Dajczer, Marcos; Rodriguez, Lucio Rigidity of real Kaehler submanifolds. (English) Zbl 0599.53005 Duke Math. J. 53, 211-229 (1986). This paper deals with isometric immersions f of 2n-dimensional Kaehler manifolds into real Euclidean \((2n+p)\)-space. If the type number is at least 3, then f is holomorphic. If f is minimal, then it is circular (i.e. \(\alpha (JX,Y)=\alpha (X,JY))\), and thus the results of the first author and D. Gromoll [J. Differ. Geom. 22, 13-28 (1985; Zbl 0587.53051)] apply. If f is minimal, then it is congruent to a holomorphic isometric immersion into \({\mathbb{C}}^{n+1}\), if such exist. For type number 3 a similar result holds in \({\mathbb{C}}^{n+q}\). The core of the proofs is a lemma on the determination of the second fundamental form by the Gauss equation generalizing a well-known result of Chern. Reviewer: D.Ferus Cited in 1 ReviewCited in 27 Documents MSC: 53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces 53B35 Local differential geometry of Hermitian and Kählerian structures 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature Keywords:real holomorphic immersions; circular immersions; flat bilinear forms; Gauss equation Citations:Zbl 0587.53051 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] L. Barbosa, M. Dajczer, and L. Jorge, Rigidity of minimal submanifolds in space forms , Math. Ann. 267 (1984), no. 4, 433-437. · Zbl 0531.53047 · doi:10.1007/BF01455960 [2] E. Calabi, Isometric imbedding of complex manifolds , Ann. of Math. (2) 58 (1953), 1-23. · Zbl 0051.13103 · doi:10.2307/1969817 [3] E. Calabi, Quelques applications de l’analyse complexe aux surfaces d’aire minima , Topics in Complex Manifolds, Univ. of Montreal, Montreal, Canada, 1968. [4] M. Dajczer, A characterization of complex hypersurfaces in \(C^m\) , · Zbl 0661.53014 [5] M. Dajczer and D. Gromoll, Real Kaehler submanifolds and uniqueness of the Gauss map , to appear in J. of Differential Geometry. · Zbl 0587.53051 [6] D. Ferus, Symmetric submanifolds of Euclidean space , Math. Ann. 247 (1980), no. 1, 81-93. · Zbl 0446.53041 · doi:10.1007/BF01359868 [7] S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol. II , Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969. · Zbl 0175.48504 [8] H. B. Lawson, Some intrinsic characterizations of minimal surfaces , J. Analyse Math. 24 (1971), 151-161. · Zbl 0251.53003 · doi:10.1007/BF02790373 [9] J. D. Moore, Submanifolds of constant positive curvature. I , Duke Math. J. 44 (1977), no. 2, 449-484. · Zbl 0361.53050 · doi:10.1215/S0012-7094-77-04421-0 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.