Talk:Laplace–Beltrami operator

Is this the same as the Laplace operator for functions in Cartesian coordinates and vector functions? — Preceding unsigned comment added by 71.244.132.130 (talk) 03:01, 11 November 2016 (UTC)

Yes. The opening sentence currently says "In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds", which I'd hope answers the question, but perhaps could be revised to make this more clear. 129.215.169.200 (talk) 15:41, 21 March 2024 (UTC)

As for today notation needs to be completly changed! A square gradient is an hessian but certainly not a Laplace-Beltrami operator, as they're not even operator taking value in the same spaces. Laplace Beltrami on functions is the trace of the hessian. As far as my knowledge goes, the Laplace-Beltrami is commonly denoted with a \Delta or \Delta_M. The Hodge de Rham laplacian can be denoted as D^2, D being defined as d+d* on the total forms space, or as square_M. — Preceding unsigned comment added by 86.212.98.179 (talk) 20:30, 13 February 2017 (UTC)

It seems like the section "Tensor Laplacian" contradicts the "Details" section, as well as many other articles that show the use of the Laplace-beltrami operator and the Hessian. Performing an explicit calculation, one can easily verify that the last equation in "Details" and the definition of the Laplace-beltrami operator in "Tensor Laplacian" do not agree. In fact, it seems that the second summand in the TL section's definition is superfluous, as its existence is not mentioned in the article on the Hessian itself, either. One only has to check a short calculation in a chart of choice to see that the definition from https://en.wikipedia.org/wiki/Hessian_matrix#Generalizations_to_Riemannian_manifolds and from "Details" agree, and those agree with other pages, as for example the use of the Laplace-Beltrami operator in https://en.wikipedia.org/wiki/Klein%E2%80%93Gordon_equation#Gravitational_interaction. The "Tensor Laplacian" version needs to have the second term removed. 2A02:810D:19C0:110C:F59B:9C06:A6ED:A8BD (talk) 16:17, 20 October 2017 (UTC)

The second term is definitely there. Please check your calculations again. Note that, without this term, there will be no terms involving the derivative of the metric (with the Laplacian acting on functions), but the expression in the details section clearly will involve the first derivative of the metric. Also, there is a second term in the Hessian: see this. Sławomir Biały (talk) 16:56, 20 October 2017 (UTC)

Self-adjoint?

Is the Laplace-Beltrami operator self-adjoint? If so, is there a citation? The section on "formal self-adjointness" shows that the operator is formally self-adjoint, but it doesn't treat the domain issues for the operator, so it doesn't answer the question of self-adjointness. The "Eigenvalues of the Laplcae-Beltrami operator (Lichnerowwicz-Obata theorem)" section then asserts that the "Formal self-adjointness" section has proved self-adjointness, not merely formal self-adjointness. 129.215.169.200 (talk) 15:39, 21 March 2024 (UTC)