Pseudo Ricci symmetric real hypersurfaces of a complex projective space

Document Type: Research Paper

Authors

1 Department of Mathematics, Sidho Kanho Birsha University, Purulia-723104, West Bengal, India.newline Department of Mathematics, Bankura University, Bankura-722155, West Bengal, India.

2 Department of Mathematics, Chuo University, Faculty of Science and Engineering, 1-13-27 Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan.

Abstract

Pseudo Ricci symmetric real hypersurfaces of a complex projective space are classified and it is proved that there are no pseudo Ricci symmetric real hypersurfaces of the complex projective space CPn for which the vector field ξ from the almost contact metric structure (φ, ξ, η, g) is a principal curvature vector field.

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Main Subjects


K. Arslan, R. Ezentas, C. Murathan and C. Ozgur, On pseudo Ricci symmetric manifolds, Balkan J. Geom. and Appl. 6 (2001), no. 2, 1--5.

A. Bejancu and S. Deshmukh, Real hypersurfaces of CPn with non-negative Ricci curvature, Proc. Amer. Math. Soc. 124 (1996), no. 1, 269--274.

T. E. Cecil and P. J. Ryan, Focal sets and real hypersurfaces in complex projective space, Trans Amer. Math. Soc. 269 (1982), no. 2, 481--499.

M. C. Chaki, On pseudo Ricci symmetric manifolds, Bulgar. J. Phys. 15 (1988), no. 6, 526--531.

M. C. Chaki and S. K. Saha, On pseudo-projective Ricci symmetric manifolds, Bulgar. J. Phys. 21 (1994), no. 1-2, 1--7.

J. T. Cho and U.-Hang Ki, Real hypersurfaces of a complex projective space in terms of the Jacobi operators, Acta Math. Hungar. 80 (1998), no. 1-2, 155--167.

S. Deshmukh, Real hypersurfaces of a complex projective space, Proc. Indian Acad. Math. Sci. 121 (2011), no. 2, 171--179.

R. Deszcz, On Ricci-pseudosymmetric warped products, Demonstratio Math. 22 (1989), no. 4, 1053--1065.

T. Hamada, On real hypersurfaces of a complex projective space with η-recurrent second fundamental tensor, Nihonkai Math. J. 6 (1995), no. 2, 153--163.

T. Hamada, On real hypersurfaces of a complex projective space with recurrent second fundamental tensor, J. Ramanujan Math. Soc. 11 (1996), no. 2, 103--107.

T. Hamada, On real hypersurfaces of a complex projective space with recurrent Ricci tensor, Glasg. Math. J. 41 (1999), no. 3, 297--302.

S. K. Hui and Y. Matsuyama, On real hypersurfaces of a complex projective space with pseudo parallel second fundamental tensor, Kobe J. Math., Accepted.

K. Ikuta, Real hypersurfaces of a complex projective space, J. Korean Math. Soc. 36 (1999), no. 4, 725--736.

U.-Hang Ki, Real hypersurfaces with parallel Ricci tensor of a complex space form, Tsukuba J. Math. 13 (1989), no. 1, 73--81.

M. Kimura, Real hypersurfaces of a complex projective space, Bull. Austral. Math. Soc. 33 (1986) no. 3, 383--387.

M. Kimura, Real hypersurfaces and complex submanifolds in a complex projective space, Trans. Amer. Math. Soc. 296 (1986), no. 1, 137--149.

M. Kimura, Sectional curvatures of holomorphic planes on a real hypersurface in CPn, Math. Ann. 27 (1987), no. 3, 487--497.

M. Kimura, Some real hypersurfaces of a complex projective space, Saitama Math. J. 5 (1987), 1--5. correction in 10 (1992) 33--34.

M. Kimura and S. Maeda, On real hypersurfaces of a complex projective space, Math. Z. 202 (1989), no. 3, 299--311.

S. Maeda, Real hypersurfaces of complex projective space, Math. Ann. 263 (1983), no. 4, 473--478.

S. Maeda, Ricci tensors of real hypersurfaces in a complex projective space, Proc. Amer. Math. Soc. 122 (1994), no. 4, 1229--1235.

Y. Maeda, On real hypersurfaces of a complex projective space, J. Math. Soc. Japan 28 (1976), no. 3, 529--540.

Y. Matsuyama, A characterization of real hypersurfaces in complex projective space, J. Institute Sci. and Eng., Chuo Univ. 2 (1996) 11--13.

Y. Matsuyama, A characterization of real hypersurfaces in complex projective space II, J. Institute Sci. and Eng., Chuo Univ. 3 (1997) 1--3.

Y. Matsuyama, A characterization of real hypersurfaces in complex projective space III, Yokohama Math. J. 46 (1999) 119--126.

M. Okumura, On some real hypersurfaces of a complex projective space, Trans. Amer. Math. Soc. 212 (1975) 355--364.

F.  Ozen, On pseudo M-projective Ricci symmetric manifolds, Int. J. Pure Appl. Math. 72 (2011), no. 2, 249--258.

E. M. Patterson, Some theorems on Ricci-recurrent spaces, J. London Math. Soc. 27 (1952) 287--295.

J. D. Perez and F. G. Santos, On the Lie derivative of structure Jacobi operator of real hypersurfaces in complex projective space, Publ. Math. Debrecen 66 (2005), no. 3-4, 269--282.

J. D. Perez and F. G. Santos, Real hypersurfaces in complex projective space with recurrent structure Jacobi operator, Differential Geom. Appl. 26 (2008), no. 2, 218--223.

J. D. Perez, F. G. Santos and Y. J. Suh, Real hypersurfaces in complex projective space whose structure Jacobi operator is Lie ζ-parallel, Differential Geom. Appl. 22 (2005), no. 2, 181--188.

J. D. Perez, F. G. Santos and Y. J. Suh, Real hypersurfaces in complex projective space whose structure Jacobi operator is D-parallel, Bull. Belg. Math. Soc. Simon Stevin 13 (2006), no. 3, 459--469.

Z. I. Szabo, Structure theorems on Riemannian spaces satisfying R(X; Y ) .R = 0. I. The local version, J. Differential Geom. 17 (1982), no. 4, 531--582.

R. Takagi, On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math. 10 (1973) 495--506.

R. Takagi, Real hypersurfaces in a complex projective space with constant principal curvatures, J. Math. Soc. Japan 27 (1975) 43--53.

R. Takagi, Real hypersurfaces in complex projective space with constant principal curvatures II, J. Math. Soc. Japan 27 (1975), no. 4, 507--516.

Q. M.Wang, Real hypersurfaces with constant principal curvatures in complex projective space I, Sci. Sinica Ser. A 26 (1983), no. 10, 1017--1024.