Vector fields on polyhedra
Author:
Michael A. Penna
Journal:
Trans. Amer. Math. Soc. 232 (1977), 131
MSC:
Primary 57D25; Secondary 58D99
MathSciNet review:
0451258
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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, evendimensional 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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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197704512586
PII:
S 00029947(1977)04512586
Keywords:
Polyhedron,
manifold,
tangent bundle,
vector field,
Hopf's theorem,
GaussBonnet formula
Article copyright:
© Copyright 1977
American Mathematical Society
