Classification of $(k,\mu )$-contact manifolds with divergence free Cotton tensor and vanishing Bach tensor
Volume 122 / 2019
Annales Polonici Mathematici 122 (2019), 153-163
MSC: Primary 53C15; Secondary 53C21, 52D10.
DOI: 10.4064/ap180228-13-11
Published online: 8 March 2019
Abstract
We first prove that a $(k,\mu )$-contact manifold of dimension $2n+1$ with divergence free Cotton tensor is flat in dimension $ 3 $, and in higher dimensions, locally isometric to $S^n(4)\times E^{n+1}$. Finally, we show that a Bach flat non-Sasakian $(k,\mu )$-contact manifold is flat in dimension 3, and in each higher dimension, there is a unique $(k,\mu )$-contact manifold locally isometric, up to a $D$-homothetic deformation, to the unit tangent sphere bundle of a space of constant curvature $\not =1$. This result provides an example of a Bach flat metric that is neither Einstein nor conformally flat.
