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CR products in locally conformal Kähler manifolds. (English) Zbl 1027.53085

Let \({\tilde M}^{2n}\) be a complex \(n\)-dimensional locally conformal Kähler (l.c.K.) manifold, with the complex structure \(J\) and the Hermitian metric \(\tilde g.\) A CR-submanifold of a Hermitian manifold \(({\tilde M}^{2n}, J, \tilde g)\) is a submanifold \(M^m\) endowed with a distribution \(D\) such that \(D\) is invariant (i.e. \(J_x(D)_x=D_x, x\in M^m\)) and its orthogonal complement \(D^{\bot }\) in \(T(M^m)\) is anti-invariant (i.e. \(J_xD^{\bot }_x\subseteq T(M^m)^{\bot }_x, x\in M^m\)). A CR-submanifold \(M^m\) of a Hermitian manifold \(\tilde M^{2n}\) is a CR-product if it is locally a Riemannian product \(M^{2p}\times M^q\) of a complex submanifold \(M^{2p}\) and an anti-invariant submanifold \(M^q\) of \({\tilde M}^{2n}.\)
In the present paper the authors study CR-products in l.c.K. manifolds. In particular, they prove that:
(a) 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 \(D\) and \(D^{\bot }\) are parallel and \(D\) has complex dimension \(1\) whenever \(M^m\) is not orthogonal to the Lee field);
(b) if \(M^m\) is a standard rough CR-product of a complex Hopf manifold (i.e. \(M^m\) is a CR-submanifold of which local CR-manifolds \(\{M_i\}_{i\in I}\) are CR-products), then each leaf of the Levi foliation, orthogonal to the Lee field, is isometric to the sphere \(S^2\);
(c) any warped product CR-submanifold \(M^m=M^{\bot }\times _{f} M^{\top }\), with \(M^{\bot }\) anti-invariant and \(M^{\top }\) invariant is a CR-product, provided that the tangential component of the Lee field is orthogonal to \(M^{\bot }\).

MSC:

53C56 Other complex differential geometry
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C40 Global submanifolds
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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