Odd dimensional tori and contact structure
Let \(M\) be a metric contact Riemannian manifold with structure \((\phi,\xi,\eta,g)\). \(M\) is called a K-contact Riemannian manifold if the structure vector field \(\xi\) is a Killing field with respect to \(g\). In this paper, the author shows that if a compact manifold of dimension \(2n+1\) satisfies the cohomological condition \(\bigwedge^{2n+1} H^1(M,{\mathbb R}) = H^{2n+1}(M, {\mathbb R})\), then \(M\) cannot be a K-contact Riemannian manifold. And since the odd-dimensional torus \(T^{2n+1}\) satisfies the above cohomological condition, it cannot carry a K-contact structure. The author also shows that \(T^5\) cannot have a contact form \(\eta\) of the form \(\eta= \cos z dx + \sin z dy + f du + g dv\), where \(f,g\) satisfy either \(\partial f/\partial y = \partial g/\partial x = 0\) or \(\partial f/\partial x = \partial g/\partial y= 0\).
scientific article; zbMATH DE number 1057078