Classification of hypersurfaces in the four dimensional Thurston geometry Nil^3 × R

Abstract

 We investigate hypersurfaces in the four-dimensional Thurston geometry Nil³ × R, by giving a complete classification of hypersurfaces whose second fundamental form is a Codazzi tensor, they are either parallel or totally geodesic. Furthermore, we prove that the totally umbilical hypersurfaces in Nil³ × R are totally geodesic.