Laplaciantype

Laplaciantype, also known as a Laplace-type operator, is a class of second-order elliptic differential operators that generalize the Laplace–Beltrami operator. These operators act on sections of a vector bundle E over a Riemannian manifold M. A Laplaciantype operator D has principal symbol equal to the metric, meaning for each cotangent vector ξ the leading part is |ξ|^2 times the identity on E. Equivalently, D can be written locally as D = ∇*∇ + E, where ∇ is a connection on E and E is a smooth endomorphism of E. Different sign conventions may be used in the literature.

Typical examples include the rough Laplacian ∇*∇ on sections of E and the Laplace–Beltrami operator on functions (up to sign). The Hodge Laplacian Δ on differential forms is another canonical Laplaciantype operator. The square of a Dirac operator D on a spinor bundle is also Laplace-type in light of the Weitzenböck formula D^2 = ∇*∇ + F, with F a curvature term.

Properties of Laplaciantype operators include ellipticity and, with suitable boundary conditions, self-adjointness. They generate heat kernels and semigroups, making them central in spectral geometry and analysis. Their spectral data encode geometric information about the underlying manifold and bundle, and they feature prominently in index theory, heat kernel asymptotics, and geometric PDEs.

See also: Laplace–Beltrami operator, Bochner Laplacian, Hodge Laplacian, Dirac operator, Weitzenböck formula.