Lie flows on contact manifolds (Q2800391)

Let \(M\) be a closed manifold of dimension \((2n+1)\), equipped with a flow \(\mathcal F\). It is said that \(\mathcal F\) is a contact flow if it is the characteristic flow of some contact \(1\)-form \(\alpha\) on \(M\). A natural problem is to give interesting characterizations of the contact flows on \(M\). The author gives a solution to this problem when \(\mathcal F\) is a Lie \(\mathfrak g\)-flow; i.e., when there is a Lie algebra \(\mathfrak g\) of transverse vector fields whose evaluation at every point \(x\in M\) defines a linear isomorphism of \(\mathfrak g\) to the normal space \(T_xM/T_x{\mathcal F}\). In this case, it is proved that \(\mathcal F\) is contact if and only if (a)~any orbit of \(\mathcal F\) is closed, and (b)~there exists a symplectic form \(\omega\) on the leaf space \(N=M/{\mathcal F}\) such that \([\omega]\in H^2(N;{\mathbb Z})\) and \(e({\mathcal F})=-[\omega]\), where \(e({\mathcal F})\) is the Euler class of \(\mathcal F\). If moreover \(\omega\) can be chosen to be algebraic in~(b), then it is also shown that \(\mathcal F\) is a homogeneous Lie \(\mathfrak g\)-flow, and the contact form \(\alpha\) can be chosen to be homogeneous. Here, the homogeneity condition on \(\mathcal F\) means that there is a surjective homomorphism of connected Lie groups, \(D:H\to G\), where the Lie algebra of \(G\) is \(\mathfrak g\), and there exists a lattice \(\Delta\subset H\) such that \((M,{\mathcal F})\) is diffeomorphic to \(\Delta\backslash H\) equipped with the foliation defined by the fibers of \(D\). If moreover \(\alpha\) corresponds to a form on \(\Delta\backslash H\) whose lift to \(H\) is left-invariant, then it is called homogeneous. These results are extensions of theorems proved by \textit{M. Llabrés} and \textit{A. Reventós} when \(M\) is of dimension \(3\) [Publ. Mat., Barc. 33, No. 3, 517--531 (1989; Zbl 0765.57021)].