CONTACT METRIC MANIFOLDS SATISFYING FISCHER-MARSDEN EQUATION
Vol. 9(2) July 2021, No. 11, pp. 143-150
D. KAR AND P. MAJHI
In this paper we study contact metric manifolds satisfying FischerMarsden equation. We obtain an expression of the Riemannian curvature tensor of a contact metric manifold satisfying Fischer-Marsden equation. We consider the potential function λ of the Fischer-Marsden equation as the eigenvalue of the Ricci operator Q in the direction which is orthogonal to the Reeb vector field ξ. In this case we prove that the solution of the Fischer-Marsden equation is trivial. Also, we prove the Ricci operator has only the eigenvalue 0 corresponding to the eigen vectors orthogonal to ξ. Moreover, we prove a necessary and sufficient condition of the Ricci operator to be Reeb flow invariant. Finally, we show a necessary and sufficient condition of the Ricci operator to be parallel with the Riemannian connection.