Gradient Ricci solitons and Fischer-Marsden equation on cosymplectic manifolds (Q2081239)

The Fischer-Marsden Conjecture [\textit{A. E. Fischer} and \textit{J. E. Marsden}, Bull. Am. Math. Soc. 80, 479--484 (1974; Zbl 0288.53040)] states that a compact Riemannian manifold that admits a non-trivial solution of the Fischer-Marsden equation is necessarily an Einstein manifold. Though it was proved by \textit{O. Kobayashi} [J. Math. Soc. Japan 34, 665--675 (1982; Zbl 0486.53034)] that the conjecture is not valid for general case, the conjecture is partially true in special cases like the manifolds admitting certain additional structures (see [\textit{U. C. De} and \textit{K. Mandal}, Quaest. Math. 43, No. 1, 25--33 (2020; Zbl 1433.53077)]). Recently this topic has become quite popular. In the present paper the authors investigate the existence of non-trivial solutions for the Fischer-Marsden equation within the framework of \((2n+1)\)-dimensional cosymplectic manifolds. It is shown that the existence of such solutions forces the metric to be a gradient \(\eta\)-Ricci soliton. The authors also study gradient Ricci solitons on \(\eta\)-Einstein cosymplectic manifolds.