Eigenvalues of Compact Riemannian Manifold without Boundary are Nonnegative

Theorem

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

Let $\lambda \in \R$ be an eigenvalue of $M$.

Then $0$ is an eigenvalue of $M$, and the rest of eigenvalues are strictly positive.

Proof

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Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Problems