82.130.84.4: typo corrected

2014-08-11T08:10:59Z

typo corrected

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en>Monkbot: /* Further reading */Fix CS1 deprecated date parameter errors

2014-02-04T12:42:17Z

Further reading: Fix CS1 deprecated date parameter errors

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2001:6B0:1:1E00:6D62:6528:A51C:8FDA: /* Collaborating jondos */ typos: added some missing whitespaces

2013-03-21T16:31:49Z

Collaborating jondos: typos: added some missing whitespaces

← Older revision		Revision as of 18:31, 21 March 2013
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	~~Oscar~~ is ~~what my wife loves to contact me~~ and ~~I completely dig that name~~. ~~Playing baseball~~ is the ~~hobby he will by no means quit doing~~. ~~Years~~ in the ~~past we moved~~ to ~~North Dakota and I adore each working day living here~~. ~~He used~~ to ~~be unemployed but now he~~ is a ~~pc operator but his promotion never arrives~~.<br><br>~~Here~~ is ~~my website~~ - ~~home std test (~~[http://www.~~videokeren~~.~~com~~/~~user~~/~~BFlockhar our website~~])		In [[mathematics]], the '''Lagrangian Grassmannian''' is the [[smooth manifold]] of [[Lagrangian subspace]]s of a real [[symplectic vector space]] ''V''. Its dimension is ''n(n+1)/2'' (where the dimension of ''V'' is ''2n''). It may be identified with the [[homogeneous space]]

			:''U''(''n'')/''O''(''n''),

			where ''U''(''n'') is the [[unitary group]] and ''O''(''n'') the [[orthogonal group]]. Following [[Vladimir Arnold]] it is denoted by Λ(''n''). The Lagrangian Grassmannian is a submanifold of the ordinary [[Grassmannian]] of '''V'''.

			A ''' complex Lagrangian Grassmannian''' is the [[homogeneous space\|complex homogeneous manifold]] of [[Lagrangian subspace]]s of a complex [[symplectic vector space]] ''V'' of dimension 2''n''. It may be identified with the [[homogeneous space]] of complex dimension ''n(n+1)/2''

			:''Sp''(''n'')/''U''(''n''),

			where ''Sp''(''n'') is the [[symplectic group\|compact symplectic group]].

			==Topology==
			The stable topology of the Lagrangian Grassmannian and complex Lagrangian Grassmannian is completely understood, as these spaces appear in the [[Bott periodicity theorem]]: <math>\Omega(Sp/U) \simeq U/O</math>, and <math>\Omega(U/O) \simeq Z\times BO</math> – they are thus exactly the [[Orthogonal_group#Homotopy_groups\|homotopy groups of the stable orthogonal group]], up to a shift in indexing (dimension).

			In particular, the [[fundamental group]] of <math>U/O</math> is [[infinite cyclic]], with a distinguished generator given by the square of the [[determinant]] of a [[unitary matrix]], as a mapping to the [[unit circle]]. Its first [[homology group]] is therefore also infinite cyclic, as is its first cohomology group. Arnold showed that this leads to a description of the '''Maslov index''', introduced by [[V. P. Maslov]].

			For a [[Lagrangian submanifold]] ''M'' of ''V'', in fact, there is a mapping

			:''M'' → Λ(''n'')

			which classifies its [[tangent space]] at each point (cf. [[Gauss map]]). The Maslov index is the pullback via this mapping, in

			:''H''<sup>1</sup>(''M'', '''Z''')

			of the distinguished generator of

			:''H''<sup>1</sup>(Λ(''n''), '''Z''').

			==Maslov index==

			A path of [[symplectomorphism]]s of a symplectic vector space may be assigned a '''Maslov index''', named after [[V. P. Maslov]]; it will be an integer if the path is a loop, and a half-integer in general.

			If this path arises from trivializing the [[symplectic vector bundle]] over a periodic orbit of a [[Hamiltonian vector field]] on a [[symplectic manifold]] or the [[Reeb vector field]] on a [[contact manifold]], it is known as the [[Conley-Zehnder index]]. It computes the [[spectral flow]] of the [[Cauchy-Riemann equations\|Cauchy-Riemann]]-type operators that arise in [[Floer homology]]{{Citation needed\|date=October 2013}}.

			It appeared originally in the study of the [[WKB approximation]] and appears frequently in the study of [[Bohr-Sommerfeld_quantization\|quantization]] and in [[symplectic geometry]] and topology. It can be described as above in terms of a Maslov index for linear Lagrangian submanifolds.

			==References==

			*V. I. Arnold, '' Characteristic class entering in quantization conditions'', Funktsional'nyi Analiz i Ego Prilozheniya, '''1967''', 1,1, 1-14, {{doi\|10.1007/BF01075861}}.
			*[[V. P. Maslov]], ''Théorie des perturbations et méthodes asymptotiques''. 1972
			*{{citation\|url=http://www.maths.ed.ac.uk/~aar/maslov.htm \|title=The Maslov index home page \|first=Andrew\|last= Ranicki}} Assorted source material relating to the Maslov index.

			[[Category:Symplectic geometry]]
			[[Category:Topology of homogeneous spaces]]
			[[Category:Mathematical quantization]]

en>Spinningspark: Reverted 1 edit by 115.248.149.18 (talk): Rv test. (TW)

2012-04-17T11:31:18Z

Reverted 1 edit by 115.248.149.18 (talk): Rv test. (TW)

New page

Oscar is what my wife loves to contact me and I completely dig that name. Playing baseball is the hobby he will by no means quit doing. Years in the past we moved to North Dakota and I adore each working day living here. He used to be unemployed but now he is a pc operator but his promotion never arrives.<br><br>Here is my website - home std test ([http://www.videokeren.com/user/BFlockhar our website])

Crowds - Revision history

82.130.84.4: typo corrected

en>Monkbot: /* Further reading */Fix CS1 deprecated date parameter errors

2001:6B0:1:1E00:6D62:6528:A51C:8FDA: /* Collaborating jondos */ typos: added some missing whitespaces

en>Spinningspark: Reverted 1 edit by 115.248.149.18 (talk): Rv test. (TW)