Talk:Laplace operators in differential geometry
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[Untitled]
editAlso 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)
- There is the more general notion of a Laplacian on a chain complex...?Billlion (talk) 08:25, 13 May 2008 (UTC)
I'd like to include a section and table that look something like the following, but it needs some work.
Comparisons
editBelow 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)
What about Hecke operators? —Preceding unsigned comment added by 132.206.150.237 (talk) 19:43, 30 August 2010 (UTC)
Overlap with other articles
editI 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)
Pseudo Reimannian metrics?
editThe 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)