On the scalar curvature of hypersurfaces in spaces with a Killing field

TL;DR

This paper derives integral formulas for hypersurfaces in spaces with a Killing field, characterizing slices with constant scalar curvature or Gaussian curvature under certain conditions, in both Riemannian and Lorentzian settings.

Contribution

It introduces new integral formulas and characterizations of hypersurfaces with constant scalar curvature in product spaces with a Killing field, extending previous results.

Findings

Slices are the only compact hypersurfaces with constant scalar curvature under specified conditions.

Characterization of surfaces with constant Gaussian curvature in product spaces.

Extension of results to Lorentzian product spaces.

Abstract

We consider compact hypersurfaces in an $(n+1)$-dimensional either Riemannian or Lorentzian space $N^{n+1}$ endowed with a conformal Killing vector field. For such hypersurfaces, we establish an integral formula which, especially in the simpler case when $N=M^n\times R$ is a product space, allows us to derive some interesting consequences in terms of the scalar curvature of the hypersurface. For instance, when $n=2$ and $M^2$ is either the sphere $\mathbb{S}^2$ or the real projective plane $\mathbb{RP}^2$, we characterize the slices of the trivial totally geodesic foliation $M^2\times\{t\}$ as the only compact two-sided surfaces with constant Gaussian curvature in the Riemannian product $M^2\times\mathbb{R}$ such that its angle function does not change sign. When $n\geq 3$ and $M^n$ is a compact Einstein Riemannian manifold with positive scalar curvature, we also characterize the slices as the only compact two-sided hypersurfaces with constant scalar curvature in the Riemannian product $M^n\times\mathbb{R}$ whose angle function does not change sign. Similar results are also established for spacelike hypersurfaces in a Lorentzian product $\mathbb{M}\times\mathbb{R}_1$.