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

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 $n\ge 3$ 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 $\mathbb{M}\times {\mathbb{R}}_1$.

First version (April 2008). Final version (July 2008). To appear in Advances in Geometry

Cited by in corpus (1)

• A Hilbert-type theorem for spacelike surfaces with constant Gaussian curvature in ${\mathbb{H}}^2\times {\mathbb{R}}_1$