A study on K-paracontact and (κ,μ)-paracontact manifold admitting vanishing Cotton tensor and Bach tensor

Abstract

The object of the present paper is to study K-paracontact manifold admitting parallel Cotton tensor, vanishing Cotton tensor and to study Bach flatness on K-paracontact manifold.
In that we prove for a K-paracontact metric manifold $M^{2n+1}$ has parallel Cotton tensor if and only if $M^{2n+1}$ is an $\eta$-Einstein manifold and r=-2n(2n+1). Further we show that if g is an $\eta$-Einstein K-paracontact metric and if $g$ is Bach flat then $g$ is an Einstein. Also we study vanishing Cotton tensor on (k,\mu)-paracontact manifold for both k>-1 and k<-1. Finally, we prove that if $M^{2n+1}$ is a (k,\mu)-paracontact manifold for $k\neq -1$ and if $M^{2n+1}$ has vanishing Cotton tensor for $\mu \neq k$, then $M^{2n+1}$ is an $\eta$-Einstein manifold.