Definition:Harmonic Function/Riemannian Manifold

Definition

Let $\struct {M, g}$ be a compact Riemannian manifold with or without boundary.

Let $\map {C^\infty} M$ be the smooth function space.

Let $u \in \map {C^\infty} M$ be a smooth real function on $M$.

Let $\nabla^2$ be the Laplace-Beltrami operator.

Then $u$ is said to be harmonic if and only if:

$\nabla^2 u = 0$

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Problems