CR PRODUCTS IN LOCALLY CONFORMAL KÄHLER MANIFOLDS

We study CR products (in the sense of Chen (J. Diff. Geom. 16 (1981), 305-322, 493-509)) in locally conformal Kähler (l.c.K.) manifolds. We show that a CR submanifold Mm of a l.c.K. manifold has a parallel f-structure P if and only if it is a restricted CR product (i.e. both the holomorphic and totally real distributions D and D⊥ are parallel and D has complex dimension 1 whenever Mm is not orthogonal to the Lee field). We study rough CR products, i.e. CR submanifolds in a l.c.K. manifold whose local CR manifolds {Mi}i∈I are CR products (relative to the local Kähler metrics {gi}i∈I of the ambient space). If Mm is a standard rough CR product of a complex Hopf manifold, each leaf of the Levi foliation, orthogonal to the Lee field, is shown to be isometric to the sphere S2. Any warped product CR submanifold Mm = M⊥ ×f MT, with M⊥ anti-invariant and MT invariant is shown to be a CR product, provided that the tangential component of the Lee field (of the ambient l.c.K. manifold) is orthogonal to M⊥.