Contactomorphism group with the $$L^2$$ L 2 metric on stream functions

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.