Static perfect fluid spacetime on Riemannian manifolds admitting concurrent-recurrent vector field with Bach tensor

Abstract In this paper, we first consider the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>𝕊</m:mi> <m:mo>⁢</m:mo> <m:mi>ℙ</m:mi> <m:mo>⁢</m:mo> <m:mi>𝔽</m:mi> </m:mrow> </m:math> {\mathbb{SPF}} equation on a Riemannian CRVF-manifold M and show that either M is Einstein or the potential function is pointwise collinear with ζ on an open set U of M . Next, we show that if a Riemannian CRVF-manifold M is the spatial factor of a <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>𝕊</m:mi> <m:mo>⁢</m:mo> <m:mi>ℙ</m:mi> <m:mo>⁢</m:mo> <m:mi>𝔽</m:mi> </m:mrow> </m:math> {\mathbb{SPF}} with a Batch tensor then it is a Batch flat space-time manifold.