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) x R. Further, we prove that if the metric of a complete almost Kenmotsu manifold with conformal Reeb foliation satises the Fischer-Marsden conjecture, then the manifold is Einstein provided the scalar curvature r ≠ -2n(2n + 1).

Mathematics Subject Classification (2010): 53C25, 53D15.

Keywords: Almost Kenmotsu manifold, nullity distribution, The Fischer-Marsden conjecture, Einstein manifold