Harmonic morphism information

In mathematics, a harmonic morphism is a (smooth) map $\phi :(M^{m},g)\to (N^{n},h)$ 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 i.e. those that are horizontally (weakly) conformal.^[1]

In local coordinates, $x$ on $M$ and $y$ on $N$ , the harmonicity of $\phi$ is expressed by the non-linear system

\tau (\phi )=\sum _{i,j=1}^{m}g^{ij}\left({\frac {\partial ^{2}\phi ^{\gamma }}{\partial x_{i}\partial x_{j}}}-\sum _{k=1}^{m}{\hat {\Gamma }}_{ij}^{k}{\frac {\partial \phi ^{\gamma }}{\partial x_{k}}}+\sum _{\alpha ,\beta =1}^{n}\Gamma _{\alpha \beta }^{\gamma }\circ \phi {\frac {\partial \phi ^{\alpha }}{\partial x_{i}}}{\frac {\partial \phi ^{\beta }}{\partial x_{j}}}\right)=0,

where $\phi ^{\alpha }=y_{\alpha }\circ \phi$ and ${\hat {\Gamma }},\Gamma$ are the Christoffel symbols on $M$ and $N$ , respectively. The horizontal conformality is given by

\sum _{i,j=1}^{m}g^{ij}(x){\frac {\partial \phi ^{\alpha }}{\partial x_{i}}}(x){\frac {\partial \phi ^{\beta }}{\partial x_{j}}}(x)=\lambda ^{2}(x)h^{\alpha \beta }(\phi (x)),

where the conformal factor $\lambda :M\to \mathbb {R} _{0}^{+}$ 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.

^ "Harmonic Morphisms Between Riemannian Manifolds". Oxford University Press.

[autogenerated1-1] "Harmonic Morphisms Between Riemannian Manifolds". Oxford University Press.

Harmonic morphism information

and 25 Related for: Harmonic morphism information

Harmonic morphism

Harmonic function

Harmonic map

Harmonic

Holomorphic function

Minimal surface

Sigmundur Gudmundsson

De Rham cohomology

Consonance and dissonance

Eugenio Calabi

Additive synthesis

Dynamic tonality

Representation theory

Function of several complex variables

Graph homomorphism

Equal temperament

Langlands program

List of types of functions

Group theory

Algebraic variety

Group representation

Diffeomorphism

Pontryagin duality

Drinfeld module

Locally compact abelian group