Orthogonality of Eigenfunctions of Compact Riemannian Manifold without Boundary

Theorem

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

Let $u, v \in \map {C^\infty} M : M \to \R$ be smooth mappings on $M$.

Suppose $u$ and $v$ are eigenfunctions of $M$ with distinct eigenvalues.

Then:

$\ds \int_M u v \rd V_g = 0$

where $\rd V_g$ is the Riemannian volume form.

Proof

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Sources

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