Definition:Laplacian/Riemannian Manifold

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

Let $f \in \map {C^\infty} M : M \to \R$ 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:


 * $\nabla^2 f := \map {\operatorname {div} } {\grad f}$

Also denoted as
The Laplace-Beltrami operator can be seen as $\Delta f$, particularly in older texts.

The $\nabla^2$ form is preferred on as it allows less opportunity for ambiguity and misunderstanding.

Also defined as
Sometimes the Laplacian-Beltrami operator is defined with a minus sign to make its eigenvalues nonnegative.