Definition:Eigenvalue of Compact Riemannian Manifold without Boundary
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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 $M$.
Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Problems