On a class of Einstein hypersurfaces immersed in a Riemannian manifold

Abstract

Let x:M→\(\) be an isometric immersion of a hypersurface M into an (n+1)-dimensional Riemannian manifold \(\) and let ρ i (i∈{1,...,n}) be the principal curvatures of M. We denote by E and P the distinguished vector field and the curvature vector field of M, respectively, in the sense of [8].¶If M is structured by a P-parallel connection [7], then it is Einsteinian. In this case, all the curvature 2-forms are exact and other properties induced by E and P are stated.¶The principal curvatures ρ i are isoparametric functions and the set (ρ1,...,ρ n ) defines an isoparametric system [10].¶In the last section, we assume that, in addition, M is endowed with an almost symplectic structure. Then, the dual 1-form π=P♭ of P is symplectic harmonic. If M is compact, then its 2nd Betti number b2≥1.