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

"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 η-Einstein manifold and r=-2n(2n+1). Further we show that if g is an η-Einstein K-paracontact metric and if g is Bach flat then g is an Einstein. Also we study vanishing Cotton tensor on (k,μ)-paracontact manifold for both k>-1 and k<-1. Finally, we prove that if M^{2n+1} is a (k,μ)-paracontact manifold for k ≠ -1 and if M^{2n+1} has vanishing Cotton tensor for μ ≠ k, then M^{2n+1} is an η-Einstein manifold."