A homogeneous submanifold with nonzero parallel mean curvature vector in a Euclidean sphere

Abstract

We show that every sufficiently high dimensional Euclidean sphere admits an odd dimensional Riemannian submanifold M having the properties: (1) M is a homogeneous submanifold with nonzero parallel mean curvature vector in the ambient sphere; (2) M is a Berger sphere; (3) M is a Sasakian space form of constant \({\phi}\) -sectional curvature. Note that our manifold M is diffeomorphic but not isometric to a Euclidean sphere.