Abstract
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 a generalization to the \(\mathcal {D}(S^1)\), the diffeomorphism group of the circle, we show that its sectional curvature is always non-negative and that the Riemannian exponential map is not locally \(C^1\). Lastly, we show that the quantomorphism group is a totally geodesic submanifold of \(\mathcal {D}_\theta (M)\) and talk about its Riemannian submersion onto the symplectomorphism group of the Boothby-Wang quotient of M.
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References
Arnold, V.I.: On the differential geometry of infinite-dimensional Lie groups and its application to the hydrodynamics of perfect fluids. In: Arnold V.I. Collected works, vol. 2, Springer, New York (2014)
Arnold, V.I., Khesin, B.: Topological methods in hydrodynamics. Springer, New York (1998)
Bauer, M., Bruveris, M., Harms, P., Michor, P.W.: Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group. Ann. Glob. Anal. Geom. 44 (2013)
Blair, D.: Riemannian geometry of contact and symplectic manifolds. Springer, New York (2010)
Chhay, B., Preston, S.C.: Geometry of the contactomorphism group. Diff. Geo. Appl. 40 (2015)
Contantin, A. Kolev, B.: On the geometric approach to the motion of inertial mechanical systems. J. Phys. A 35 (2002)
Ebin, D.G., Marsden, J.: Diffeomorphism groups and the motion of an incompressible fluid. Ann. Math. 92 (1970)
Ebin, D.G., Misiołek, G., Preston, S.C.: Singularities of the exponential map on the volume-preserving diffeomorphism group. Geom. Funct. Anal. 16 (2006)
Ebin, D. G., Preston, S.C.: Riemannian geometry of the contactomorphism group. Arnold Math. J. 1 (2014)
Kriegl, A., Michor, P.W.: The convenient setting of global analysis. American Mathematical Society, Providence (1997)
Michor, P. W., Mumford, D.: Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms. Doc. Math. 10 (2005)
Misiołek, G., Preston, S.C.: Fredholm properties of Riemannian exponential maps on diffeomorphism groups. Invent. Math. 179 (2009)
Shelukhin, E.: The Hofer norm of a contactomorphism. arXiv:1411.1457. math.SG (2015)
Smolentsev, N.K.: A biinvariant metric on the group of symplectic diffeomorphisms and equations $(\partial /\partial t)\Delta F=\{\Delta F,F\}$, Siberian Math. J. 27 (1986)
Smolentsev, N.K.: Diffeomorphism groups of compact manifolds. J. Math. Sci. 146 (2007)
Acknowledgments
I would like to thank Martin Bauer for suggesting this idea and also my advisor Stephen C. Preston for the helpful discussion. I gratefully acknowledge the support of ESI in Vienna and the Simons Foundation Collaboration Grant #318969.
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Chhay, B. Contactomorphism group with the \(L^2\) metric on stream functions. Ann Glob Anal Geom 49, 205–211 (2016). https://doi.org/10.1007/s10455-015-9488-7
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DOI: https://doi.org/10.1007/s10455-015-9488-7