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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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