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Levi Harmonic Maps of Contact Riemannian Manifolds

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Abstract

We study Levi harmonic maps, i.e., C ∞ solutions f:M→M′ to \(\tau_{\mathcal{H}} (f) \equiv \operatorname{trace}_{g} ( \varPi_{\mathcal{H}}\beta_{f} ) = 0\), where (M,η,g) is an (almost) contact (semi) Riemannian manifold, M′ is a (semi) Riemannian manifold, β f is the second fundamental form of f, and \(\varPi_{\mathcal{H}} \beta_{f}\) is the restriction of β f to the Levi distribution \({\mathcal{H}} = \operatorname{Ker}(\eta)\). Many examples are exhibited, e.g., the Hopf vector field on the unit sphere S 2n+1, immersions of Brieskorn spheres, and the geodesic flow of the tangent sphere bundle over a Riemannian manifold of constant curvature 1 are Levi harmonic maps. A CR map f of contact (semi) Riemannian manifolds (with spacelike Reeb fields) is pseudoharmonic if and only if f is Levi harmonic. We give a variational interpretation of Levi harmonicity. Any Levi harmonic morphism is shown to be a Levi harmonic map.

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Authors and Affiliations

  1. Dipartimento di Matematica, Informatica ed Economia, Università degli Studi della Basilicata, Via dell’Ateneo Lucano 10, 85100, Potenza, Italy

    Sorin Dragomir

  2. Dipartimento di Matematica e Fisica Ennio De Giorgi, Università del Salento, Via Provinciale Lecce-Arnesano, 73100, Lecce, Italy

    Domenico Perrone

Authors
  1. Sorin Dragomir
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  2. Domenico Perrone
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Correspondence to Sorin Dragomir.

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Communicated by Marco M. Peloso.

Supported by Università del Salento and M.I.U.R. (within P.R.I.N.).

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Dragomir, S., Perrone, D. Levi Harmonic Maps of Contact Riemannian Manifolds. J Geom Anal 24, 1233–1275 (2014). https://doi.org/10.1007/s12220-012-9371-8

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