Juan de Dios PÉREZ, David PÉREZ-LÓPEZ

New results on derivatives of the shape operator of a real hypersurface in a complex projective space

We consider real hypersurfaces M in complex projective space equipped with both the Levi-Civita and generalized Tanaka-Webster connections. For any nonnull real number k and any symmetric tensor field of type (1,1) L on M we can define a tensor field of type (1,2) on M, L(k) F , related to both connections. We study symmetry and skewsymmetry of the tensor A(k) F associated to the shape operatorA of M.

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[1] Blair DE. Almost contact manifolds with Killing structure tensor. Pacific Journal of Mathematics 1971; 39: 285-292.
[2] Blair DE. Riemannian Geometry of contact and symplectic manifolds. Progress in Mathematics. Birkhauser. Boston 2002; 203.
[3] Cecil TE, Ryan PJ. Focal sets and real hypersurfaces in complex projective space. Transactions of the American Mathematical Society 1982; 269: 481-499.
[4] Cho JT. CR-structures on real hypersurfaces of a complex space form. Publicationes Mathematicae Debrecen 1999; 54: 473-487.
[5] Cho JT. Pseudo-Einstein CR-structures on real hypersurfaces in a complex space form. Hokkaido Mathematical Journal 2008; 37: 1-17.
[6] Kimura M. Real hypersurfaces and complex submanifolds in complex projective space. Transactions of the American Mathematical Society 1986; 296: 137-149.
[7] Kimura M. Sectional curvatures of holomorphic planes of a real hypersurface in Pn(C) . Mathematische Annalen 1987; 276: 487-497.
[8] Kobayashi S, Nomizu K. Foundations on Differential Geometry. Volume 1. New York, NY, USA: Interscience, 1963.
[9] Lohnherr M, Reckziegel H. On ruled real hypersurfaces in complex space forms. Geometriae Dedicata 1999; 74: 267-286.
[10] Maeda Y. On real hypersurfaces of a complex projective space. Journal of the Mathematical Society of Japan 1976; 28: 529-540.
[11] Niebergall R, Ryan PJ. Real hypersurfaces in complex space forms. Tight and Taut Submanifolds. MSRI Publications 1997; 32: 233-305.
[12] Okumura M. On some real hypersurfaces of a complex projective space. Transactions of the American Mathematical Society 1975; 212: 355-364
[13] Pérez JD, Suh YJ. Generalized Tanaka-Webster and covariant derivatives on a real hypersurface in a complex projective space. Monatshefte für Mathematik 2015; 177: 637-647.
[14] Takagi R. On homogeneous real hypersurfaces in a complex projective space. Osaka Journal of Mathematics 1973; 10: 495-506.
[15] Takagi R. Real hypersurfaces in complex projective space with constant principal curvatures. Journal of the Mathematical Society of Japan 1975; 27: 43-53.
[16] Takagi R. Real hypersurfaces in complex projective space with constant principal curvatures II. Journal of the Mathematical Society of Japan 1975; 27: 507-516.
[17] Tanaka N. On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections. Japanese Journal of Mathematics 1976; 2: 131-190.
[18] Tanno S. Variational problems on contact Riemannian manifolds. Transactions of the American Mathematical Society 1989; 314: 349-379.
[19] Webster SM. Pseudohermitian structures on a real hypersurface, Journal of Differential Geometry 1978; 13: 25-41.