Ideal einstein, conformally flat and semi-symmetric immersions

Abstract

Recently, B. Y. Chen introduced a new intrinsic invariant of a manifold, and proved that everyn-dimensional submanifold of real space formsR m (e) of constant sectional curvature e satisfies a basic inequality δ(n 1,…,n k )≤c(n 1,…,n k )H 2+b(n 1,…,n k )e, whereH is the mean curvature of the immersion, andc(n 1,…,n k ) andb(n 1,…,n k ) are constants depending only onn 1,…,n k ,n andk. The immersion is calledideal if it satisfies the equality case of the above inequality identically for somek-tuple (n 1,…,n k ). In this paper, we first prove that every ideal Einstein immersion satisfyingn≥n 1+…+n k +1 is totally geodesic, and that every ideal conformally flat immersion satisfyingn≥n 1+…+n k +2 andk≥2 is also totally geodesic. Secondly we completely classify all ideal semi-symmetric hypersurfaces in real space forms.