Cotton tensor, Bach tensor and Kenmotsu manifolds

Abstract

It is known that Einstein metrics are Bach flat. So, the question may arises whether there exists any Riemannian metric which is Bach flat but not Einstein. In this paper, we answer this by constructing several example on certain class of warped product manifold. Indeed the warped product spaces $\mathbb {R}\times _f \mathbb {CP}^n$ and $\mathbb {R}\times _f \mathbb {CH}^n$ with the warping function $f(t) = ke^t$ (where k is a non zero constant) are non Einstein Bach flat manifolds. These spaces are commonly known as Kenmotsu manifold. Moreover, we prove that a Kenmotsu manifold with parallel Cotton tensor is $\eta$-Einstein and Bach flat. Next, we establish that any $\eta$-Einstein Kenmotsu manifold of dimension $>3$ is Bach flat.