Vector Fields on Manifolds
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 v
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180. SITZUNG AM 4. JUNI 1969 IN DüSSELDORF
ARBEITSGEMEINSCHAFT FüR FORSCHUNG DES LANDES NORDRHEIN-WESTFALEN NATUR-, INGENIEUR- UND GESELLSCHAFTSWISSENSCHAFTEN HEFT 200
MICHAEL F. ATIYAH Vector Fie1ds on Manifolds
HERAUSGEGEBEN IM AUFTRAGE DES MINISTERPRASIDENTEN HEINZ KüHN VON STAATSSEKRETAR PROFESSOR Dr. h. c. Dr. E. h. LEO BRANDT
MICHAEL F. ATIYAH
Vector Fields on Manifolds
Springer Fachmedien Wiesbaden GmbH
ISBN 978-3-322-97941-4 ISBN 978-3-322-98503-3 (eBook) DOI 10.1007/978-3-322-98503-3
© 1970 by Springer Fachmedien Wiesbaden Ursprünglich erschienen bei Westdeutscher Verlag, Köln und Opladen 1970 Gcsamtherstel1ung: WcatdcutscheJ: Verlag GmbH •
Inhalt Michael F. Atiyah, Ph. D., F.R.S., Savilian Professor of Geometry, Oxford University*
Vector Fields on Manifolds §1 §2 §3 §4 §5
Inttoduction .......................................... Clifford algebras and differential forms .................... Euler characteristic and signature ........................ Kervaire semi-characteristic ............................. Vector fields with finite singularities ......................
7 8 11 15 18
References ...................................................
24
Zusammenfassung ............................................
25
Resume ................................. ,...................
26
* Now at the Institute for Advanced Study,
Princeton.
§ 1 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 existence 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 extensions 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). Then the Euler characteristic of X is wen and is congruent to the signature of X modulo 4. Of course, as a corollary of (1.1), we deduce that the existence of 2 independent vector fields implies that the signature of X is divisible by 4. Generalizing this
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