Classification of (k, μ)-contact manifolds with divergence free Cotton tensor and vanishing Bach tensor

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) x En+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 equal 1. This result provides an example of a Bach flat metric that is neither Einstein nor conformally flat.