Some remarks on \(R\)-contact flows
A flow \({\mathfrak F} = (M,{\mathfrak F})\) (tangentially oriented 1-dimensional foliation) is said to be Riemannian if there exists a Riemannian metric \(g\) on \(M\) such that for any vector field \(X\) tangent to \(\mathfrak F\) and \(U\), \(V\) orthogonal to \(\mathfrak F\), one has \((L\times g)(U,V) = 0\). Then the metric \(g\) is called a bundle-like metric. A contact manifold \((M,\alpha)\) is said to be an \(R\)-contact manifold if its characteristic vector field \(\xi\) of \(\alpha\) defines a Riemannian flow. The author obtains several results concerning the structures of the set \(F_ \xi\) of periodic points of \(\xi\) and the curvature of the \(R\)-metrics. Furthermore, the author proves that no torus \(T^{2n + 1}\) can carry an \(R\)-contact metric, which generalizes a result of \textit{D. E. Blair} [Contact manifolds in Riemannian geometry, Lect. Notes Math. 509, Springer Verlag (1976; Zbl 0319.53026)].
scientific article; zbMATH DE number 727087
scientific article