Abstract
We discuss some of the possibilities of endowing the diffeomorphism group of the circle with Riemannian structures arising from right-invariant metrics.
Mathematics Subject Classification (2010): 35Q35, 58B25
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Notes
- 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 In a Fréchet space, such classical results like the Cauchy–Lipschitz theorem or the local inverse theorem are no longer valid in general as they are in on Banach manifold.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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Dedicated to the memory of Professor Serge Lang
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Constantin, A., Kolev, B. (2012). On the geometry of the diffeomorphism group of the circle. In: Goldfeld, D., Jorgenson, J., Jones, P., Ramakrishnan, D., Ribet, K., Tate, J. (eds) Number Theory, Analysis and Geometry. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-1260-1_7
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