Definition:Harmonic Function/Riemannian Manifold

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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:

$\nabla^2 u = 0$