Harmonic morphism

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

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

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

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

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

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