The Fischer-Marsden conjecture on almost Kenmotsu manifolds

Abstract

The purpose of this paper is to investigate the Fischer-Marsden conjecture on almost Kenmotsu manifolds. First, we prove that if a three-dimensional non-Kenmotsu (k, µ)^′-almost Kenmotsu manifold satisfies the Fischer-Marsden conjecture, then the manifold is locally isometric to the product space ℍ²(−4) × ℝ. Further, we prove that if the metric of a complete almost Kenmotsu manifold with conformal Reeb foliation satisfies the Fischer-Marsden conjecture, then the manifold is Einstein provided the scalar curvature r ≠ −2n(2n + 1).