The dimension of leaf closures of K-contact flows
Let \(M\) be a \((2n + 1)\)-dimensional manifold with a contact structure \((\xi, \alpha, g)\). The author considers special vector fields, mainly those which are infinitesimal isometries. They arise naturally in the setting of isometric flows where they describe the leaf closures. By considering the abelian Lie subalgebra of Killing vector fields containing the characteristic vector field \(\xi\) the author shows that a leaf closure of a \(K\)-contact \((2n + 1)\)-dimensional compact manifold has dimension at most \(n + 1\). It is known that the basic first cohomology of a \(K\)-contact flow on a compact manifold is isometric to its first de Rham cohomology. It is therefore expected that the de Rham cohomology will affect the topology of those flows. The author shows how the first Betti number of the manifold puts some restrictions on the maximum dimension of leaf closures. Using the first cohomological rank \(r_1 (M)\) of\(M\), that is the rank of the vector space of harmonic vector fields on \(M\), the author proves that a leaf closure of the \(K\)-contact flow on a compact \(K\)-contact \((2n + 1)\)-dimensional manifold \(M\) with \(K\)-contact metric \(g\) has dimension at most \(\inf \{n + 1, 2n + 1 - r_1 (M)\}\).
scientific article; zbMATH DE number 762854
scientific article