Laplace operators in differential geometry

In differential geometry there are a number of second-order, linear, elliptic differential operators bearing the name Laplacian. This article provides an overview of some of them.

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, ${\displaystyle \nabla }$ is the Levi-Civita connection associated to the metric, and the trace is taken with respect to the metric. 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 connection of interest is Levi-Civita connection one can find a convenient formula for Laplacian of scalar function in terms of partial derivatives with respect to chosen coordinates:

where ${\displaystyle \phi }$ is scalar function, ${\displaystyle |g|}$ is absolute value of determinant of metric (the use of absolute value is necessary in Pseudo Riemmanian case, for example in General Relativity) and ${\displaystyle g^{\mu \nu }}$ denotes inverse of the metric tensor

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