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⊥.
抄録全体を表示