Romero, Alfonso An extension of Calabi’s rigidity theorem to complex submanifolds of indefinite complex space forms. (English) Zbl 0617.53055 Manuscr. Math. 59, 261-276 (1987). 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 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. Cited in 1 ReviewCited in 3 Documents MSC: 53C40 Global submanifolds 53C55 Global differential geometry of Hermitian and Kählerian manifolds 32C25 Analytic subsets and submanifolds Keywords:complex space form; full holomorphic isometric immersions; indefinite Kähler manifold; rigidity theorem Citations:Zbl 0051.131 PDFBibTeX XMLCite \textit{A. Romero}, Manuscr. Math. 59, 261--276 (1987; Zbl 0617.53055) Full Text: DOI EuDML References: [1] M. Barros, A. Romero, ?Indefinite K?hler manifolds?, Math. Ann.261, 55-62 (1982) · Zbl 0487.53021 · doi:10.1007/BF01456410 [2] S. Bochner, ?Curvature in Hermitian metric?, Bull. Amer. Math. Soc.53, 179-195 (1947) · Zbl 0035.10403 · doi:10.1090/S0002-9904-1947-08778-4 [3] E. Calabi, ?Isometric imbeddings of complex manifolds?, Ann. of Math.58, 1-23 (1953) · Zbl 0051.13103 · doi:10.2307/1969817 [4] S. Montiel, A. Romero, ?Complex Einstein hypersurfaces of indefinite complex space forms?, Math. Proc. Camb. Phil. Soc.94 495-508 (1983) · Zbl 0536.53024 · doi:10.1017/S0305004100000888 [5] K. Nomizu, B. Smyth, ?Differential geometry of complex hypersurfaces II?, J. Math. Soc. Japan20, 498-521 (1968) · Zbl 0181.50103 · doi:10.2969/jmsj/02030498 [6] A. Romero, ?Some examples of complete indefinite complex Einstein hypersurfaces not locally symmetric?, Proc. Amer. Math. Soc.98, 283-286 (1986) · Zbl 0603.53028 · doi:10.1090/S0002-9939-1986-0854034-6 [7] A. Romero, ?On a certain class of complex Einstein hypersurfaces in indefinite complex space forms?, Math. Zeitschrift192, 627-635 (1986) · Zbl 0579.53019 · doi:10.1007/BF01162709 [8] J. A. Wolf, Spaces of constant curvature, McGraw-Hill, 1967 · Zbl 0162.53304 [9] H. Wu, ?The Bochner technique?, Proc. of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, Science Press, Beijing, People’s Republic of China, Vol 2 929-1071 (1982) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.