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Harmonic maps of foliated Riemannian manifolds

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Abstract

We study \({({\mathcal{F}}, {\mathcal{G}})}\)-harmonic maps between foliated Riemannian manifolds \({(M, {\mathcal{F}}, g)}\) and \({(N, {\mathcal{G}}, h)}\) i.e. smooth critical points ϕ : M → N of the functional \({E_T (\phi ) = \frac{1}{2} \int_M \| d_T \phi \|^2 \,d \, v_g}\) with respect to variations through foliated maps. In particular we study \({({\mathcal{F}}, {\mathcal{G}})}\)-harmonic morphisms i.e. smooth foliated maps preserving the basic Laplace equation Δ B u =  0. We show that CR maps of compact Sasakian manifolds preserving the Reeb flows are weakly stable \({({\mathcal{F}}, {\mathcal{G}})}\)-harmonic maps. We study \({({\mathcal{F}}, {\mathcal{G}}_0 )}\)-harmonic maps into spheres and give foliated analogs to Solomon’s (cf., J Differ Geom 21:151–162, 1985) results.

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Authors and Affiliations

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

    Sorin Dragomir

  2. Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, 40126, Bologna, Italy

    Andrea Tommasoli

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  1. Sorin Dragomir
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  2. Andrea Tommasoli
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Correspondence to Sorin Dragomir.

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Dragomir, S., Tommasoli, A. Harmonic maps of foliated Riemannian manifolds. Geom Dedicata 162, 191–229 (2013). https://doi.org/10.1007/s10711-012-9723-3

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