Harmonic Maps

  • Idea: Given $(\mathcal{M}, g_{\mu\nu})$ and $(\mathcal{N}, h_{\alpha\beta})$ Riemannian manifolds with their metrics, a smooth map $\psi: \mathcal{M} \to \mathcal{N}$ is called Harmonic if its coordinate representative (say $\phi(x)$) satisfy a certain nonlinear PDE.
  • Definition: $\psi$ is called Harmonic map if its laplacian vanishes. i.e. $\Delta_{x} \phi(x) = 0$ where the Laplacian in-general is given by Voss-Weyl formula $$ \Delta_{x} := \frac{1}{\sqrt{g(x)}} \frac{\partial}{\partial x^{\mu}} ( \sqrt{g(x)} g^{\mu\nu}(x) ) \frac{\partial}{\partial x^{\nu}} $$
  • Definition: $\psi$ is called totally geodesic if its hessian vanishes.

Permalink at https://www.physicslog.com/math-notes/harmonic-maps

Published on Jul 5, 2021

Last revised on Apr 3, 2022