Reduction in contact geometry (Q2718656)

A contact structure on the \((2n+1)\)-dimensional manifold \(C\) is defined by a distribution \(H\subset TC\) of maximal rank. The author defines the notion of a moment map for contact manifolds. He proves the following theorem. Let \((C,H)\) be a contact manifold and let \(G\) be a Lie group acting by contactomorphisms freely and properly on \(C\). Let \(L=TC/H\), \(\Phi:C\to Y=g^*\otimes L\) be the induced moment map and \(0\subseteq Y\) be the image of the zero section of \(Y\to C\). Then the following holds. (a) \(M=\Phi^{-1}(0)\) is a smooth invariant submanifold of \(C\); (b) \(i^\#H=i_*^{-1} (i^*H)\subseteq TM\) defines a distribution of constant rank on \(M\), where \(i:M\hookrightarrow C\) denotes the inclusion map; (c) there exists a unique distribution \(H_0\subseteq TC_0\) of hyperplanes with \(\pi^\# H_0=i^\#H\), where \(\pi:M\to C_0=M/G\) denotes the natural \(G\)-fibration. Moreover \(H_0\) is nondegenerate and defines a contact structure on \(C_0\). Similar results are obtained in the case of exact contact manifolds (when the distribution \(H\) is defined as the kernel of a globally defined 1-form \(\alpha\) on \(C)\).