N(κ)-paracontact metric manifolds admitting the Fischer-Marsden conjecture

Abstract

We characterize $ N(\kappa) $-paracontact metric manifolds (NKPMM) $ M^{2n+1} $ satisfying the Fischer-Marsden conjecture. We demostrate that, if an $ M^{2n+1} $ satisfies the Fischer-Marsden equation, then either $ M^{2n+1} $ with $ \kappa > -1 $ is a non-Einstein manifold or $ M^{2n+1} $ is locally isometric to $ \mathbb{E}^{n+1} \times \mathbb{H}^{n}(-4) $ for $ n > 1 $. For the $ 3 $-dimensional case, we show that $ M^3 $ is an Einstein manifold.

Keywords

• $ n(\kappa) $-paracontact metric manifolds
• fischer-marsden conjecture
• einstein manifold