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Erschienen in:
Raynaud’s group-scheme and reduction of coverings Kapitel lesen Erstes Kapitel lesen

2012 | OriginalPaper | Buchkapitel

On the geometry of the diffeomorphism group of the circle

verfasst von : Adrian Constantin, Boris Kolev

Erschienen in: Number Theory, Analysis and Geometry

Verlag: Springer US

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Abstract

We discuss some of the possibilities of endowing the diffeomorphism group of the circle with Riemannian structures arising from right-invariant metrics.

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Vorheriges Kapitel Weyl group multiple Dirichlet series of type A 2
Nächstes Kapitel Harmonic representatives for cuspidal cohomology classes
Fußnoten
1
A is symmetric if (Au, v) = (Av, u) for all \(u,v \in{\mathfrak{g}}^{{_\ast}}\), where the round brackets stand for the pairing of elements of the dual spaces \(\mathfrak{g}\) and \({\mathfrak{g}}^{{_\ast}}\).
 
2
Notice that geodesics issuing from some g 0 ∈ G are obtained via left translation by g 0 from geodesics issuing from e.
 
3
The coadjoint action of G on \({\mathfrak{g}}^{{_\ast}}\) is defined by \(({\mathrm{Ad}}_{g}^{{_\ast}}\,m,u) = (m,{\mathrm{Ad}}_{{g}^{-1}}\,u)\).
 
4
The strong version does not require ω to be an exact form. It only assumes that ω is G-invariant and that the symplectic group action of G on M has a moment map.
 
5
It is defined by \((\mathrm{{ad}}_{\omega }^{{_\ast}}\,m,u) = -(m,\mathrm{{ ad}}_{\omega }\,u)\).
 
6
A topological vector space E has a canonical uniform structure. When this structure is complete and when the topology of E may be given by a countable family of semi-norms, we say that E is a Fréchet vector space. In a Fréchet space, such classical results like the Cauchy–Lipschitz theorem or the local inverse theorem are no longer valid in general as they are in on Banach manifold. the typical example of a Fréchet space is the space of smooth functions on a compact manifold where semi-norms are just the C k -norms (\(k = 0, 1,\ldots \)).
 
7
That is, for all \(u,v \in \mathrm{{C}}^{\infty }({\mathbb{S}}^{1})\), \(\mathrm{Supp}(u) \cap \mathrm{Supp}(v) = \varnothing \Rightarrow \mathbf{a}(u,v) = 0\).
 
8
Indeed, this map is not locally surjective. Otherwise, every diffeomorphism sufficiently near to the identity (for the C ∞ topology) would have a square root. However one can build (see [19]) diffeomorphisms arbitrary near to the identity which have exactly 1 periodic orbit of period 2n. But the number of periodic orbits of even periods of the square of a diffeomorphism is always even. Therefore, such a diffeomorphism cannot have a square root.
 
9
If \(\mathfrak{g}\) is the Lie algebra of a Lie group G, this structure corresponds to the reduction of the canonical symplectic structure on T  ∗  G by the left action of G on T  ∗  G.
 
10
A special case occurs when this cocycle γ is a coboundary i.e. γ(u, v) = m 0([u, v]) for some \({m}_{0} \in{\mathfrak{g}}^{{_\ast}}\) (freezing structure).
 
11
By Gel’fand, Dorfman, Magri. See the review [23].
 
12
The composition in the Virasoro group \(\mathrm{Vir} = \mathrm{Diff}({\mathbb{S}}^{1}) \times\mathbb{R}\) is given by
$$\begin{array}{rcl} (\phi ,\alpha ) \circ(\psi ,\beta ) = \Big{(}\phi\circ\psi ,\alpha+ \beta+ B(\phi ,\psi )\Big{)}& & \\ \end{array}$$
where
$$\begin{array}{rcl} B(\phi ,\psi ) = -\frac{1} {2}{ \int\nolimits \nolimits }_{0}^{1}\log {(\phi (\psi (x)))}_{x}\;d\log {\psi }_{x}(x)& & \\ \end{array}$$
is the Bott cocycle.
 
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Adams RA, Sobolev spaces, Academic Press, 1975.
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Arnold VI, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble) 16 (1966), 319–361.
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Arnold VI and Khesin BA, Topological methods in hydrodynamics, Springer-Verlag, New York, 1998.MATH
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Constantin A and Kolev B, On the geometric approach to the motion of inertial mechanical systems, J. Phys. A 35(2002), R51–R79.
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Constantin A and Kolev B, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv. 78 (2003), 787–804.
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Constantin A and Kolev B, Integrability of invariant metrics on the diffeomorphism group of the circle, J. Nonlinear Sci. 16 (2006), 109–122.
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Constantin A, Kolev B and Lenells J, Integrability of invariant metrics on the Virasoro group, Phys. Lett. A 350 (2006), 75–80.
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Constantin A, Kappeler T, Kolev B, and Topalov P, On geodesic exponential maps of the Virasoro group, Ann. Glob. Anal. Geom., 31 (2007), 155–180.MathSciNetMATHCrossRef
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Constantin A and McKean HP, A shallow water equation on the circle, Comm. Pure Appl. Math. 52 (1999), 949–982.
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Euler L, Theoria motus corporum solidorum seu rigidorum ex primiis nostrae cognitionis principiis stabilita et ad onmes motus qui inhuiusmodi corpora cadere possunt accomodata, Mémoires de l’Académie des Sciences Berlin (1765).
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Hamilton R, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. 7 (1982), 66–222.
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Khesin B and Misiolek G, Euler equations on homogeneous spaces and Virasoro orbits. Adv. Math. 176 (2003), no. 1, 116–144.
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Kolev B, Bi-Hamiltonian systems on the dual of the Lie algebra of vector fields of the circle and periodic shallow water equations, Phil. Trans. Roy. Soc. London, 365 (2007), 2333–2357.MathSciNetMATHCrossRef
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Kouranbaeva S, The Camassa-Holm equation as a geodesic flow on the diffeomorphism group, J. Math. Phys. 40 (1999), 857–868.
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Milnor J, Remarks on infinite-dimensional Lie groups, in Relativity, Groups and Topology, pp. 1009–1057, (1984), North-Holland, Amsterdam.
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Misiołek G, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys. 24 (1998), 203–208.
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Ovsienko V and Roger C, Looped cotangent Virasoro algebra and non-linear integrable systems in dimension 2+1, Comm. Math. Phys. 273 (2007), 357–378.
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Peetre J, Une caractérisation abstraite des opérateurs différentiels, Math. Scand. 7 (1959), 211–218.
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Praught J and Smirnov RG, Andrew Lenard: a mystery unraveled, SIGMA 1 (2005), 7pp.
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Souriau JM, Structure of Dynamical Systems Birkhäuser, Boston, 1997.MATH
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Metadaten
Titel
On the geometry of the diffeomorphism group of the circle
verfasst von
Adrian Constantin
Boris Kolev
Copyright-Jahr
2012
Verlag
Springer US
Buch
Number Theory, Analysis and Geometry

Print ISBN: 978-1-4614-1259-5

Electronic ISBN: 978-1-4614-1260-1

Copyright-Jahr: 2012

DOI
https://doi.org/10.1007/978-1-4614-1260-1_7