A zoo of diffeomorphism groups on $$\mathbb{R }^{n}$$Rn
@article{Michor2012AZO, title={A zoo of diffeomorphism groups on \$\$\mathbb\{R \}^\{n\}\$\$Rn}, author={Peter W. Michor and David Mumford}, journal={Annals of Global Analysis and Geometry}, year={2012}, volume={44}, pages={529-540} }
We consider the groups $${\mathrm{Diff }}_\mathcal{B }(\mathbb{R }^n)$$DiffB(Rn), $${\mathrm{Diff }}_{H^\infty }(\mathbb{R }^n)$$DiffH∞(Rn), and $${\mathrm{Diff }}_{\mathcal{S }}(\mathbb{R }^n)$$DiffS(Rn) of smooth diffeomorphisms on $$\mathbb{R }^n$$Rn which differ from the identity by a function which is in either $$\mathcal{B }$$B (bounded in all derivatives), $$H^\infty = \bigcap _{k\ge 0}H^k$$H∞=⋂k≥0Hk, or $$\mathcal{S }$$S (rapidly decreasing). We show that all these groups are smooth…
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References
SHOWING 1-10 OF 23 REFERENCES
Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group
- Mathematics
- 2011
We study Sobolev-type metrics of fractional order s ≥ 0 on the group Diffc(M) of compactly supported diffeomorphisms of a manifold M. We show that for the important special case M = S1, the geodesic…
Homogeneous Sobolev Metric of Order One on Diffeomorphism Groups on Real Line
- MathematicsJ. Nonlinear Sci.
- 2014
It is proved that the spaceequipped with the homogeneous Sobolev metric of order one is a flat space in the sense of Riemannian geometry, as it is isometric to an open subset of a mapping space equipped with the flat L2-metric.
Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group. II
- Mathematics
- 2013
The geodesic distance vanishes on the group $$\text{ Diff }_c(M)$$ of compactly supported diffeomorphisms of a Riemannian manifold $$M$$ of bounded geometry, for the right invariant weak Riemannian…
Sobolev metrics on diffeomorphism groups and the derived geometry of spaces of submanifolds
- Mathematics
- 2013
Given a finite-dimensional manifold , the group of diffeomorphisms diffeomorphism of which decrease suitably rapidly to the identity, acts on the manifold of submanifolds of of diffeomorphism-type…
REGULAR INFINITE DIMENSIONAL LIE GROUPS
- Mathematics
- 1998
Regular Lie groups are innite dimensional Lie groups with the property that smooth curves in the Lie algebra integrate to smooth curves in the group in a smooth way (an 'evolution operator' exists).…
Théorie des distributions
- Mathematics
- 1966
II. Differentiation II.2. Examples of differentiation. The case of one variable (n = 1). II.2.3. Pseudofunctions. Hadamard finite part. We calculate the derivative of a function f(x) which is equal…
Weighted diffeomorphism groups of Banach spaces and weighted mapping groups
- Mathematics
- 2012
In this work, we construct and study certain classes of infinite dimensional Lie groups that are modelled on weighted function spaces. In particular, we construct a Lie group of weighted…
A convenient setting for differential geometry and global analysis II
- Mathematics
- 1984
© Andrée C. Ehresmann et les auteurs, 1984, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les…
Vanishing geodesic distance for the Riemannian metric with geodesic equation the KdV-equation
- Mathematics, Physics
- 2011
The Virasoro-Bott group endowed with the right-invariant L2-metric (which is a weak Riemannian metric) has the KdV-equation as geodesic equation. We prove that this metric space has vanishing…
Topics in Differential Geometry
- Mathematics
- 2008
Manifolds and vector fields Lie groups and group actions Differential forms and de Rham cohomology Bundles and connections Riemann manifolds Isometric group actions or Riemann $G$-manifolds…