Definition:Eigenvalue of Compact Riemannian Manifold without Boundary

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

Let $u \in \map {C^\infty} M : M \to \R$ be a smooth mapping on $M$.

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

Let $\lambda \in \R$ be a real number.

Suppose:


 * $\ds \nabla^2 u + \lambda u = 0$

Then $\lambda$ is called an eigenvalue of $M$.