 Laplace operators in differential geometry

This family includes the Laplace-Beltrami operator on functions, the Hodge Laplacian on differential forms, and the rough Laplacian associated with a connection on a vector bundle. Their domains and sign conventions must be specified when comparing them.

In differential geometry, the concept of the Laplace operator, often denoted as \(\Delta\) or \(\nabla^2\), is a generalization of the Laplacian from classical analysis to manifolds. It plays a significant role in understanding the geometric and analytical properties of functions defined on a manifold.