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, 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.
Read more about this topic: Laplace Operators In Differential Geometry
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