Geometriae Dedicata

, Volume 162, Issue 1, pp 191–229 | Cite as

Harmonic maps of foliated Riemannian manifolds

Original paper


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 : MN 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.


Foliated Riemannian manifold \({({\mathcal{F}}, {\mathcal{G}})}\)-harmonic map \({({\mathcal{F}}, {\mathcal{G}})}\)-harmonic morphism Reeb flow 

Mathematics Subject Classification

53C12 58E20 


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© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Dipartimento di Matematica e InformaticaUniversità degli Studi della BasilicataPotenzaItaly
  2. 2.Dipartimento di MatematicaUniversità di BolognaBolognaItaly

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