IR evaluation - Revision history

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IR Evaluation

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	~~In [[differential geometry]] there are~~ a ~~number of second~~-~~order~~, ~~linear, [[elliptic operator\|elliptic]] [[differential operators]] bearing~~ the ~~name '''Laplacian'~~''. ~~This article provides an overview~~ of ~~some~~ of ~~them~~.		Next - GEN Gallery is a full incorporated Image Gallery plugin for Word - Press which has a Flash slideshow option. Online available for hiring are most qualified, well knowledgeable and talented Wordpress developer India from offshore Wordpress development services company. Change the site's theme and you have essentially changed the site's personality. If you need a special plugin for your website , there are thousands of plugins that can be used to meet those needs. By using this method one can see whether the theme has the potential to become popular or not and is their any scope of improvement in the theme. <br><br>

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	~~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 metric\|Riemannian]]- or [[Pseudo-Riemannian manifold\|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:~~

	:<~~math~~>~~\Delta T= -\text{tr}\;\nabla^2 T,~~<~~/math>~~
	~~where ''T'' is any tensor, <math>\nabla</math~~> 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~~

	~~:<math>\nabla^2_{X,Y} T = -\left(\nabla_X \nabla_Y T~~ - ~~\nabla_{\nabla_X Y} T\right)~~.~~</math>~~

	~~Note~~ that ~~with this definition, the connection Laplacian~~ has ~~negative [[Spectrum of an operator\|spectrum]]~~. ~~On functions, it agrees with~~
	~~the operator given as the divergence of the gradient.~~

	If ~~connection of interest is~~ [~~[Levi-Civita connection]] one can find convenient formula for Laplacian of scalar function in terms of partial derivatives with respect to chosen coordinates~~:

	~~:<math>\Delta \phi = \|g\|^{-1~~/~~2} \partial_\mu\left( \|g\|^{1~~/~~2} g^{\mu\nu} \partial_\nu\right)\phi <~~/~~math>~~

	where <math>\phi</math> is scalar function, <math>\|g\|</math> is absolute value of determinant of metric (the use of absolute value is necessary in [[Pseudo-Riemannian manifold\|Pseudo Riemmanian case]], for example in [[General Relativity]]) and <math>g^{\mu \nu}</math> denotes [[Metric tensor#Inverse_metric\|inverse of the metric tensor]]

	~~== Hodge Laplacian =~~=
	~~The '''Hodge Laplacian''', also known as the '''Laplace–de Rham operator''', is differential operator on acting on [[differential forms]~~]~~. (Abstractly,~~
	~~it is a second order operator on each exterior power of~~ the ~~[[cotangent bundle]]~~.~~) This operator is defined on any manifold equipped with~~
	~~a [[Riemannian metric\|Riemannian]]- or [[Pseudo-Riemannian manifold\|pseudo-Riemannian]] metric.~~

	:~~<math>\Delta= \mathrm{d}\delta+\delta\mathrm{d} =~~ (~~\mathrm{d}+\delta~~)^2,~~\;</math>~~

	~~where d is the [[exterior derivative]] or differential~~ and ~~δ is the [[codifferential]]. The Hodge Laplacian on a compact manifold has nonnegative [[Spectrum of an operator\|spectrum]].~~

	~~The connection Laplacian may also be taken~~ to ~~act on differential forms by restricting it to act on skew~~-~~symmetric tensors. The connection Laplacian differs from the Hodge Laplacian by means of a [[Weitzenböck identity]].~~

	~~== Bochner Laplacian ==~~
	~~The '''Bochner Laplacian''' is defined differently from the connection Laplacian~~, ~~but the two will turn out to differ only by a sign~~, ~~whenever the former is defined. Let ''M'' be~~ a ~~compact, oriented manifold equipped with a metric~~. ~~Let ''E'' be a vector bundle over ''M'' equipped a fiber metric and a compatible connection,~~ <~~math~~>~~\nabla~~<~~/math~~>. ~~This connection gives rise to a differential operator~~
	~~::<math>\nabla:\Gamma(E)\rightarrow \Gamma(T^*M\otimes E)</math>~~
	~~where <math>\Gamma(E)</math> denotes smooth sections of ''E'~~', and ~~''T''<sup>*</sup>M is the [[cotangent bundle]] of ''M''~~. ~~It is possible to take~~ the ~~<math>L^2</math>~~-~~adjoint of <math>\nabla</math>, giving a differential operator~~
	~~::<math>\nabla^:\Gamma(T^M\otimes E)\rightarrow \Gamma(E).</math>~~
	~~The '''Bochner Laplacian''' is given by~~
	~~::<math>\Delta=\nabla^*\nabla</math>~~
	~~which is~~ a ~~second order operator acting~~ on ~~sections of~~ the ~~vector bundle ''E''~~. ~~Note that the connection Laplacian and Bochner Laplacian differ only by a sign~~:
	~~::<math> \nabla^* \nabla = - \text{tr}\, \nabla^2</math>~~

