Definition:Laplacian/Riemannian Manifold

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Let $\struct {M, g}$ be a Riemannian manifold.

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

Let $\grad$ be the gradient operator.

Let $\operatorname {div}$ be the divergence operator.

The Laplacian of $f$ is defined as:

$\Delta f := \map {\operatorname {div} } {\grad f}$

Also defined as

Sometimes the Laplacian is defined with a minus sign to make its eigenvalues nonnegative.