Right-invariant Sobolev metrics ${H}^{s}$ on the diffeomorphisms group of the circle

Details Right-invariant Sobolev metrics ${H}^{s}$ on the diffeomorphisms group of the circle Right-invariant Sobolev metrics ${H}^{s}$ on the diffeomorphisms group of the circle

2012-02-23

arxiv.org/abs/1202.5122v1

Physics

Mathematical Physics

35 pages. arXiv admin note: significant text overlap with arXiv:1010.4844

Scientific paper

We study the geodesic flow on the diffeomorphisms group of the circle with respect to the right-invariant metric induced by the fractional Sobolev norm $H^s$ for $s\ge1/2$. We show that the corresponding initial value problem possesses a maximal solution in the smooth category and that the Riemannian exponential mapping is a smooth diffeomorphism from a neighbourhood of 0 in $C^{\infty}(S)$ onto a neighbourhood of the identity in $Diff^{\infty}(S)$.

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