An extension of Calabi's rigidity theorem to complex submanifolds of indefinite complex space forms
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.
scientific article; zbMATH DE number 4000933