# **Harmonic morphisms between Riemannian manifolds **

## ** Paul Baird and John C. Wood **

## * London Mathematical Society Monographs, No. 29, *

Oxford University Press (2003)

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**Description**

*Harmonic morphisms* are maps which preserve Laplace's equation. More explicitly, a map
between Riemannian manifolds is called a harmonic morphism if its composition
with any locally defined harmonic function on the codomain
is a harmonic function on the domain; it thus `pulls back' germs of harmonic functions to germs
of harmonic functions. Examples
include *harmonic functions*, *conformal mappings in the plane*, *holomorphic
mappings with values in a Riemann surface* and certain submersions arising from
*Killing fields* and *geodesic fields*. Their study involves many different
branches of mathematics: we shall discuss aspects of the theory of foliations,
polynomials induced by Clifford systems and orthogonal multiplications, twistor
and mini-twistor spaces and Hermitian structures. We shall explore relations with
topology, including Seifert fibre spaces and circle actions, and with isoparametric functions and the Beltrami
fields equation of hydrodynamics.
This monograph represents the first attempt to
document the theory of harmonic morphisms between Riemannian manifolds in book form.
Chapter 1 gives the flavour of the subject in a way which should be
understandable to a competent final-year undergraduate student.
For the rest of the book, we shall assume that the reader is familiar with the basic
concepts of differential geometry as found, for example, in the first two volumes of M. Spivak's book:
*A comprehensive introduction to Riemannian geometry*,
though we shall remind readers of key concepts at appropriate points.
Apart from some occasional technicalities of analysis or partial differential equations for which we refer the
reader to the appropriate literature, this book is self-contained; in particular, it is not necessary
to know anything about harmonic maps or partial differential equations to read it.
We hope that it will be a useful book both for graduate students wishing to learn about
harmonic morphisms and related topics on harmonic maps,
and for researchers in the field or in related fields.

As shown by B. Fuglede and T. Ishihara in 1978/9, a harmonic morphism is a
*harmonic map* with an additional property called *horizontal weak
conformality* (or *semiconformality*). However, there are major differences with
studies of the general theory of harmonic
mappings. Firstly, Fuglede established that *any non-constant harmonic morphism is an
open mapping*, in particular, it must *preserve* or *decrease*
dimension. Secondly, the second order equation of harmonicity and the first
order equation of horizontal weak conformality form an overdetermined system of
equations: solutions are *not* guaranteed even locally; their existence depends
in an essential way on the topology of the manifolds and the Riemannian structures on
them, for example, *there is no harmonic morphism from the (**n*+1)-sphere to the *n*-sphere
whatever metrics they have, when n > 3. This contrasts with the theory of harmonic mappings, where a theorem of
Eells and Ferreira (1991) shows that, *provided the dimension of the domain
manifold is at least 3, any homotopy class of maps between Riemannian manifolds contains a
(smooth) harmonic representative with respect to a metric on the domain conformally
equivalent to the given one*.

On the other hand, we find a remarkable duality between the theory of harmonic
morphisms *to* a surface and the theory of conformal minimal immersions *from* a surface
giving *Weierstrass representations for harmonic morphisms* and *Bernstein theorems*, for example, the Hopf
map is essentially the only harmonic morphism from the 3-sphere to the 2-sphere, both with their
standard metrics.