Abstract
On a manifold with a given nowhere vanishing vector field, we examine the squared \(L^2\)-norm of the integrability tensor of the orthogonal complement of the field, as a functional on the space of Riemannian metrics of fixed volume. We compute the first variation of this action and prove that its only critical points locally are metrics with integrable orthogonal complement of the field, or metrics of contact metric structures rescaled by a function. Moreover, in dimensions other than 5, that function is constant and the above characterization is global. We examine the second variation of the functional at the critical points and estimate it for some geometrically meaningful sets of variations.
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Author would like to thank the anonymous referee, whose thorough review and thoughtful remarks helped to significantly improve this paper.
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Zawadzki, T. A variational characterization of contact metric structures. Ann Glob Anal Geom 62, 129–166 (2022). https://doi.org/10.1007/s10455-022-09842-4
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DOI: https://doi.org/10.1007/s10455-022-09842-4