Talk:Laplace operators in differential geometry

[Untitled][edit]

Also discuss

Harmonic map laplacian
Rough laplacian (c.f. Besse, pg. 52)
${\bar {\partial }}$ -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[edit]

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]

Overlap with other articles[edit]

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?[edit]

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]