The Hodge Laplacian, also known as the Laplace–de Rham operator, is differential operator on acting on differential forms. (Abstractly, it is a second order operator on each exterior power of the cotangent bundle.) This operator is defined on any manifold equipped with a Riemannian- or pseudo-Riemannian metric.
where d is the exterior derivative or differential and δ is the codifferential. The Hodge Laplacian on a compact manifold has nonnegative spectrum.
The connection Laplacian may also be taken to act on differential forms by restricting it to act on skew-symmetric tensors. The connection Laplacian differs from the Hodge Laplacian by means of a Weitzenböck identity.
Read more about this topic: Laplace Operators In Differential Geometry