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Vector Fields on Manifolds

  • Authors
  • Michael F. Atiyah

Part of the Arbeitsgemeinschaft für Forschung des Landes Nordrhein-Westfalen book series (AFLNW, volume 200)

Table of contents

  1. Front Matter
    Pages 1-5
  2. Michael F. Atiyah
    Pages 7-26
  3. Back Matter
    Pages 27-30

About this book

Introduction

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).

Keywords

Area DEX Euler characteristic Invariant Signatur Vector field boundary element method character differential operator eXist index theory manifold symmetry theorem vector bundle

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-322-98503-3
  • Copyright Information Springer Fachmedien Wiesbaden GmbH 1970
  • Publisher Name VS Verlag für Sozialwissenschaften, Wiesbaden
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-322-97941-4
  • Online ISBN 978-3-322-98503-3
  • Buy this book on publisher's site
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