Cauchy-Riemann submanifolds of locally conformal Kähler manifolds. (English) Zbl 0659.53041

Dragomir, Sorin

Geom. Dedicata 28, No. 2, 181-197 (1988).

The concept of CR-submanifold [see the reviewer, Geometry of CR- submanifolds (1986; Zbl 0605.53001)] in a locally conformal Kähler manifold (l.c.K.) [see I. Vaisman, Trans. Am. Math. Soc. 262, 533- 542 (1980; Zbl 0446.53048)] is considered by the author of the present paper. With respect to the differential geometry of a CR-submanifold M of a l.c.K. manifold \(\tilde M\) the author obtains the following results: 1. A characterization of CR-products provided \(\tilde M\) has negative holomorphic bisectional curvature [in the Kähler case the result is due to B. Y. Chen, J. Differ. Geom. 16, 305-322 (1981; Zbl 0431.53048)]. 2. Any leaf of the totally real distribution is totally geodesic in M, provided M is mixed totally geodesic and the Lee field of \(\tilde M\) is normal to M. The other results are concerned with invariant, anti-invariant or arbitrary submanifolds of l.c.K. manifolds.
Remark of the reviewer: The Theorem 1 which asserts that an invariant submanifold M is minimal if and only if the Lee field of M is tangent to M has been obtained before by K. Matsumoto [Bull. Yamagata Univ., Nat. Sci. 11, 33-38 (1984; Zbl 0603.53029)].

Reviewer: A.Bejancu

MSC:

53C40

Global submanifolds

Keywords:

CR-submanifold; locally conformal Kähler manifold; CR-products; totally geodesic; Lee field

Citations:

Zbl 0468.53046; Zbl 0605.53001; Zbl 0446.53048; Zbl 0431.53048; Zbl 0603.53029