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Critical associated metrics on contact manifolds. II

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

Published online by Cambridge University Press:  09 April 2009

D. E. Blair  and
A. J. Ledger
D. E. Blair
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, Michigan 48824, U.S.A.
A. J. Ledger
Affiliation:
Department of Pure Mathematics, University of Liverpool, Liverpool L69 3BX, United Kingdom
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Abstract

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The study of the integral of the scalar curvature, ∫MRdVg, as a function on the set of all Riemannian metrics of the same total volume on a compact manifold is now classical, and the critical points are the Einstein metrics. On a compact contact manifold we consider this and ∫M (R − R* − 4n2) dv, with R* the *-scalar curvature, as functions on the set of metrics associated to the contact structure. For these integrals the critical point conditions then become certain commutativity conditions on the Ricci operator and the fundamental collineation of the contact metric structure. In particular, Sasakian metrics, when they exist, are maxima for the second function.

MSC classification

Secondary: 58E11: Critical metrics 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 41 , Issue 3 , December 1986 , pp. 404 - 410
Copyright
Copyright © Australian Mathematical Society 1986

References

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