Abstract
The aim of this paper is to answer the question if there are three-dimensional real hypersurfaces in non-flat complex space forms whose covariant derivative with respect to the kth generalized Tanaka–Webster connection of a tensor field of type (1, 1) coincides with the Lie derivative of it either in direction of any vector field orthogonal to \(\xi \) or in direction of \(\xi \). The answer is given, in case the tensor field is the shape operator of a real hypersurface or the structure Jacobi operator of it.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berndt, J.: Real hypersurfaces with constant principal curvatures in complex hyperbolic space. J. Reine Angew. Math. 395, 132–141 (1989)
Blair, D.E.: Riemannian geometry of contact and symplectic manifolds. In: Progress in Mathematics, vol. 203. Birkhauser, Boston (2002)
Cho, J.T.: CR-structures on real hypersurfaces of a complex space form. Publ. Math. Debr. 54, 473–487 (1999)
Ivey, T.A., Ryan, P.J.: The structure Jacobi operator for real hypersurfaces in \(\mathbb{C}P^{2}\) and \(\mathbb{C}H^{2}\). Results Math. 56, 473–488 (2009)
Ivey, T.A., Ryan, P.J.: Hopf hypersurfaces of small Hopf principal curvature in \(\mathbb{C}H^{2}\). Geom. Dedic. 141, 147–161 (2009)
Ki, U.-H., Suh, Y.J.: On real hypersurfaces of a complex space form. Math. J. Okayama Univ. 32, 207–221 (1990)
Kimura, M.: Sectional curvatures of holomorphic planes of a real hypersurface in \(P^n(\mathbb{C})\). Math. Ann. 276, 487–497 (1987)
Lohnherr, M., Reckziegel, H.: On ruled real hypersurfaces in complex space forms. Geom. Dedic. 74, 267–286 (1999)
Maeda, Y.: On real hypersurfaces of a complex projective space. J. Math. Soc. Jpn. 28, 529–540 (1976)
Montiel, S.: Real hypersurfaces of a complex hyperbolic space. J. Math. Soc. Jpn. 35, 515–535 (1985)
Montiel, S., Romero, A.: On some real hypersurfaces of a complex hyperbolic space. Geom. Dedic. 20, 245–261 (1986)
Niebergall, R., Ryan, P.J.: Real hypersurfaces in complex space forms: in tight and taut submanifolds. MSRI Publ. 32, 233–305 (1997)
Okumura, M.: On some real hypersurfaces of a complex projective space. Trans. Am. Math. Soc. 212, 355–364 (1975)
Panagiotidou, K.: The structure Jacobi and the shape operator of real hypersurfaces in \(\mathbb{C}P^{2}\) and \(\mathbb{C}H^{2}\). Beitr. Algebra Geom. 55, 1991–1998 (2013)
Panagiotidou, K., Pérez, J.D.: Commuting conditions of the \(k\)-th Cho operator with the structure Jacobi operator of real hypersurfaces in complex space forms. Open Math. 13, 321–332 (2015)
Panagiotidou, K., Pérez, J.D.: Commutativity of the \(k\)-th Cho operator with the shape operator of real hypersurfaces in complex space forms (work in progress)
Panagiotidou, K., Xenos, Ph.J.: Real hypersurfaces in \(\mathbb{C}P^{2}\) and \(\mathbb{C}H^{2}\) equipped with structure Jacobi operator \({\cal{L}}_\xi l=\nabla _{\xi }l\). Adv. Pure Math. 2, 1–5 (2012)
Panagiotidou, K., Xenos, PhJ: Real hypersurfaces in \(\mathbb{C}P^{2}\) and \(\mathbb{C}H^{2}\) whose structure Jacobi operator is Lie \(\mathbb{D}\)-parallel. Note Math. 32, 89–99 (2012)
Pérez, J.D.: Commutativity of Cho and structure Jacobi operators of a real hypersurface in a complex projective space. Ann. Math. 194, 1781–1794 (2015)
Pérez, J.D.: Comparing Lie derivatives on real hypersurfaces in complex projective space. Mediter. J. Math. (2015). doi:10.1007/s00009-015-0601-8
Pérez, J.D., Santos, F.G.: Real hypersurfaces in complex projective space whose structure Jacobi operator satisfies \(\mathfrak{L}_\xi R_{\xi }=\nabla _{\xi }R_{\xi }\). Rocky Mt. J. Math. 39, 1293–1301 (2009)
Pérez, J.D., Suh, Y.J.: Generalized Tanaka–Webster and covariant derivatives on a real hypersurface in a complex projective space. Monatsh. Math. 177, 637–647 (2015)
Takagi, R.: Real hypersurfaces in complex projective space with constant principal curvatures. J. Math. Soc. Jpn. 27, 43–53 (1975)
Takagi, R.: On homogeneous real hypersurfaces in a complex projective space. Osaka J. Math. 10, 495–506 (1973)
Author information
Authors and Affiliations
Corresponding author
Additional information
J. de Dios Pérez is partially supported by MINECO-FEDER Grant MTM2013-47828-C2-1-P.
Rights and permissions
About this article
Cite this article
Kaimakamis, G., Panagiotidou, K. & de Dios Pérez, J. Derivatives on Real Hypersurfaces of Two-Dimensional Non-flat Complex Space Forms. Mediterr. J. Math. 14, 74 (2017). https://doi.org/10.1007/s00009-017-0850-9
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00009-017-0850-9