# Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group

@article{Bauer2011GeodesicDF, title={Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group}, author={Martin Bauer and Martins Bruveris and Philipp Harms and Peter W. Michor}, journal={Annals of Global Analysis and Geometry}, year={2011}, volume={44}, pages={5-21} }

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 distance on Diffc(S1) vanishes if and only if $${s\leq\frac12}$$. For other manifolds, we obtain a partial characterization: the geodesic distance on Diffc(M) vanishes for $${M=\mathbb{R}\times N, s < \frac12}$$ and for $${M=S^1\times N, s\leq\frac12}$$, with N being a compact Riemannian manifold…

## 44 Citations

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.

Vanishing geodesic distance for right-invariant Sobolev metrics on diffeomorphism groups

- MathematicsAnnals of Global Analysis and Geometry
- 2019

We study the geodesic distance induced by right-invariant metrics on the group $${\text {Diff}}_\text {c}({\mathcal {M}})$$Diffc(M) of compactly supported diffeomorphisms, for various Sobolev norms…

Geodesic distance for right-invariant metrics on diffeomorphism groups: critical Sobolev exponents

- MathematicsAnnals of Global Analysis and Geometry
- 2019

We study the geodesic distance induced by right-invariant metrics on the group \({\text {Diff}}_\text {c}(\mathcal {M})\) of compactly supported diffeomorphisms of a manifold \(\mathcal {M}\) and…

Can we run to infinity? The diameter of the diffeomorphism group with respect to right-invariant Sobolev metrics

- Mathematics
- 2019

The group $${\text {Diff}}({\mathcal {M}})$$ Diff ( M ) of diffeomorphisms of a closed manifold $${\mathcal {M}}$$ M is naturally equipped with various right-invariant Sobolev norms $$W^{s,p}$$ W s ,…

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…

Contactomorphism group with the $$L^2$$L2 metric on stream functions

- Mathematics
- 2016

Here we investigate some geometric properties of the contactomorphism group $$\mathcal {D}_\theta (M)$$Dθ(M) of a compact contact manifold with the $$L^2$$L2 metric on the stream functions. Viewing…

Sobolev Metrics on Shape Space, II: Weighted Sobolev Metrics and Almost Local Metrics

- Mathematics
- 2011

In continuation of [3] we discuss metrics of the form $$ G^P_f(h,k)=\int_M \sum_{i=0}^p\Phi_i(\Vol(f)) \g((P_i)_fh,k) \vol(f^*\g) $$ on the space of immersions $\Imm(M,N)$ and on shape space…

Contactomorphisms with $L^2$ metric on stream functions

- Mathematics
- 2015

Here we investigate some geometric properties of the contactomorphism group $\mathcal{D}_\theta(M)$ of a compact contact manifold with the $L^2$ metric on the stream functions. Viewing this group as…

Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle

- Mathematics
- 2014

In this paper, we study the geodesic flow of a right-invariant metric induced by a general Fourier multiplier on the diffeomorphism group of the circle and on some of its homogeneous spaces. This…

Sub-Riemannian Geometry on Infinite-Dimensional Manifolds

- Mathematics
- 2012

We generalize the concept of sub-Riemannian geometry to infinite-dimensional manifolds modeled on convenient vector spaces. On a sub-Riemannian manifold $$M$$M, the metric is defined only on a…

## References

SHOWING 1-10 OF 54 REFERENCES

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 shape space of surfaces

- Mathematics
- 2012

Let $M$ and $N$ be connected manifolds without boundary with
$\dim(M) < \dim(N)$, and let $M$ compact.
Then shape space in this work is either the manifold of submanifolds of $N$ that are …

The geometry of a vorticity model equation

- Mathematics
- 2010

We show that the modified Constantin-Lax-Majda equation modeling vortex and quasi-geostrophic dynamics [27] can be recast as the geodesic flow on the subgroup $\mathrm{Diff}_{1}^{\infty}(\mathbb{S})$…

Sobolev metrics on diffeomorphism groups and the derived geometry of spaces of submanifolds

- Mathematics
- 2013

Given a finite-dimensional manifold , the group of diffeomorphisms of which decrease suitably rapidly to the identity, acts on the manifold of submanifolds of of diffeomorphism-type , where is a…

VANISHING GEODESIC DISTANCE ON SPACES OF SUBMANIFOLDS AND DIFFEOMORPHISMS

- Mathematics
- 2004

The L 2 -metric or Fubini-Study metric on the non-linear Grassmannian of all submanifolds of type M in a Riemannian man- ifold (N;g) induces geodesic distance 0. We discuss another metric which…

Riemannian Geometries on Spaces of Plane Curves

- Mathematics
- 2003

We study some Riemannian metrics on the space of regular smooth curves in the plane, viewed as the orbit space of maps from the circle to the plane modulo the group of diffeomorphisms of the circle,…

On geodesic exponential maps of the Virasoro group

- Mathematics
- 2007

We study the geodesic exponential maps corresponding to Sobolev type right-invariant (weak) Riemannian metrics μ(k) (k≥ 0) on the Virasoro group Vir and show that for k≥ 2, but not for k = 0,1, each…

Global Analysis on Open Manifolds

- Mathematics
- 2007

Preface Introduction A Setting of Linear Analysis Basics of Riemannian Geometry Tools from Hilbert Space Theory Sobolev Spaces on Open Manifolds Uniform Pseudo-differential and Fourier Integral…

Geodesic flow on the diffeomorphism group of the circle

- Mathematics
- 2003

Abstract
We show that certain right-invariant metrics endow the
infinite-dimensional Lie group of all smooth
orientation-preserving diffeomorphisms of the circle with a
Riemannian structure. The…

Geometry of the Virasoro-Bott group

- Mathematics
- 1998

We consider a natural Riemannian metric on the innite dimensional manifold of all embeddings from a manifold into a Riemannian manifold, and derive its geodesic equation in the case Emb(R;R) which…