ISBN-10:
3322979415
ISBN-13:
9783322979414
Pub. Date:
01/01/1970
Publisher:
VS Verlag für Sozialwissenschaften
Vector Fields on Manifolds

Vector Fields on Manifolds

by Michael Francis AtiyahMichael Francis Atiyah

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Overview

This paper is a contribution to the topological study of vector fields on manifolds. In particular we shall be concerned with the problems of exist­ ence of r linearly independent vector fields. For r = 1 the classical result of H. Hopf asserts that the vanishing of the Euler characteristic is the necessary and sufficient condition, and our results will give partial extens­ ions of Hopf's theorem to the case r > 1. Arecent article by E. Thomas [10] gives a good survey of work in this general area. Our approach to these problems is based on the index theory of elliptic differential operators and is therefore rather different from the standard topological approach. Briefly speaking, what we do is to observe that certain invariants of a manifold (Euler characteristic, signature, etc. ) are indices of elliptic operators (see [5]) and the existence of a certain number of vector fields implies certain symmetry conditions for these operators and hence corresponding results for their indices. In this way we obtain certain necessary conditions for the existence of vector fields and, more generally , for the existence of fields of tangent planes. For example, one of our results is the following THEOREM (1. 1). Let X be a compact oriented smooth manifold 0/ dimension 4 q, and assume that X possesses a tangent fteld of oriented 2-planes (that is, an oriented 2-dimensional sub-bundle 0/ the tangent vector bundle).

Product Details

ISBN-13: 9783322979414
Publisher: VS Verlag für Sozialwissenschaften
Publication date: 01/01/1970
Series: Arbeitsgemeinschaft für Forschung des Landes Nordrhein-Westfalen , #200
Edition description: 1970
Pages: 30
Product dimensions: 6.69(w) x 9.61(h) x 0.00(d)

Table of Contents

Vector Fields on Manifolds.- § 1 Introductio.- § 2 Clifford algebras and differential forms.- § 3 Euler characteristic and signature.- § 4 Kervaire semi-characteristic.- § 5 Vector fields with finite singularities.- References.- Zusammenfassung.- Résumé.

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