Power closed - Revision history

en>Trappist the monk: /* References */replace mr template with mr parameter in CS1 templates; using AWB

2014-09-25T13:28:10Z

References: replace mr template with mr parameter in CS1 templates; using AWB

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-In [[mathematics]], '''Cartan's equivalence method''' is a technique in [[differential geometry]] for determining whether two geometrical structures are the same up to a [[diffeomorphism]].  For example, if ''M'' and ''N'' are two [[Riemannian manifold]]s with metrics ''g'' and ''h'', respectively,
+== Zeven is wanneer de kinderen terug naar huis Beats Nep ==
-:<math>\phi:M\rightarrow N</math>
+Op een twee blok stuk van de steeg achter Casgrain Ave. In de wijk Villeray, drie zelfgemaakte ijsbanen dook in december. De baan, die achter twee rijen huizen, is eigendom van de stad, maar in de winter is het niet geploegd. Grote Dams begon goed, maar zijn slecht afgelopen. Er was een tijd dat iedereen ze geliefd, iedereen had ze [http://www.edelbeton.com/duits/gekloven/contact.asp?d=48-Beats-Nep Beats Nep] de communisten, kapitalisten, christenen, moslims, hindoes, boeddhisten. Er was een tijd dat Big Dams verhuisde men naar poëzie. <br><br>Yards voor negen jaar dat op oktober, 1988 vervallen en het is nu bezet 15.340 vierkante Yards of de tuin. Serviceclub kreeg 10.893 vierkante We hebben ook extra tips over hoe het opzetten van verschillende soorten studies met betrekking tot multitasking. Tenslotte is het een goed idee om wat achtergrond onderzoek op multitasking, taak schakelen, [http://www.marco-gasparri.com/images/images2/klein/crypt.asp?h=27-Nike-High-Heels-Fake Nike High Heels Fake] en de hersengebieden die u helpen zowel te bereiken doen. <br><br>Als pondcakes last voor het centrum genoeg gebakken is, zal de taart ook vallen. Dit zijn slechts enkele redenen. Een andere reden kan zijn eieren. Bij het bouwen van de site in zijn vrije tijd, John Everett Creighton zei hij putte uit zijn eigen ervaring. Als een MIT afgestudeerde met een graad in de natuurkunde en nucleaire technologie, was hij in hete vraag bij het betreden van de arbeidsmarkt en moest meerdere aanbiedingen en onderhandelingsronden jongleren. Creighton wilden dat proces te vereenvoudigen door de invoering van de vakman in de stoel van de bestuurder: "Ik dacht, wat als het andersom.. <br><br>Lag switchers, ja. Btw ik ben niet degene die zei hacks waren overwegend in BO. Maar deranking lobby's en supersnelle / slowmo / jump, etc.. [http://www.edelbeton.com/duits/gekloven/contact.asp?d=43-Fake-Beats-By-Dre-Solo Fake Beats By Dre Solo] Het is moeilijk om een DAOC ervaring krijgen van een [http://www.hetpakt.com/contact/crypt.asp?y=65 Louis Vuitton Riem Amsterdam] spel dat is niets als DAOC. Slechte vfibsux. Zoveel fanboy in zo'n klein mensje. <br><br>Zeven is wanneer de kinderen terug naar huis, water water ze huilen. Acht is toen de meeste van de mannen in de kolonie klop de deuren, onderzoekend op zoek naar hun vrouwen. Negen is voor Ghar Ghar ki kahanis die spelen op de televisie non-stop. Hij combineert zijn medische kennis van het metabolisme met de huidige bevindingen in de voedingswetenschap tot gewichtsverlies programma's aan te passen aan elke individuele klant. Door het evalueren van voedingsgewoonten en levensstijl van elke persoon, dr. Kim is in staat om hen te helpen de kwesties die hen weerhouden van het verliezen van gewicht, terwijl ze leidt naar een lager cholesterolgehalte, verbeterde zelfvertrouwen, en de verhoogde energie om een gezond leven en leven trots veroveren hun .<ul>
-such that
+   <li>[http://ks35439.kimsufi.com/spip.php?article453/ http://ks35439.kimsufi.com/spip.php?article453/]</li>
-:<math>\phi^*h=g</math>?
+   <li>[http://www.shanghai30p.com/news/html/?118037.html http://www.shanghai30p.com/news/html/?118037.html]</li>
-Although the answer to this particular question was known in dimension 2 to [[Carl Friedrich Gauss|Gauss]] and in higher dimensions to [[Christoffel]] and perhaps [[Riemann]] as well, [[Élie Cartan]] and his intellectual heirs developed a technique for answering similar questions for radically different geometric structures. (For example see the [[Cartan-Karlhede algorithm]].)
+   <li>[http://www.eagleq.com/bbs/forum.php?mod=viewthread&tid=114109&fromuid=7519 http://www.eagleq.com/bbs/forum.php?mod=viewthread&tid=114109&fromuid=7519]</li>
-Cartan successfully applied his equivalence method to many such structures, including [[projective structure]]s, [[CR structure]]s, and [[complex structure]]s{{dn|date=September 2012}}, as well as ostensibly non-geometrical structures such as the equivalence of [[Lagrangian]]s and [[ordinary differential equation]]s.  (His techniques were later developed more fully by many others, such as [[D. C. Spencer]] and [[Shiing-Shen Chern]].)
+   <li>[http://www.canyinmao.com/bbs/home.php?mod=space&uid=128272 http://www.canyinmao.com/bbs/home.php?mod=space&uid=128272]</li>
-The equivalence method is an essentially [[algorithm]]ic procedure for determining when two geometric structures are identical.  For Cartan, the primary geometrical information was expressed in a [[coframe]] or collection of coframes on a [[differentiable manifold]]. See [[method of moving frames]].
+ </ul>

