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On Compact Riemannian Manifolds with Zero Ricci Curvature

Published online by Cambridge University Press:  20 January 2009

T. J. Willmore
T. J. Willmore
Affiliation:
The Mathematical Institute, University of Liverpool.
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In this note I prove the following result:—

Theorem. A compact, orientable, Riemannian manifold Mn, with positive definite metric and zero Ricci curvature, is flat if the first Betti number R1 exceeds n — 4.

In this statement of the theorem it is assumed that the dimensions of Mn are not less than four. If this is not the case, the result is still valid but appears as a purely local result and is true for a metric of arbitrary signature.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 10 , Issue 3 , January 1956 , pp. 131 - 133
Copyright
Copyright © Edinburgh Mathematical Society 1956

References

Page 131 note 1 Eiscnhart, L. P., Riemannian Geonutry (1926), p. 91.Google Scholar

Page 132 note 1 Bochner, S., Annals of Math., 49 (1948), 379–390.CrossRefGoogle Scholar

Page 132 note 2 Lichnerowicz, A., C. B. Acad. Sci., 226 (1948), 1678–80.Google Scholar

Page 132 note 3 Lichnerowicz, A., C. R. Acad. Sci., 230 (1950), 2146–8.Google Scholar

Page 133 note 1 Walker, A. G., Proc. London Math. Soc., 52 (1950), 36–64.CrossRefGoogle Scholar