Generalized harmonic morphisms and horizontally weakly conformal biharmonic maps

Abstract

Harmonic morphisms are maps between Riemannian manifolds that pull back harmonic functions to harmonic functions. These maps are characterized as horizontally weakly conformal harmonic maps and they have many interesting links and applications to several areas in mathematics (see the book [2] by Baird and Wood for details). In this paper, we study generalized harmonic morphisms which are defined to be maps between Riemannian manifolds that pull back harmonic functions to biharmonic functions. We obtain some characterizations of generalized harmonic morphisms into a Euclidean space and give two methods of constructions that can be used to produce many examples of generalized harmonic morphisms which are not harmonic morphisms. We also give a complete classification of generalized harmonic morphisms among the projections of a warped product space, which provides infinitely many examples of proper biharmonic Riemannian submersions and conformal submersions from a warped product manifold.Date: 12/08/2017. 1991 Mathematics Subject Classification. 58E20, 53C43.

Complex-Valued (p, q)-Harmonic Morphisms from Riemannian Manifolds

“…They give a characterisation of these objects between Riemannian manifolds. In general this is rather complicated, see Theorem 2.2 of [4]. In our context, of complex-valued functions, it is the following.…”

“…In their paper [4], the authors introduce the notion of generalised harmonic morphisms between Riemannian manifolds. These are exactly the (2, 1)-harmonic morphisms in the sense of our Definition 2.4.…”

“…In their paper [4] the authors construct the following first known proper (2, 1)harmonic morphism. This was basically the only known example before this current study.…”

Biharmonic Riemannian submersions

“…In a recent paper [5], the authors found many examples of biharmonic Riemannian submersions in their study of generalized harmonic morphisms which are maps between Riemannian manifolds that pull back local harmonic functions to local biharmonic functions.…”

“…is a proper biharmonic Riemannian submersion with the mean curvature vector field of fibers being basic vector field. In fact, using cylindrical coordinates (r , θ, ϕ, x 4 ) the Riemannian submersion can be expressed as φ(r , θ, ϕ, Example 2 (see [5] for details) Let R 2 + = {(x, y) ∈ R 2 : y > 0} denote the upper-half plane and C be a positive constant. Then, the Riemannian submersion defined by the projection of the warped product…”