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

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

2011-05-02

arxiv.org/abs/1105.0327v2

Mathematics

Differential Geometry

16 pages. Results improved, case s=1/2 solved. Title changed

Scientific paper

We study Sobolev-type metrics of fractional order on the group of compactly supported diffeomorphisms $\Diff_c(M)$, where $M$ is a Riemannian manifold of bounded geometry. We prove that the geodesic distance, induced by the Riemannian metric, vanishes if the order $s$ satisfies $0\le s< \frac 12$. For $M\neq \R$ we show the vanishing of the geodesic distance also for $s=\frac 12$, and for $\dim(M)=1$ we show that the distance is positive for $\frac 12

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