	~~== Lichnerowicz Laplacian ==~~
	The '''Lichnerowicz Laplacian'''<ref>{{Citation \| last1=Chow \| first1=Bennett \| last2=Lu \| first2=Peng \| last3=Ni \| first3=Lei \| title=Hamilton's Ricci flow \| publisher=[[American Mathematical Society]] \| location=Providence, R.I. \| series=Graduate Studies in Mathematics \| isbn=978-0-~~8218-4231-7; 978-0-8218-4231-7 \| id={{MathSciNet \| id = 2274812}} \| year=2006 \| volume=77}}~~
	~~</ref> is defined~~ on ~~symmetric tensors by taking <math>\nabla : \Gamma(\operatorname{Sym}^k(TM))\to \Gamma(\operatorname{Sym}^{k+1}(TM))</math> to be~~ the ~~symmetrized covariant derivative~~. ~~The Lichnerowicz Laplacian is then defined by~~ <~~math~~>~~\Delta_L = \nabla^\nabla~~<~~/math~~>, ~~where <math>\nabla^</math> is the formal adjoint. The Lichnerowicz Laplacian differs from the usual tensor Laplacian by a [[Weitzenbock formula]] involving the [[Riemann curvature tensor]]~~, ~~and has natural applications in the study~~ of ~~[[Ricci flow]]~~ and ~~the [[prescribed Ricci curvature problem]]~~.

	~~== Conformal Laplacian ==~~
	~~On a [[Riemannian manifold]], one can define the '''conformal Laplacian''' as an operator on smooth functions;~~ it ~~differs~~ from the ~~Laplace–Beltrami operator by a term involving~~ the ~~[[scalar curvature]] of the underlying metric~~. ~~In dimension ''n'' ≥ 3, the conformal Laplacian, denoted ''L'', acts on a smooth function ''u'' by~~

	~~:<math>Lu = -4\frac{n-1}{n-2} \Delta u + Ru,</math>~~

	~~where Δ is the Laplace-Beltrami operator (of negative spectrum), and ''R''~~ is ~~the scalar curvature. This operator often makes an appearance when studying how the scalar curvature behaves under~~ a ~~conformal change~~ of ~~a Riemannian metric~~. ~~If ''n'' ≥ 3 and ''g'' is~~ a ~~metric and ''u'' is~~ a ~~smooth, positive function, then~~ the ~~[[Conformal map\|conformal]] metric~~

	~~: <math>\tilde g = u^{4/(n-2)} g \, </math>~~

	~~has scalar curvature given by~~

	~~:<math>\tilde R = u^{-(n+2)/(n-2)} L u~~. ~~\, </math>~~

	~~==See also==~~
	*[[Weitzenböck identity]]

	~~==References==~~
	~~<references/>~~

	~~[[Category:Differential operators]]~~
	~~[[Category:Differential geometry]]~~

en>FrescoBot: Bot: fixing section wikilinks

2011-02-03T11:17:13Z

Bot: fixing section wikilinks

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In [[differential geometry]] there are a number of second-order, linear, [[elliptic operator|elliptic]] [[differential operators]] bearing the name '''Laplacian'''. This article provides an overview of some of them.

== Connection Laplacian ==
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 metric|Riemannian]]- or [[Pseudo-Riemannian manifold|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:

:<math>\Delta T= -\text{tr}\;\nabla^2 T,</math>
where ''T'' is any tensor, <math>\nabla</math> 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

:<math>\nabla^2_{X,Y} T = -\left(\nabla_X \nabla_Y T - \nabla_{\nabla_X Y} T\right).</math>

Note that with this definition, the connection Laplacian has negative [[Spectrum of an operator|spectrum]]. On functions, it agrees with
the operator given as the divergence of the gradient.

