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Real hypersurfaces and complex submanifolds in complex projective space. (English) Zbl 0597.53021

It is shown that a real hypersurface M in \(P^ n({\mathbb{C}})\) has constant principal curvatures and \(J\xi\) is principal where J is the complex structure and \(\xi\) is a normal vector if and only if M is congruent to an open subset of a homogeneous real hypersurface. Furthermore the complex submanifolds whose principal curvatures neither depend on the point of the submanifold nor on the normal vector are classified.
Reviewer: G.Thorbergsson

MSC:

53B25 Local submanifolds
53B35 Local differential geometry of Hermitian and Kählerian structures
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[1] Uwe Abresch, Notwendige Bedingungen für isoparametrische Hyperflächen in Sphären mit mehr als drei verschiedenen Hauptkrümmungen, Bonner Mathematische Schriften [Bonn Mathematical Publications], vol. 146, Universität Bonn, Mathematisches Institut, Bonn, 1982 (German). U. Abresch, Isoparametric hypersurfaces with four or six distinct principal curvatures. Necessary conditions on the multiplicities, Math. Ann. 264 (1983), no. 3, 283 – 302. · Zbl 0505.53027 · doi:10.1007/BF01459125
[2] M. Buchner, K. Fritzsche, and Takashi Sakai, Geometry and cohomology of certain domains in the complex projective space, J. Reine Angew. Math. 323 (1981), 1 – 52. · Zbl 0447.32003
[3] Thomas E. Cecil, Geometric applications of critical point theory to submanifolds of complex projective space, Nagoya Math. J. 55 (1974), 5 – 31. · Zbl 0291.53014
[4] Thomas E. Cecil and Patrick J. Ryan, Focal sets and real hypersurfaces in complex projective space, Trans. Amer. Math. Soc. 269 (1982), no. 2, 481 – 499. · Zbl 0492.53039
[5] Yoshiaki Maeda, On real hypersurfaces of a complex projective space, J. Math. Soc. Japan 28 (1976), no. 3, 529 – 540. · Zbl 0324.53039 · doi:10.2969/jmsj/02830529
[6] Hans Friedrich Münzner, Isoparametrische Hyperflächen in Sphären, Math. Ann. 251 (1980), no. 1, 57 – 71 (German). · Zbl 0417.53030 · doi:10.1007/BF01420281
[7] Hans Friedrich Münzner, Isoparametrische Hyperflächen in Sphären. II. Über die Zerlegung der Sphäre in Ballbündel, Math. Ann. 256 (1981), no. 2, 215 – 232 (German). · Zbl 0438.53050 · doi:10.1007/BF01450799
[8] Hisao Nakagawa and Ryoichi Takagi, On locally symmetric Kaehler submanifolds in a complex projective space, J. Math. Soc. Japan 28 (1976), no. 4, 638 – 667. · Zbl 0328.53009 · doi:10.2969/jmsj/02840638
[9] K. Nomizu, Elie Cartan’s work on isoparametric families of hypersurfaces, Proc. Sympos. Pure Math., vol. 27, Part I, Amer. Math. Soc., Providence, R. I., 1974, pp. 191-200.
[10] Koichi Ogiue, Differential geometry of Kaehler submanifolds, Advances in Math. 13 (1974), 73 – 114. · Zbl 0275.53035 · doi:10.1016/0001-8708(74)90066-8
[11] Masafumi Okumura, Submanifolds of real codimension of a complex projective space, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 58 (1975), no. 4, 544 – 555 (English, with Italian summary). · Zbl 0345.53039
[12] Hideki Ozeki and Masaru Takeuchi, On some types of isoparametric hypersurfaces in spheres. I, Tôhoku Math. J. (2) 27 (1975), no. 4, 515 – 559. , https://doi.org/10.2748/tmj/1178240941 Hideki Ozeki and Masaru Takeuchi, On some types of isoparametric hypersurfaces in spheres. II, Tôhoku Math. J. (2) 28 (1976), no. 1, 7 – 55. · Zbl 0359.53012 · doi:10.2748/tmj/1178240877
[13] Brian Smyth, Differential geometry of complex hypersurfaces, Ann. of Math. (2) 85 (1967), 246 – 266. · Zbl 0168.19601 · doi:10.2307/1970441
[14] Hitoshi Takagi, A condition for isoparametric hypersurfaces of \?\(^{n}\) to be homogeneous, Tohoku Math. J. (2) 37 (1985), no. 2, 241 – 250. · Zbl 0566.53050 · doi:10.2748/tmj/1178228681
[15] Ryoichi Takagi, On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math. 10 (1973), 495 – 506. · Zbl 0274.53062
[16] Ryoichi Takagi, Real hypersurfaces in a complex projective space with constant principal curvatures, J. Math. Soc. Japan 27 (1975), 43 – 53. · Zbl 0292.53042 · doi:10.2969/jmsj/02710043
[17] Ryoichi Takagi, Real hypersurfaces in a complex projective space with constant principal curvatures. II, J. Math. Soc. Japan 27 (1975), no. 4, 507 – 516. · Zbl 0311.53064 · doi:10.2969/jmsj/02740507
[18] Qi Ming Wang, Isoparametric hypersurfaces in complex projective spaces, Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, Vol. 1, 2, 3 (Beijing, 1980) Sci. Press Beijing, Beijing, 1982, pp. 1509 – 1523.
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