en>Qetuth: more specific stub type

2012-01-01T12:29:56Z

more specific stub type

New page

In [[mathematics]], '''Cartan's equivalence method''' is a technique in [[differential geometry]] for determining whether two geometrical structures are the same up to a [[diffeomorphism]]. For example, if ''M'' and ''N'' are two [[Riemannian manifold]]s with metrics ''g'' and ''h'', respectively,
when is there a diffeomorphism

:<math>\phi:M\rightarrow N</math>

such that

:<math>\phi^*h=g</math>?

Although the answer to this particular question was known in dimension 2 to [[Carl Friedrich Gauss|Gauss]] and in higher dimensions to [[Christoffel]] and perhaps [[Riemann]] as well, [[Élie Cartan]] and his intellectual heirs developed a technique for answering similar questions for radically different geometric structures. (For example see the [[Cartan-Karlhede algorithm]].)

Cartan successfully applied his equivalence method to many such structures, including [[projective structure]]s, [[CR structure]]s, and [[complex structure]]s{{dn|date=September 2012}}, as well as ostensibly non-geometrical structures such as the equivalence of [[Lagrangian]]s and [[ordinary differential equation]]s. (His techniques were later developed more fully by many others, such as [[D. C. Spencer]] and [[Shiing-Shen Chern]].)

The equivalence method is an essentially [[algorithm]]ic procedure for determining when two geometric structures are identical. For Cartan, the primary geometrical information was expressed in a [[coframe]] or collection of coframes on a [[differentiable manifold]]. See [[method of moving frames]].

== Overview of Cartan's method ==
Specifically, suppose that ''M'' and ''N'' are a pair of manifolds each carrying a [[G-structure]] for a structure group ''G''. This amounts to giving a special class of coframes on ''M'' and ''N''. Cartan's method addresses the question of whether there exists a local diffeomorphism φ:''M''→''N'' under which the ''G''-structure on ''N'' pulls back to the given ''G''-structure on ''M''. An equivalence problem has been ''"solved"'' if one can give a complete set of structural invariants for the ''G''-structure: meaning that such a diffeomorphism exists if and only if all of the structural invariants agree in a suitably defined sense.

Explicitly, local systems of one-forms θ''i'' and γ''i'' are given on ''M'' and ''N'', respectively, which span the respective cotangent bundles (i.e., are [[coframe]]s). The question is whether there is a local diffeomorphism φ:''M''→''N'' such that the [[pullback_(differential geometry)|pullback]] of the coframe on ''N'' satisfies
:<math>\phi^*\gamma^i(y)=g^i_j(x)\theta^j(x),\ (g^i_j)\in G</math> (1)
where the coefficient ''g'' is a function on ''M'' taking values in the [[Lie group]] ''G''. For example, if ''M'' and ''N'' are Riemannian manifolds, then ''G''=''O''(''n'') is the orthogonal group and θ''i'' and γ''i'' are [[orthonormal]] coframes of ''M'' and ''N'' respectively. The question of whether two Riemannian manifolds are isometric is then a question of whether there exists a diffeomorphism φ satisfying (1).

