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Here we investigate some geometric properties of the contactomorphism group$\mathcal{D}_\theta(M)$ of a compact contact manifold with the $L^2$ metric onthe stream functions. Viewing this group as a generalization to the$\mathcal{D}(S^1)$, the diffeomorphism group of the circle, we show that itssectional curvature is always non-negative and that the the Riemannianexponential map is not locally $C^1$. Lastly, we show that the quantomorphismgroup is a totally geodesic submanifold of $\mathcal{D}_\theta(M)$ and talkabout its Riemannian submersion onto the symplectomorphism group of theBoothby-Wang quotient of $M$.

Contactomorphisms with $L^2$ metric on stream functions