\begin{definition}[Definition:Harmonic Function/Riemannian Manifold]
Let $\struct {M, g}$ be a compact Riemannian manifold with or without boundary.
Let $\map {\CC^\infty} M$ be the smooth function space.
Let $u \in \map {\CC^\infty} M$ be a smooth real function on $M$.
Let $\Delta$ be the Laplacian.
Then $u$ is said to be '''harmonic''' if $\Delta u = 0$
\end{definition}