The connection Laplacian, also known as the rough Laplacian, is a differential operator acting on the various tensor bundles of a manifold, defined in terms of a Riemannian- or pseudo-Riemannian metric.
When applied to functions (i.e. tensors of rank 0), the connection Laplacian is often called the Laplace–Beltrami operator.
It is defined as the trace of the second covariant derivative: where T is any tensor,
Recall that the second covariant derivative of T is defined as Note that with this definition, the connection Laplacian has negative spectrum.
On functions, it agrees with the operator given as the divergence of the gradient.
If the connection of interest is the Levi-Civita connection one can find a convenient formula for the Laplacian of a scalar function in terms of partial derivatives with respect to a coordinate system: where
(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.
Let M be a compact, oriented manifold equipped with a metric.
Let E be a vector bundle over M equipped with a fiber metric and a compatible connection,
This connection gives rise to a differential operator where
denotes smooth sections of E, and T*M is the cotangent bundle of M. It is possible to take the
The Lichnerowicz Laplacian differs from the usual tensor Laplacian by a Weitzenbock formula involving the Riemann curvature tensor, and has natural applications in the study of Ricci flow and the prescribed Ricci curvature problem.
On a Riemannian manifold, one can define the conformal Laplacian as an operator on smooth functions; it differs from the Laplace–Beltrami operator by a term involving the scalar curvature of the underlying metric.
In dimension n ≥ 3, the conformal Laplacian, denoted L, acts on a smooth function u by
where Δ is the Laplace-Beltrami operator (of negative spectrum), and R is the scalar curvature.
This operator often makes an appearance when studying how the scalar curvature behaves under a conformal change of a Riemannian metric.
More generally, the action of the conformal Laplacian of g̃ on smooth functions φ can be related to that of the conformal Laplacian of g via the transformation rule
This operator acts on complex-valued functions of a complex variable.
It is essentially the complex conjugate of the ordinary partial derivative with respect to.
Below is a table summarizing the various Laplacian operators, including the most general vector bundle on which they act, and what structure is required for the manifold and vector bundle.