# On Completeness of Groups of Diffeomorphisms

@article{Bruveris2014OnCO, title={On Completeness of Groups of Diffeomorphisms}, author={Martins Bruveris and Franccois-Xavier Vialard}, journal={arXiv: Differential Geometry}, year={2014} }

We study completeness properties of the Sobolev diffeomorphism groups $\mathcal D^s(M)$ endowed with strong right-invariant Riemannian metrics when the underlying manifold $M$ is $\mathbb R^d$ or compact without boundary. The main result is that for $s > \dim M/2 + 1$, the group $\mathcal D^s(M)$ is geodesically and metrically complete with a surjective exponential map. We then present the connection between the Sobolev diffeomorphism group and the large deformation matching framework in order…

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## References

SHOWING 1-10 OF 65 REFERENCES

Geodesics On The Symplectomorphism Group

- Mathematics
- 2012

Let M be a compact manifold with a symplectic form ω and consider the group $${\mathcal{D}_\omega}$$ consisting of diffeomorphisms that preserve ω. We introduce a Riemannian metric on M which is…

Fredholm properties of Riemannian exponential maps on diffeomorphism groups

- Mathematics
- 2009

We prove that exponential maps of right-invariant Sobolev Hr metrics on a variety of diffeomorphism groups of compact manifolds are nonlinear Fredholm maps of index zero as long as r is sufficiently…

Curvatures of Sobolev metrics on diffeomorphism groups

- Mathematics
- 2011

Many conservative partial differential equations correspond to geodesic equations on groups of diffeomorphisms. Stability of their solutions can be studied by examining sectional curvature of these…

On the Regularity of the Composition of Diffeomorphisms

- Mathematics
- 2011

For $M$ a closed manifold or the Euclidean space $\mathbb{R}^n$, the authors present a detailed proof of regularity properties of the composition of $H^s$-regular diffeomorphisms of $M$ for $s >…

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…

Riemannian Exponential Maps of the Diffeomorphism Group of T2

- Mathematics
- 2008

We study the exponential maps induced by right-invariant weak Riemannian metrics of Sobolev type of order k 0 on the Lie group of smooth, orientation preserving dieomorphisms of the two dimensional…

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 …

Geodesic completeness for Sobolev Hs-metrics on the diffeomorphism group of the circle

- Mathematics
- 2013

We prove that the weak Riemannian metric induced by the fractional Sobolev norm Hs on the diffeomorphism group of the circle is geodesically complete, provided that s > 3/2.

On the well-posedness of the incompressible Euler Equation

- Mathematics
- 2013

In this thesis we prove that the homogeneous incompressible Euler equation of hydrodynamics on the Sobolev spaces $H^s(\R^n)$, $n \geq 2$ and $s > n/2+1$, can be expressed as a geodesic equation on…

GEODESIC COMPLETENESS FOR SOBOLEV METRICS ON THE SPACE OF IMMERSED PLANE CURVES

- MathematicsForum of Mathematics, Sigma
- 2014

Abstract We study properties of Sobolev-type metrics on the space of immersed plane curves. We show that the geodesic equation for Sobolev-type metrics with constant coefficients of order 2 and…