977-0451258-6

Keywords:
Polyhedron,
manifold,
tangent bundle,
vector field,
Hopf's theorem,
Gauss-Bonnet formula

Article copyright:
© Copyright 1977
American Mathematical Society