If connection of interest is [[Levi-Civita connection]] one can find convenient formula for Laplacian of scalar function in terms of partial derivatives with respect to chosen coordinates:

:<math>\Delta \phi = |g|^{-1/2} \partial_\mu\left( |g|^{1/2} g^{\mu\nu} \partial_\nu\right)\phi </math>

where <math>\phi</math> is scalar function, <math>|g|</math> is absolute value of determinant of metric (the use of absolute value is necessary in [[Pseudo-Riemannian manifold|Pseudo Riemmanian case]], for example in [[General Relativity]]) and <math>g^{\mu \nu}</math> denotes [[Metric tensor#Inverse_metric|inverse of the metric tensor]]

== Hodge Laplacian ==
The '''Hodge Laplacian''', also known as the '''Laplace–de Rham operator''', is differential operator on acting on [[differential forms]]. (Abstractly,
it is a second order operator on each exterior power of the [[cotangent bundle]].) This operator is defined on any manifold equipped with
a [[Riemannian metric|Riemannian]]- or [[Pseudo-Riemannian manifold|pseudo-Riemannian]] metric.

:<math>\Delta= \mathrm{d}\delta+\delta\mathrm{d} = (\mathrm{d}+\delta)^2,\;</math>

where d is the [[exterior derivative]] or differential and δ is the [[codifferential]]. The Hodge Laplacian on a compact manifold has nonnegative [[Spectrum of an operator|spectrum]].

The connection Laplacian may also be taken to act on differential forms by restricting it to act on skew-symmetric tensors. The connection Laplacian differs from the Hodge Laplacian by means of a [[Weitzenböck identity]].

== Bochner Laplacian ==
The '''Bochner Laplacian''' is defined differently from the connection Laplacian, but the two will turn out to differ only by a sign, whenever the former is defined. Let ''M'' be a compact, oriented manifold equipped with a metric. Let ''E'' be a vector bundle over ''M'' equipped a fiber metric and a compatible connection, <math>\nabla</math>. This connection gives rise to a differential operator
::<math>\nabla:\Gamma(E)\rightarrow \Gamma(T^*M\otimes E)</math>
where <math>\Gamma(E)</math> denotes smooth sections of ''E'', and ''T''<sup>*</sup>M is the [[cotangent bundle]] of ''M''. It is possible to take the <math>L^2</math>-adjoint of <math>\nabla</math>, giving a differential operator
::<math>\nabla^*:\Gamma(T^*M\otimes E)\rightarrow \Gamma(E).</math>
The '''Bochner Laplacian''' is given by
::<math>\Delta=\nabla^*\nabla</math>
which is a second order operator acting on sections of the vector bundle ''E''. Note that the connection Laplacian and Bochner Laplacian differ only by a sign:
::<math> \nabla^* \nabla = - \text{tr}\, \nabla^2</math>

== Lichnerowicz Laplacian ==
The '''Lichnerowicz Laplacian'''<ref>{{Citation | last1=Chow | first1=Bennett | last2=Lu | first2=Peng | last3=Ni | first3=Lei | title=Hamilton's Ricci flow | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Graduate Studies in Mathematics | isbn=978-0-8218-4231-7; 978-0-8218-4231-7 | id={{MathSciNet | id = 2274812}} | year=2006 | volume=77}}
</ref> is defined on symmetric tensors by taking <math>\nabla : \Gamma(\operatorname{Sym}^k(TM))\to \Gamma(\operatorname{Sym}^{k+1}(TM))</math> to be the symmetrized covariant derivative. The Lichnerowicz Laplacian is then defined by <math>\Delta_L = \nabla^*\nabla</math>, where <math>\nabla^*</math> is the formal adjoint. The Lichnerowicz Laplacian differs from the usual tensor Laplacian by a [[Weitzenbock formula]] involving the [[Riemann curvature tensor]], and has natural applications in the study of [[Ricci flow]] and the [[prescribed Ricci curvature problem]].

== Conformal Laplacian ==
On a [[Riemannian manifold]], one can define the '''conformal Laplacian''' as an operator on smooth functions; it differs from the Laplace–Beltrami operator by a term involving the [[scalar curvature]] of the underlying metric. In dimension ''n'' ≥ 3, the conformal Laplacian, denoted ''L'', acts on a smooth function ''u'' by

:<math>Lu = -4\frac{n-1}{n-2} \Delta u + Ru,</math>

where Δ is the Laplace-Beltrami operator (of negative spectrum), and ''R'' is the scalar curvature. This operator often makes an appearance when studying how the scalar curvature behaves under a conformal change of a Riemannian metric. If ''n'' ≥ 3 and ''g'' is a metric and ''u'' is a smooth, positive function, then the [[Conformal map|conformal]] metric

: <math>\tilde g = u^{4/(n-2)} g \, </math>

has scalar curvature given by

:<math>\tilde R = u^{-(n+2)/(n-2)} L u. \, </math>

==See also==
*[[Weitzenböck identity]]

==References==
<references/>

[[Category:Differential operators]]
[[Category:Differential geometry]]