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2013-03-15T16:35:14Z

Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q1779479

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	Hi there. ~~Let me start~~ by ~~introducing~~ the ~~author~~, ~~her name~~ is ~~Sophia Boon but she~~ by ~~no means really liked~~ that ~~title. My working day occupation~~ is a ~~travel agent~~. ~~Her family lives in Ohio. My spouse doesn~~'~~t like it~~ the ~~way I do but what I really like doing~~ is ~~caving but I don~~'~~t have~~ the ~~time lately~~.<br><br>~~Here~~ is ~~my page~~ ... ~~best psychic~~ ([~~http~~://~~srncomm~~.~~com~~/~~blog~~/~~2014~~/~~08/25/relieve~~-~~that~~-~~stress~~-~~find~~-a-~~new~~-~~hobby/ Click Link~~])		{{distinguish\|Weitzenböck's inequality}}

			In [[mathematics]], in particular in [[differential geometry]], [[mathematical physics]], and [[representation theory]] a '''Weitzenböck identity''', named after [[Roland Weitzenböck]], expresses a relationship between two second-order [[elliptic operator]]s on a [[manifold (mathematics)\|manifold]] with the same leading symbol. (The origins of this terminology seem doubtful, however, as there does not seem to be any evidence that such identities ever appeared in Weitzenböck's work.) Usually Weitzenböck formulae are implemented for ''G''-invariant self-adjoint operators between vector bundles associated to some [[principal bundle\|principal ''G''-bundle]], although the precise conditions under which such a formula exists are difficult to formulate. Instead of attempting to be completely general, then, this article presents three examples of Weitzenböck identities: from Riemannian geometry, spin geometry, and complex analysis.

			==Riemannian geometry==
			In [[Riemannian geometry]] there are two notions of the [[Laplacian]] on [[differential forms]] over an oriented compact Riemannian manifold ''M''. The first definition uses the [[divergence\|divergence operator]] δ defined as the formal adjoint of the de Rham operator ''d'':
			::<math>\int_M \langle \alpha,\delta\beta\rangle := \int_M\langle d\alpha,\beta\rangle</math>
			where α is any ''p''-form and β is any (''p'' + 1)-form, and <math>\langle -,-\rangle</math> is the metric induced on the bundle of (''p'' + 1)-forms. The usual '''form Laplacian''' is then given by

			::Δ = dδ + δd.

			On the other hand, the [[Levi-Civita connection]] supplies a differential operator

			::<math>\nabla:\Omega^pM\rightarrow T^*M\otimes\Omega^pM</math>

			where Ω<sup>p</sup>''M'' is the bundle of ''p''-forms and ''T''<sup>*</sup>M is the [[cotangent bundle]] of ''M''. The '''Bochner Laplacian''' is given by

			::<math>\Delta'=\nabla^*\nabla</math>

			where <math>\nabla^*</math> is the adjoint of <math>\nabla</math>.

			The Weitzenböck formula then asserts that

			::<math>\Delta' - \Delta = A</math>

			where ''A'' is a linear operator of order zero involving only the curvature.

			The precise form of ''A'' is given, up to an overall sign depending on curvature conventions, by

			::<math>A=\frac{1}{2}\langle R(\theta,\theta)\#,\#\rangle + \operatorname{Ric}(\theta,\#) \, </math>

			where

			:*''R'' is the Riemann curvature tensor,
			:* Ric is the Ricci tensor,
			:* <math>\theta:T^*M\otimes\Omega^pM\rightarrow\Omega^{p+1}M</math> is the map that takes the wedge product of a 1-form and p-form and gives a (p+1)-form,
			:* <math>\#:\Omega^{p+1}M\rightarrow T^*M\otimes\Omega^pM</math> is the universal derivation inverse to θ on 1-forms.

			==Spin geometry==
			If ''M'' is an oriented [[spin manifold]] with [[Dirac operator]] ð, then one may form the spin Laplacian Δ = ð<sup>2</sup> on the spin bundle. On the other hand, the Levi-Civita connection extends to the spin bundle to yield a differential operator
			:<math>\nabla:SM\rightarrow T^*M\otimes SM.</math>
			As in the case of Riemannian manifolds, let <math>\Delta'=\nabla^*\nabla</math>. This is another self-adjoint operator and, moreover, has the same leading symbol as the spin Laplacian. The Weitzenböck formula yields:

			:<math>\Delta'-\Delta=-\frac{1}{4}Sc</math>

			where ''Sc'' is the scalar curvature. This result is also known as the [[Lichnerowicz formula]].

			==Complex differential geometry==
			If ''M'' is a compact [[Kähler manifold]], there is a Weitzenböck formula relating the <math>\bar{\partial}</math>-Laplacian (see [[Dolbeault complex]]) and the Euclidean Laplacian on [[complex differential form\|(''p'',''q'')-forms]]. Specifically, let

			: <math>\Delta=\bar{\partial}^\bar{\partial}+\bar{\partial}\bar{\partial}^</math>, and
			: <math>\Delta'=-\sum_k\nabla_k\nabla_{\bar{k}}</math> in a unitary frame at each point.

			According to the Weitzenböck formula, if α ε Ω<sup>(''p'',''q'')</sup>''M'', then

			: Δ'α − Δα = ''A''(α)

			where ''A'' is an operator of order zero involving the curvature. Specifically,
			::if <math>\alpha=\alpha_{i_1i_2\dots i_p\bar{j}_1\bar{j}_2\dots\bar{j}_q}</math> in a unitary frame, then
			::<math>A(\alpha)=-\sum_{k,j_s} \operatorname{Ric}_{\bar{j}_\alpha}^{\bar{k}}\alpha_{i_1i_2\dots i_p\bar{j}_1\bar{j}_2\dots\bar{k}\dots\bar{j}_q}</math> with ''k'' in the ''s''-th place.

			==Other Weitzenböck identities==
			*In [[conformal geometry]] there is a Weitzenböck formula relating a particular pair of differential operators defined on the [[tractor bundle]]. See Branson, T. and Gover, A.R., "Conformally Invariant Operators, Differential Forms, Cohomology and a Generalisation of Q-Curvature", ''Communications in Partial Differential Equations'', '''30''' (2005) 1611–1669.

			==See also==
			*[[Bochner identity]]
			*[[Bochner–Kodaira–Nakano identity]]
			*[[Laplacian operators in differential geometry]]

			==References==
			*{{citation\|title=Principles of algebraic geometry\|first1=Philip\|last1=Griffiths\|first2=Joe\|last2=Harris\|authorlink1=Philip A. Griffiths\|authorlink2=Joe Harris (mathematician)\|publisher=Wiley-Interscience\|publication-date=1994\|isbn=978-0-471-05059-9\|year=1978}}

			{{DEFAULTSORT:Weitzenbock identity}}
			[[Category:Mathematical identities]]
			[[Category:Differential operators]]
			[[Category:Differential geometry]]

Jitse Niesen: mention Gauss–Legendre method

2012-03-23T15:24:39Z

mention Gauss–Legendre method

New page

Hi there. Let me start by introducing the author, her name is Sophia Boon but she by no means really liked that title. My working day occupation is a travel agent. Her family lives in Ohio. My spouse doesn't like it the way I do but what I really like doing is caving but I don't have the time lately.<br><br>Here is my page ... best psychic ([http://srncomm.com/blog/2014/08/25/relieve-that-stress-find-a-new-hobby/ Click Link])

Collocation method - Revision history

en>Addbot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q1779479

Jitse Niesen: mention Gauss–Legendre method