Transversal harmonic transformations for Riemannian foliations

Abstract

The main purpose of the present paper is to study geometric properties of transversal (infinitesimal) harmonic transformations for Riemannian foliations. For the point foliation these notions are discussed in [14]. Especially we treat transversal infinitesimal harmonic transformations from the standpoint of λ-automorphisms. Our results extend those obtained in [6,7,15] for the case of harmonic foliations. (2000): Primary 53C20, Secondary 57R30. Mathematics Subject Classifications

Riemannian foliations admitting transversal conformal fields

“…Hence the metric defined by (2.2) induces an identification [13] V (F) ∼ = 1 B (F). (2.11) For the later use, we recall the transversal divergence theorem [20] …”

Liouville type theorem for (F;F')p-harmonic maps on foliations

“…Let V (F ) be the space of all transversal infinitesimal automorphisms Y of F , i.e., [Y, Z] ∈ ΓL for all Z ∈ ΓL [11]. Let [19]. For later use, we recall the transversal divergence theorem [28] on a foliated Riemannian manifold.…”

On Transversally Harmonic Maps of Foliated Riemannian Manifolds

“…Then the operator ∇ * tr ∇ tr is positive definite and formally self adjoint on the space of basic forms [3]. Let V (F) be the space of all transversal infinitesimal automorphisms Y of [10]. For later use, we recall the transversal divergence theorem [16] on a foliated Riemannian manifold.…”