Conformal geometry on a class of embedded hypersurfaces in spacetimes

TL;DR

This paper investigates the geometric properties of embedded hypersurfaces in certain spacetimes, focusing on conformal transformations, curvature conditions, and classification results, especially relating to the 3-sphere in the context of Riemannian geometry.

Contribution

It provides new conditions under which hypersurfaces in $1+1+2$ decomposed spacetimes are isomorphic to the 3-sphere, extending understanding of conformal geometry in these settings.

Findings

Hypersurfaces are either Einstein or have vanishing twist.

Compact Einstein hypersurfaces with conformal symmetry are isomorphic to the 3-sphere.

Non-Einstein hypersurfaces with certain curvature conditions can also be isomorphic to the 3-sphere.

Abstract

In this work, we study various properties of embedded 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 a non-vanishing positivity scalar curvature of the induced metric, we impose that the scalar curvature of the conformal metric is non-negative and that the associated conformal factor satisfies $\hat{\varphi}^2+2\hat{\hat{\varphi}}>0$, where $\hat{\ast}$ denotes derivative along $e^{\mu}$. Firstly, it is demonstrated that such hypersurface is either Einstein or the twist vanishes on it, and that the scalar curvature of the induced metric is constant. It is then proved that if the hypersurface is compact and of Einstein, and admits a proper conformal transformation, then the hypersurface must be isomorphic to the $3$-sphere, where we make use of well known results on Riemannian manifolds admitting conformal transformations. If the hypersurface is not Einstein and has 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 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 context of locally rotationally symmetric spacetimes and show that, if the hypersurfaces are compact and not of Einstein type, then under specified conditions the hypersurface is isomorphic to the $3$-sphere, where we constructed explicit examples of proper conformal Killing vector fields along $e^{\mu}$.