\[ \DeclareMathOperator{\div}{div} \DeclareMathOperator{\grad}{grad} \]

Let \((M,g)\) be a Riemannian manifold. The (geometric) Laplacian is a linear map \(\Delta\colon C^\infty(M)\to C^\infty(M)\) defined by \(\Delta u = -\div(\grad u)\).

Links Link to heading

• Laplacian in local coordinates
• Greens identities
• eigenfunctions

Other identities for \(-\Delta\) Link to heading

• geometric Laplacian in terms of the covariant Hessian

Related Definitions Link to heading

• divergence
• gradient
• Laplacian