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Almost contact metric manifolds whose Reeb vector field is a harmonic section

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Abstract

We investigate almost contact metric manifolds whose Reeb vector field is a harmonic unit vector field, equivalently a harmonic section. We first consider an arbitrary Riemannian manifold and characterize the harmonicity of a unit vector field ξ, when ∇ξ is symmetric, in terms of Ricci curvature. Then, we show that for the class of locally conformal almost cosymplectic manifolds whose Reeb vector field ξ is geodesic, ξ is a harmonic section if and only if it is an eigenvector of the Ricci operator. Moreover, we build a large class of locally conformal almost cosymplectic manifolds whose Reeb vector field is a harmonic section. Finally, we exhibit several classes of almost contact metric manifolds where the associated almost contact metric structures σ are harmonic sections, in the sense of Vergara-Diaz and Wood [25], and in some cases they are also harmonic maps.

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  1. Dipartimento di Matematica “E. De Giorgi”, Università del Salento, 73100, Lecce, Italy

    Domenico Perrone

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  1. Domenico Perrone
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Correspondence to Domenico Perrone.

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Supported by funds of the University of Salento.

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Perrone, D. Almost contact metric manifolds whose Reeb vector field is a harmonic section. Acta Math Hung 138, 102–126 (2013). https://doi.org/10.1007/s10474-012-0228-1

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  • DOI: https://doi.org/10.1007/s10474-012-0228-1

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