CR PRODUCTS IN LOCALLY CONFORMAL KAHLER MANIFOLDS

概要

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We study CR products (in the sense of Chen (J. Diff. Geom. $ \mathbf{16} $ (1981), 305-322, 493-509)) in locally conformal Kahler (l.c.K.) manifolds. We show that a CR submanifold $ M^m $ 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 $ \mathcal{D} $ and $ \mathcal{D}^{\perp} $ are parallel and $ \mathcal{D} $ has complex dimension 1 whenever $ M^m $ is not orthogonal to the Lee field). We study $ \mathit{rough} $ CR products, i.e. CR submanifolds in a l.c.K. manifold whose local CR manifolds $ \{M_i\}_{i \in I} $ are CR products (relative to the local Kahler metrics $ \{ \tilde{g}_i\}_{i \in I} $ of the ambient space). If $ M^m $ is a $ \mathit{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 $ S^2 $. Any warped product CR submanifold $ M^m={M^\perp} \times _f M^T $, with $ M^\perp $ anti-invariant and $ M^T $ 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^\perp $.
Faculty of Mathematics, Kyushu Universityの論文