ON THE LIE DERIVATIVE OF REAL HYPERSURFACES IN ℂP2 AND ℂH2 WITH RESPECT TO THE GENERALIZED TANAKA-WEBSTER CONNECTION

Title & Authors
ON THE LIE DERIVATIVE OF REAL HYPERSURFACES IN ℂP2 AND ℂH2 WITH RESPECT TO THE GENERALIZED TANAKA-WEBSTER CONNECTION
PANAGIOTIDOU, KONSTANTINA; PEREZ, JUAN DE DIOS;

Abstract
In this paper the notion of Lie derivative of a tensor field T of type (1,1) of real hypersurfaces in complex space forms with respect to the generalized Tanaka-Webster connection is introduced and is called generalized Tanaka-Webster Lie derivative. Furthermore, three dimensional real hypersurfaces in non-flat complex space forms whose generalized Tanaka-Webster Lie derivative of 1) shape operator, 2) structure Jacobi operator coincides with the covariant derivative of them with respect to any vector field X orthogonal to $\small{{\xi}}$ are studied.
Keywords
real hypersurface;structure Jacobi operator;shape operator;Lie derivative;generalized Tanaka-Webster connection;
Language
English
Cited by
References
1.
J. Berndt, Real hypersurfaces with constant principal curvatures in complex hyperbolic space, J. Reine Angew. Math. 395 (1989), 132-141.

2.
J. T. Cho, CR-structures on real hypersurfaces of a complex space form, Publ. Math. Debrecen 54 (1999), no. 3-4, 473-487.

3.
T. A. Ivey and P. J. Ryan, The structure Jacobi operator for real hypersurfaces in ${\mathbb{C}}P^2$ and ${\mathbb{C}}H^2$, Results Math. 56 (2009), no. 1-4, 473-488.

4.
Y. Maeda, On real hypersurfaces of a complex projective space, J. Math. Soc. Japan 28 (1976), no. 3, 529-540.

5.
S. Montiel, Real hypersurfaces of a complex hyperbolic space, J. Math. Soc. Japan 35 (1985), no. 3, 515-535.

6.
R. Niebergall and P. J. Ryan, Real hypersurfaces in complex space forms, Tight and taut submanifolds (Berkeley, CA, 1994), 23300305, Math. Sci. Res. Inst. Publ., 32, Cambridge Univ. Press, Cambridge, 1997.

7.
K. Panagiotidou and J. D. Perez, Commuting conditions of the k-th Cho operator with structure Jacobi operator of real hypersurfaces in complex space forms, Open Math. 13 (2015), 321-332.

8.
K. Panagiotidou and J. D. Perez, Commutativity of shape operator with the k-th Cho operator of real hypersur-faces in complex space forms, Preprint.

9.
K. Panagiotidou and Ph. J. Xenos, Real hypersurfaces in ${\mathbb{C}}P^2$ and ${\mathbb{C}}H^2$ whose structure Jacobi operator is Lie ${\mathbb{D}}$-parallel, Note Mat. 32 (2012), no. 2, 89-99.

10.
J. D. Perez, Lie and generalized Tanaka-Webster derivatives on real hypersurfaces in complex projective spaces, Internat. J. Math. 25 (2014), no. 12, 1450115, 13 pp.

11.
J. D. Perez, Commutativity of Cho and structure Jacobi operators of a real hypersurface in a complex projective space, Ann. di Mat. DOI 10.1007/s10231-014-0444-0 (to appear).

12.
J. D. Perez and Y. J. Suh, Generalized Tanaka-Webster and covariant derivatives on a real hypersurface in a complex projective space, Monatsh. Math. 177 (2015), no. 4, 637-647.

13.
R. Takagi, On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math. 10 (1973), 495-506.

14.
S. Tanno, Variational problems on contact Riemannian manifolds, Trans. Amer. Math. Soc. 314 (1989), no. 1, 349-379.