Harmonic morphism explained
In mathematics, a harmonic morphism is a (smooth) map
between
Riemannian manifolds that pulls back real-valued
harmonic functions on the
codomain to harmonic functions on the domain. Harmonic morphisms form a special class of
harmonic maps, namely those that are horizontally (weakly) conformal.
[1] In local coordinates,
on
and
on
, the
harmonicity of
is expressed by the
non-linear system
\tau(\phi
gij\left(
| \partial2\phi\gamma |
\partialxi\partialxj |
| \partial\phi\gamma |
\partialxk |
| n\Gamma |
+\sum | |
| \alpha,\beta=1 |
| \gamma | |
| | \circ\phi
|
| \alpha\beta | |
| \partial\phi\alpha |
\partialxi |
| \partial\phi\beta |
\partialxj |
\right)=-1,
where
| \alpha=y |
\phi | |
| \alpha\circ\phi |
and
are the
Christoffel symbols on
and
, respectively. The
horizontal conformality is given by
ij(x)
| \partial\phi\alpha | (x) |
\partialxi |
| \partial\phi\beta |
\partialxj |
(x)=λ2(x)h\alpha\beta(\phi(x)),
where the conformal factor
is a continuous function called the
dilation. Harmonic morphisms are therefore solutions to
non-linear over-determined systems of
partial differential equations, determined by the geometric data of the
manifolds involved. For this reason, they are difficult to find and have no general existence theory, not even locally.
Complex analysis
When the codomain of
is a
surface, the system of
partial differential equations that we are dealing with, is invariant under conformal changes of the metric
. This means that, at least for local studies, the
codomain can be chosen to be the
complex plane with its standard flat metric. In this situation a complex-valued
function
is a harmonic morphisms if and only if
\DeltaM(\phi)=\DeltaM(u)+i\DeltaM(v)=0
and
g(\nabla\phi,\nabla\phi)=\|\nablau\|2-\|\nablav\|2+2ig(\nablau,\nablav)=0.
This means that we look for two real-valued harmonic functions
with
gradients
that are orthogonal and of the same norm at each point. This shows that complex-valued harmonic morphisms
from
Riemannian manifolds generalise
holomorphic functions
from
Kähler manifolds and possess many of their highly interesting properties. The theory of harmonic morphisms can therefore be seen as a generalisation of
complex analysis.
[1] Minimal surfaces
In differential geometry, one is interested in constructing minimal submanifolds of a given ambient space
. Harmonic morphisms are useful tools for this purpose. This is due to the fact that every regular fibre
of such a map
with values in a
surface is a minimal submanifold of the domain with codimension 2.
[1] This gives an attractive method for manufacturing whole families of
minimal surfaces in 4-dimensional
manifolds
, in particular,
homogeneous spaces, such as
Lie groups and
symmetric spaces.
Examples
are harmonic morphisms.
,
and
are harmonic morphisms.
the standard Riemannian
fibration
is a harmonic morphism.
External links
Notes and References
- Web site: Harmonic Morphisms Between Riemannian Manifolds. Oxford University Press.