Fischer-Marsden conjecture on K-paracontact manifolds and quasi-para-Sasakian manifolds

The aim of this paper is to study of the non-trivial solutions of Fischer-Marsden conjecture on K-paracontact manifolds and 3-dimensional quasi-para-Sasakian manifolds. We prove that if a semi-Riemannian manifold of dimension $2n+1$ admits a non-trivial solution of Fischer-Marsden equation, then it has constant scalar curvature. We give a comprehensive classification for a $(2n+1)$-dimensional K-paracontact manifold which admits a non-trivial solution of Fischer-Marsden equation. We consider 3-dimensional quasi-para-Sasakian manifolds with $\beta$ constant which admits Fischer-Marsden equation and prove that there are two possibilities. The first one is the scalar curvature $r = −6\beta^2$ and $M^3$ is Einstein. The second one is the manifold is paracosymplectic manifold and η-Einstein.