Transversally Harmonic Maps between Manifolds with Riemannian Foliations

Abstract

We consider leaf preserving maps between manifolds equipped with Riemannian foliations. We construct a transversal tension field for such maps and define transversally harmonic maps. Then we give some examples of such maps using the suspension construction.

Some Remarks on Transversally Harmonic Maps

“…In [24] we have demonstrated that the mapping f is transversally harmonic if and only if the mappingf is harmonic; cf. Theorem 3.1.…”

“…In our paper [24], we generalise the notion of a harmonic map to foliated maps between manifolds with Riemannian foliations. The definition of a transversally harmonic map arises naturally in the context of the transverse geometry of foliations.…”

“…Hence the induced mapping between the transverse manifolds is (anti)holomorphic, the domain is cosymplectic, and the target manifold is (1, 2)-symplectic, so that this mapping is harmonic and therefore the foliated map f : M 1 → M 2 is transversally harmonic; cf. [24].…”

“…The remark above, Theorem 3.3 and the suspension construction can be used to provide examples of transversally harmonic maps in the singular case. See [24] for interesting examples.…”

“…The right choice of representations in the examples of [24] can provide us with interesting examples.…”