Constant mean curvature hypersurfaces with constant angle in semi-Riemannian space forms

Abstract

We study constant angle semi-Riemannian hypersurfaces M immersed in semi-Riemannian space forms, where the constant angle is defined in terms of a closed and conformal vector field Z in the ambient space form. We show that such hypersurfaces belong to the class of hypersurfaces with a canonical principal direction. This property is a type of rigidity. We further specialize to the case of constant mean curvature (CMC) hypersurfaces and characterize them in two relevant cases: when the hypersurface is orthogonal to Z then it is totally umbilical, whereas if Z is tangent to the hypersurface then it has zero Gauss–Kronecker curvature and either its mean curvature vanishes or the ambient is a semi-Euclidean space. We also treat in detail the surface case, giving a full characterization of the constant angle CMC surfaces immersed in all three dimensional space forms. They are isoparametric surfaces with constant principal curvatures when the ambient is flat. If the mean curvature of the surface is not ±2/3 they are either totally umbilic or totally geodesic. In particular when the surface has zero mean curvature it is totally geodesic.