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Critical associated metrics on contact manifolds III

Part of: Variational problems in infinite-dimensional spaces Global differential geometry

Published online by Cambridge University Press:  09 April 2009

David E. Blair
David E. Blair
Affiliation:
Michigan State UniversityEast Lansing, Michigan 48824-1027, U.S.A.
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Abstract

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In the first paper of this series we studied on a compact regular contact manifold the integral of the Ricci curvature in the direction of the characteristic vector field considered as a functional on the set of all associated metrics. We showed that the critical points of this functional are the metrics for which the characteristic vector field generates a 1-parameter group of isometries and conjectured that the result might be true without the regularity of the contact structure. In the present paper we show that this conjecture is false by studying this problem on the tangent sphere bundle of a Riemannian manifold. In particular the standard associated metric is a critical point if and only if the base manifold is of constant curvature +1 or −1; in the latter case the characteristic vector field does not generate a 1-parameter group of isometries.

MSC classification

Secondary: 58E11: Critical metrics 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.)

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 50 , Issue 2 , April 1991 , pp. 189 - 196
Copyright
Copyright © Australian Mathematical Society 1991

References

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