Conformal geometry on a class of embedded hypersurfaces in spacetimes

Abstract In this work, we study various geometric properties of embedded spacelike hypersurfaces in $1+1+2$ decomposed spacetimes with a preferred spatial direction, denoted $e^{\mu}$, which are orthogonal to the fluid flow velocity spacetime and admit proper conformal transformation. To ensure non-vanishing positivity scalar curvature induced metric on hypersurface, impose that is non-negative associated factor $\varphi$ satisfies $\hat{\varphi}^2+2\hat{\hat{\varphi}}>0$, where \hat{\ast} denotes derivative along direction. Firstly, it demonstrated such hypersurface either Einstein type or twist vanishes them, constant. It then proved if compact admits transformation, these must be isomorphic 3-sphere, make use some well known results Riemannian manifolds admitting transformations. If not have nowhere vanishing sheet expansion, show conclusion fails. However, additional conditions curvatures coincide, strictly negative third higher order derivatives vanish, 3-sphere follows. Furthermore, obtained under also Finally, consider our context locally rotationally symmetric that, type, certain specified constructed explicit examples several Killing vector fields $e^{\mu}$.