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Rigidity of real Kaehler submanifolds. (English) Zbl 0599.53005

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

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

Citations:

Zbl 0587.53051
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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
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