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REAL HYPERSURFACES OF NON-FLAT COMPLEX SPACE FORMS WITH GENERALIZED ξ-PARALLEL JACOBI STRUCTURE OPERATOR

Part of: Global differential geometry Symplectic geometry, contact geometry

Published online by Cambridge University Press:  23 July 2015

TH. THEOFANIDIS
TH. THEOFANIDIS*
Affiliation:
Department of Civil Engineering, School of Technology, Aristotle University of Thessaloniki, Thessaloniki, 54124, Greece e-mail: onslaught5000@hotmail.com, theotheo@civil.auth.gr
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Abstract

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The aim of the present paper is the classification of real hypersurfaces M equipped with the condition Al = lA, l = R(., ξ)ξ, restricted in a subspace of the tangent space T p M of M at a point p. This class is large and difficult to classify, therefore a second condition is imposed: (∇ξ l)X = ω(X)ξ + ψ(X)lX, where ω(X), ψ(X) are 1-forms. The last condition is studied for the first time and is much weaker than ∇ξ l = 0 which has been studied so far. The Jacobi Structure Operator satisfying this weaker condition can be called generalized ξ-parallel Jacobi Structure Operator.

MSC classification

Primary: 53C40: Global submanifolds
Secondary: 53D15: Almost contact and almost symplectic manifolds

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 58 , Issue 3 , September 2016 , pp. 677 - 687
Copyright
Copyright © Glasgow Mathematical Journal Trust 2015 

References

REFERENCES

1. Berndt, J., Real hypersurfaces with constant principal curvatures in complex hyperbolic space, J Reine Angew. Math. 395 (1989), 132–141.Google Scholar
2. Blair, D. E., Riemannian geometry of contact and symplectic manifolds, Progress in Mathematics (Birkhauser, New York, 2002).Google Scholar
3. Cho, J. T. and Ki, U-H., Real hypersurfaces in complex space forms with Reeb flow symmetric structure Jacobi operator, Canad. Math. Bull. 51 (3) (2008), 359–371.CrossRefGoogle Scholar
4. Ki, U-H. and Kurihara, H., Jacobi operators along the structure flow on real hypersurfaces in non flat complex space form II, Bull. Korean Math. Soc. 48 (6) (2011), 1315–1327.Google Scholar
5. Ki, U-H. and Nagai, S., The Ricci tensor and structure Jacobi operator of real hypersurfaces in a complex projective space, J. Geom. 94 (2009), 123–142.Google Scholar
6. Ki, U-H., De Dios Pérez, J., Santos, F. G. and Suh, Y. J., Real hypersurfaces in complex space forms with ξ-parallel Ricci tensor and structure Jacobi operator, J. Korean Math. Soc. 44 (2) (2007), 307–326.Google Scholar
7. Sadahiro, M., Geometry of the horosphere in a complex hyperbolic space, Differ. Geom. Appl. 29 (1) (2011), 246 –250.Google Scholar
8. Montiel, S. and Romero, A. On some real hypersurfaces of a complex hyperbolic space, Geom. Dedicata 20 (2) (1986), 245–261.Google Scholar
9. Niebergall, R. and Ryan, P. J., Real hypersurfaces in complex space forms, Math. Sci. Res. Inst. Publ., vol. 32 (Cambridge University Press, Cambridge, 1997), 233–305.Google Scholar
10. Okumura, M., On some real hypersurfaces of a complex projective space, Trans. Amer. Math. Soc. 212 (1975), 355–364.CrossRefGoogle Scholar
11. Ortega, M., De Dios Pérez, J. and Santos, F. G., Non-existence of real hypersurfaces with parallel structure Jacobi operator in nonflat complex space forms, Rocky Mt. J. Math. 36 (5) (2006), 1603–1613.Google Scholar
12. Takagi, R., On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math. 10 (3) (1973), 495–506.Google Scholar
13. Takagi, R., On real hypersurfaces in a complex projective space with constant principal curvatures, J. Math. Soc. Japan 27 (1975), 43–53.Google Scholar