On the Scalar Curvature of Hypersurfaces in Spaces with a Killing Field
2010 - A.L. Albujer, J.A. Aledo, L.J. Alías.
Advances in Geometry 10, 487-503 (2010).
We consider compact hypersurfaces in an (n + 1)-dimensional either Riemannian or Lorentzian space endowed with a conformal Killing vector field. For such hypersurfaces, we establish an integral formula which, especially in the simpler case when 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 is either the sphere or the real projective plane , we characterize the slices of the trivial totally geodesic foliation as the only compact two-sided surfaces with constant Gaussian curvature in the Riemannian product such that its angle function does not change sign. When n ? 3 and 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 whose angle function does not change sign. Similar results are also established for spacelike hypersurfaces in a Lorentzian product ???? × ?1.