Contactomorphisms with $L^2$ metric on stream functions

Boramey Chhay

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arXiv · 1506.08178

Contactomorphisms with $L^2$ metric on stream functions

Boramey Chhay

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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 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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Boramey Chhay. 2015-06-26. Contactomorphisms with $L^2$ metric on stream functions. https://arxiv.org/abs/1506.08178

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