\begin{definition}[Definition:Eigenvalue of Compact Riemannian Manifold without Boundary]
Let $\struct {M, g}$ be a compact Riemannian manifold without boundary.
Let $u \in \map {\CC^\infty} M : M \to \R$ be a smooth mapping on $M$.
Let $\Delta$ be the Laplacian.
Let $\lambda \in \R$ be a real number.
Suppose:
:$\ds \Delta u + \lambda u = 0$
Then $\lambda$ is called an '''eigenvalue of $M$'''.
\end{definition}