Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-23T06:49:44.342Z Has data issue: false hasContentIssue false

A Classification of Three-dimensional Real Hypersurfaces in Non-flat Complex Space Forms in Terms of their Generalized Tanaka–Webster Lie Derivative

Published online by Cambridge University Press:  20 November 2018

George Kaimakamis
Affiliation:
Faculty of Mathematics and Engineering Sciences, Hellenic Military Academy, Vari, Attiki, Greece e-mail: [email protected] e-mail: [email protected]
Konstantina Panagiotidou
Affiliation:
Faculty of Mathematics and Engineering Sciences, Hellenic Military Academy, Vari, Attiki, Greece e-mail: [email protected] e-mail: [email protected]
Juan de Dios Perez
Affiliation:
Departmento de Geometria y Topologia, Universidad de Granada, 18071, Granada Spain e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

On a real hypersurface $M$ in a non-flat complex space form there exist the Levi–Civita and the $k$-th generalized Tanaka–Webster connections. The aim of this paper is to study three dimensional real hypersurfaces in non-flat complex space forms, whose Lie derivative of the structure Jacobi operatorwith respect to the Levi–Civita connection coincides with the Lie derivative of it with respect to the $k$-th generalized Tanaka-Webster connection. The Lie derivatives are considered in direction of the structure vector field and in direction of any vector field orthogonal to the structure vector field.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2016

References

[1] Berndt, J., Real hypersurfaces with constant principal curvatures in complex hyperbolic space. J. Reine Angew. Math. 395(1989), 132141. http://dx.doi.Org/10.1515/crll.1989.395.132 Google Scholar
[2] Cho, J. T., CR structures on real hypersurfaces of a complex space form. Publ. Math. Debrecen 54(1999), no. 3-4, 473487.Google Scholar
[3] Cho, J. T., Pseudo-Einstein CR-structures on real hypersurfaces in a complex space form. Hokkaido Math. J. 37(2008), no. 1, 117. http://dx.doi.Org/10.14492/hokmj71253539581 Google Scholar
[4] Ivey, T. A. and P. Ryan, J., The structure Jacobi operator for real hypersurfaces in CP2 and CH2. Results Math. 56(2009), no. 1-4, 473488. http://dx.doi.Org/10.1007/s00025-009-0380-2 Google Scholar
[5] Ki, U.-H. and Suh, Y. J., On real hypersurfaces of a complex space form. Math. J. Okayama Univ. 32(1990), 207221.Google Scholar
[6] Kimura, M., Real hypersurfaces and complex submanifolds in complex protective space. Trans. Amer. Math. Soc. 296(1986), no. 1,137-149. http://dx.doi.Org/10.1090/S0002-9947-1986-0837803-2 Google Scholar
[7] Maeda, Y., On real hypersurfaces of a complex protective space. J. Math. Soc. Japan 28(1976), no. 3, 529540. http://dx.doi.Org/10.2969/jmsj702830529 Google Scholar
[8] Montiel, S., Real hypersurfaces of a complex hyperbolic space. J. Math. Soc. Japan 35(1985), no. 3, 515535. http://dx.doi.Org/10.2969/jmsj703730515 Google Scholar
[9] Montiel, S. and Romero, A., On some real hypersurfaces of a complex hyperbolic space. Geom. Dedicata 20(1986), no. 2, 245261. http://dx.doi.Org/10.1007/BF00164402 Google Scholar
[10] Niebergall, R. and Ryan, P. J., Real hypersurfaces in complex space forms. In: Tight and taut submanifolds, Math. Sci. Res. Inst. Publ., 32, Cambridge University Press, Cambridge, 1997, pp. 233305.Google Scholar
[11] Okumura, M., On some real hypersurfaces of a complex projective space. Trans. Amer. Math. Soc. 212(1975), 355364. http://dx.doi.Org/10.1090/S0002-9947-1975-0377787-X Google Scholar
[12] Panagiotidou, K. and Xenos, Ph. J., Real hypersurfaces in CP2 and CH2 whose structure Jacobi operator is Lie ID-parallel. Note Mat. 32(2012), no. 2, 8999.Google Scholar
[13] Takagi, R., On homogeneous real hypersurfaces in a complex projective space. Osaka J. Math. 10(1973), 495506.Google Scholar
[14] Takagi, R., Real hypersurfaces in complex projective space with constant principal curvatures. J. Math. Soc. Japan 27(1975), 4353. http://dx.doi.Org/10.2969/jmsj702710043 Google Scholar
[15] Takagi, R., Real hypersurfaces in complex projective space with constant principal curvatures. II. J. Math. Soc. Japan 27(1975), no. 4, 507516. http://dx.doi.Org/10.2969/jmsj702740507 Google Scholar
[16] Tanaka, N., On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections. Japan. J. Math. 2(1976), no. 1,131-190.Google Scholar
[17] Tanno, S., Variational problems on contact Riemannian manifolds. Trans. Amer. Math. Soc. 314(1989), 349379. http://dx.doi.Org/10.1090/S0002-9947-1989-1000553-9 Google Scholar
[18] Webster, S. M., Pseudohermitian structures on a real hypersurface. J. Diff. Geom. 13(1978), no. 1, 2541.Google Scholar