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 \(M^{\bot }\).


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
Full Text: DOI