Conformal geometry on a class of embedded hypersurfaces in spacetimes

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