CR Submanifolds of Maximal CR Dimension in a Complex Hopf Manifold

Abstract

We study CR submanifolds M in a Hopf manifold (C H N (λ), J 0, g 0) with the Boothby metric g 0,of maximal CR dimension. Any such M is a CR manifold ofhypersurface type, although embedded in higher codimension, and itsanti-invariant distribution H(M)⊥ is spanned by a unit vectorfield U. We classify the CR submanifolds M for which ξ = −J 0 Uis parallel in the normal bundle under assumptions on thespectrum of the Weingarten operator a ξ. We show that (1) ifa ξ(U) = (1/2)A (where A is the anti-Lee vector) andM fibres in tori over a CR submanifold of the complex projectivespace, then M lies on the (total space of the) pullback of the Hopf fibration via S ⊂ C P N − 1, for some geodesic hypersphere S, and (2) if a ξ(U)= 0 and Spec(a ξ) = {0, c}, for some c ∈ R ∖ {0}, then M is locally a Riemannian product of totally geodesicsubmanifolds.