Suppose \((M,g)\) is a Riemannian manifold. A function \(u\colon M\to \mathbb{C}\) is called eigenfunction of the Laplace-Beltrami operator \(-\Delta_M\) if there exists a number \(\lambda\in \mathbb{C}\) such that
\[ -\Delta_M u = \lambda u. \]The number \(\lambda\) is a eigenvalue of \(-\Delta_M\).