An extension of Calabi's rigidity theorem to complex submanifolds of indefinite complex space forms
type: Item
instance of: scholarly article
title: An extension of Calabi's rigidity theorem to complex submanifolds of indefinite complex space forms
zbMATH Open document ID: 0617.53055
DOI: 10.1007/BF01174794
author: Alfonso Romero
published in: Manuscripta Mathematica
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155250
review text: Let \(F^ N_ S(c)\) be a simply-connected indefinite complex space form with dimension N, index 2S and holomorphic sectional curvature \(c\in {\mathbb{R}}\). In this paper is proved that if f and f' are two full holomorphic isometric immersions of the same indefinite Kähler manifold M into \(F^ N_ S(c)\) and \(F^{N'}_{S'}(c)\), resp., then \(N=N'\), \(S=S'\) and there exists a unique holomorphic rigid motion \(\Phi\) of \(F^ N_ S(c)\) such that \(\Phi _ 0f=f'\). Moreover all such immersions between indefinite complex projective (and hyperbolic) spaces are explicitly obtained. These theorems generalize well known results of \textit{E. Calabi} [Ann. Math., II. Ser. 58, 1-23 (1953; Zbl 0051.131)] to indefinite Kähler metrics. The author would like to say that Prof. Nakagawa informs him that Prof. Umehara has obtained independently the first cited result.
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 32C25
zbMATH DE Number: 4000933
zbMATH Keywords: complex space form
zbMATH Keywords: full holomorphic isometric immersions
zbMATH Keywords: indefinite Kähler manifold
zbMATH Keywords: rigidity theorem
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4000933