Static perfect fluid spacetimes on f-Kenmotsu 3-manifolds

Abstract

The present article deals with static perfect fluid spacetimes on f-Kenmotsu 3-manifolds. At first, we demonstrate if a 3-dimensional f-Kenmotsu manifold with constant scalar curvature as the spatial factor of a static perfect fluid spacetime, then either it is a space of constant sectional curvature or \(grad\, \psi \) is pointwise collinear with \(\xi \) and the warping function of the static perfect fluid spacetime is given by \(\psi = k_1 t + k_2\), \(k_1 \ne 0\). As a result, we establish that if a cosymplectic manifold of dimension three with constant scalar curvature is the spatial factor of a static perfect fluid spacetime, then either it is flat or, the manifold becomes a space of constant sectional curvature. Next, we show that under certain restrictions if a 3-dimensional f-Kenmotsu manifold is the spatial factor of a static perfect fluid spacetime, then either the manifold is a space of constant sectional curvature or, the manifold is locally isometric to either the flat Euclidean space \(\mathcal {R}^3\) or the Riemannian product \(\mathcal {R}\times M^2(c)\), where \(M^2(c)\) represents a Kahler surface with constant curvature \(c\ne 0\), provided \(\xi \psi =0\) and \(\xi \tilde{f} =0\). Lastly, we have cited an example of an f-Kenmotsu manifold to validate our result.