On the Cartan-Norden theorem for affine Kähler immersions. (English) Zbl 0719.53029

Nomizu, Katsumi; Podestà, Fabio

Nagoya Math. J. 121, 127-135 (1991).

The purpose of this paper is to prove an analogue of what was called the Cartan-Norden theorem for affine immersions by the author and U. Pinkall [Math. Z. 195, 165-178 (1987; Zbl 0629.53012)] in the case of affine Kähler immersions studied by the author, U. Pinkall and F. Podestà [Nagoya Math. J. 120, 205-222 (1990; Zbl 0701.53038)]. The main result states: If a non-flat Kähler manifold \((M^ n,g)\) admits a nondegenerate affine Kähler immersion into \({\mathbb{C}}^{n+1}\), then for every point x of \(M^ n\) there is a parallel pseudokählerian metric in \({\mathbb{C}}^{n+1}\) such that f is locally isometric around x.

Reviewer: K.Nomizu (Providence)

MSC:

53C40	Global submanifolds
53A15	Affine differential geometry

Keywords:

Cartan-Norden theorem; affine Kähler immersions; parallel pseudokählerian metric

Citations:

Zbl 0629.53012; Zbl 0701.53038

References:

[1]	Nagoya Math. J. 120 pp 205– (1990) · Zbl 0701.53038 · doi:10.1017/S0027763000003342
[2]	On the geometry of affine immersions Math. Z. 195 pp 165– (1987)
[3]	DOI: 10.2969/jmsj/02030498 · Zbl 0181.50103 · doi:10.2969/jmsj/02030498
[4]	Differential Geometry on Complex and Almost Complex Spaces (1965) · Zbl 0127.12405
[5]	Ann. of Math. pp 246– (1967)
[6]	Foundations of Differential Geometry II (1969)