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Arbeitsgemeinschaft für Forschung des Landes Nordrhein-Westfalen

Vector Fields on Manifolds

Authors: Atiyah, Michael Francis

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About this book

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).

Table of contents (1 chapters)

Table of contents (1 chapters)

Buy this book

eBook $74.99
price for USA in USD
  • ISBN 978-3-322-98503-3
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $99.00
price for USA in USD
  • ISBN 978-3-322-97941-4
  • Free shipping for individuals worldwide
  • Institutional customers should get in touch with their account manager
  • Covid-19 shipping restrictions
  • Usually ready to be dispatched within 3 to 5 business days, if in stock
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Bibliographic Information

Bibliographic Information
Book Title
Vector Fields on Manifolds
Authors
Series Title
Arbeitsgemeinschaft für Forschung des Landes Nordrhein-Westfalen
Series Volume
200
Copyright
1970
Publisher
VS Verlag für Sozialwissenschaften
Copyright Holder
Springer Fachmedien Wiesbaden
eBook ISBN
978-3-322-98503-3
DOI
10.1007/978-3-322-98503-3
Softcover ISBN
978-3-322-97941-4
Edition Number
1
Number of Pages
30
Topics