Talk:Laplace operators in differential geometry

Latest comment: 10 years ago by Billlion in topic Pseudo Reimannian metrics?

[Untitled]

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Also discuss

  • Harmonic map laplacian
  • Rough laplacian (c.f. Besse, pg. 52)
  •  -Laplacian
  • square of the Dirac operator
  • Weitzenbock formulas

Also, add in references Jjauregui (talk) 15:10, 29 February 2008 (UTC)Reply

There is the more general notion of a Laplacian on a chain complex...?Billlion (talk) 08:25, 13 May 2008 (UTC)Reply

I'd like to include a section and table that look something like the following, but it needs some work.

Comparisons

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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. All of these operators are second order, linear, and elliptic.

Laplacian vector bundle required structure, base manifold required structure, vector bundle spectrum
Hodge differential forms metric, orientation induced metric and connection positive
Connection tensors metric induced metric and connection negative
Bochner any vector bundle metric, orientation fiber metric, compatible connection positive
Lichnerowicz symmetric 2-tensors metric induced connection ?
Conformal functions metric none varies

Jjauregui (talk) 20:47, 13 February 2008 (UTC)Reply

What about Hecke operators? —Preceding unsigned comment added by 132.206.150.237 (talk) 19:43, 30 August 2010 (UTC)Reply

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I like this article, but there is considerable overlap with Laplace-Beltrami operator. I thought there was another related article as well, but now I can't seem to find it. Anyway, in light of the existence of this article, it may be worth doing some top level reorganization, such as trimming away bits of the Laplace-Beltrami operator article which are already duplicated here. silly rabbit (talk) 16:40, 2 May 2008 (UTC)Reply

Pseudo Reimannian metrics?

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The article starts with "In differential geometry there are a number of second-order, linear, elliptic differential operators bearing the name Laplacian." Then in section 2 "This operator is defined on any manifold equipped with a Riemannian- or pseudo-Riemannian metric." Well if you take a Lorentzian metric the Laplacian so defined is hyperbolic. Should the article start be including pseudo-Reimannian metrics, thus dropping ellipticity as generally true? What about statements about the spectrum if we are considering non-elliptic operators?Billlion (talk) 22:01, 12 June 2014 (UTC)Reply