Extrinsic spheres in a Kähler manifold. (English) Zbl 0578.53038

Yamaguchi, Seiichi; Nemoto, Hiroaki; Kawabata, Nobuyuki

Mich. Math. J. 31, 15-19 (1984).

In [ibid. 23(1976), 327-330 (1977; Zbl 0349.53040)], the reviewer proved that a complete, connected, simply-connected, even-dimensional extrinsic sphere of a Kähler manifold is isometric to an ordinary sphere if its normal connection is flat. In the present paper, the authors generalize the reviewer’s result by omitting the condition of a flat normal connection to obtain the following result. Theorem. A complete, connected, simply-connected extrinsic sphere \(M^ n\) in a Kähler manifold is one of the following: (1) an ordinary sphere, (2) \(M^ n\) is homothetic to a Sasakian manifold, or (3) \(M^ n\) is a totally real submanifold with non-parallel f-structure in the normal bundle.

Reviewer: B.Y.Chen

MSC:

53C40

Global submanifolds

Keywords:

extrinsic sphere; Kähler manifold; Sasakian manifold; totally real submanifold; f-structure

Citations:

Zbl 0349.53040