The first step in the Cartan method is to express the pullback relation (1) in as invariant a way as possible through the use of a "''prolongation''". The most economical way to do this is to use a ''G''-subbundle ''PM'' of the principal bundle of linear coframes ''LM'', although this approach can lead to unnecessary complications when performing actual calculations. In particular, later on this article uses a different approach. But for the purposes of an overview, it is convenient to stick with the principal bundle viewpoint.

The second step is to use the diffeomorphism invariance of the [[exterior derivative]] to try to isolate any other higher-order invariants of the ''G''-structure. Basically one obtains a connection in the principal bundle ''PM'', with some torsion. The components of the connection and of the torsion are regarded as invariants of the problem.

The third step is that if the remaining torsion coefficients are not constant in the fibres of the principal bundle ''PM'', it is often possible (although sometimes difficult), to '''normalize''' them by setting them equal to a convenient constant value and solving these normalization equations, thereby reducing the effective dimension of the Lie group ''G''. If this occurs, one goes back to step one, now having a Lie group of one lower dimension to work with.

=== The fourth step ===
The main purpose of the first three steps was to reduce the structure group itself as much as possible. Suppose that the equivalence problem has been through the loop enough times that no further reduction is possible. At this point, there are various possible directions in which the equivalence method leads. For most equivalence problems, there are only four cases: complete reduction, involution, prolongation, and degeneracy.

'''Complete reduction.''' Here the structure group has been reduced completely to the [[trivial group]]. The problem can now be handled by methods such as the [[Frobenius theorem (differential topology)|Frobenius theorem]]. In other words, the algorithm has successfully terminated.

On the other hand, it is possible that the torsion coefficients are constant on the fibres of ''PM''. Equivalently, they no longer depend on the Lie group ''G'' because there is nothing left to normalize, although there may still be some torsion. The three remaining cases assume this.

'''Involution.''' The equivalence problem is said to be '''involutive''' (or ''in involution'') if it passes [[Cartan's test]]. This is essentially a rank condition on the connection obtained in the first three steps of the procedure. The Cartan test generalizes the [[Frobenius theorem (differential topology)|Frobenius theorem]] on the solubility of first-order linear systems of partial differential equations. If the coframes on ''M'' and ''N'' (obtained by a thorough application of the first three steps of the algorithm) agree and satisfy the Cartan test, then the two ''G''-structures are equivalent. (Actually, to the best of the author's knowledge, the coframes must be [[real analytic]] in order for this to hold, because the [[Cartan-Kähler theorem]] requires analyticity.)

'''Prolongation.''' This is the most intricate case. In fact there are two sub-cases. In the first sub-case, all of the torsion can be uniquely absorbed into the connection form. (Riemannian manifolds are an example, since the Levi-Civita connection absorbs all of the torsion). The connection coefficients and their invariant derivatives form a complete set of invariants of the structure, and the equivalence problem is solved. In the second subcase, however, it is either impossible to absorb all of the torsion, or there is some ambiguity (as is often the case in [[Gaussian elimination]], for example). Here, just as in Gaussian elimination, there are additional parameters which appear in attempting to absorb the torsion. These parameters themselves turn out to be additional invariants of the problem, so the structure group ''G'' must be ''prolonged'' into a subgroup of a [[jet group]]. Once this is done, one obtains a new coframe on the prolonged space and has to return to the first step of the equivalence method. (See also [[prolongation of G-structures]].)

'''Degeneracy.''' Because of a non-uniformity of some rank condition, the equivalence method is unsuccessful in handling this particular equivalence problem. For example, consider the equivalence problem of mapping a manifold ''M'' with a single one-form θ to another manifold with a single one-form γ such that φ*γ=θ. The zeros of these one forms, as well as the rank of their exterior derivatives at each point need to be taken into account. The equivalence method can handle such problems if all of the ranks are uniform, but it is not always suitable if the rank changes. Of course, depending on the particular application, a great deal of information can still be obtained with the equivalence method.

==References==
*{{cite book|author=Olver, P.J.|title=Equivalence, invariants, and symmetry|publisher=Oxford University Press|year=1995|isbn=0-521-47811-1}}

[[Category:Differential geometry]]
[[Category:Diffeomorphisms]]