Abstract
Using GTW connection, we considered a real hypersurface M in a complex two-plane Grassmannian \({G_{2}({\mathbb{C}}^{m+2})}\) when the GTW Reeb Lie derivative of the structure Jacobi operator coincides with the Reeb Lie derivative. Next using the method of simultaneous diagonalization, we prove a complete classification for a real hypersurface in \({G_{2}({\mathbb{C}}^{m+2})}\) satisfying such a condition. In this case, we have proved that M is an open part of a tube around a totally geodesic \({G_{2}({\mathbb{C}}^{m+1})}\) in \({G_{2}({\mathbb{C}}^{m+2})}\).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alekseevskii D.V.: Compact quaternion spaces. Funct. Anal. Appl. 2, 11–20 (1968)
Berndt J., Suh Y.J.: Real hypersurfaces in complex two-plane Grassmannians. Monatsh. Math. 127, 1–14 (1999)
Berndt J., Suh Y.J.: Isometric flows on real hypersurfaces in complex two-plane Grassmannians. Monatsh. Math. 137, 87–98 (2002)
Jeong I., Pak E., Suh Y.J.: Real hypersurfaces in complex two-plane Grassmannians with generalized Tanaka–Webster invariant shape operator. J. Math. Phys. Anal. Geom. 9, 360–378 (2013)
Jeong I., Pak E., Suh Y.J.: Lie invariant shape operator for real hypersurfaces in complex two-plane Grassmannians. J. Math. Phys. Anal. Geom. 9, 455–475 (2013)
Jeong I., Pérez J.D., Suh Y.J.: Real hypersurfaces in complex two-plane Grassmannians with parallel structure Jacobi operator. Acta Math. Hungar. 122, 173–186 (2009)
Jeong, I., Machado, C.J.G., Pérez, J.D., Suh, Y.J.: Real hypersurfaces in complex two-plane Grassmannians with \({{{\mathfrak{D}}^{\bot}}}\)-parallel structure Jacobi operator. Int. J. Math. 22, 655–673 (2011)
Ki U.-H., Pérez J.D., Santos F.G., Suh Y.J.: Real hypersurfaces in complex space forms with \({\xi}\)-parallel Ricci tensor and structure Jacobi operator. J. Korean Math. Soc. 44, 307–326 (2007)
Kon, M.: Real hypersurfaces in complex space forms and the generalized-Tanaka–Webster connection. In: Proceedings of the 13th International Workshop on Differential Geometry and Related Fields, Daegu, pp. 145–159. National Institute of Mathematical Sciences (2009)
Lee H., Suh Y.J.: Real hypersurfaces of type B in complex two-plane Grassmannians related to the Reeb vector. Bull. Korean Math. Soc. 47(3), 551–561 (2010)
Lee H., Suh Y.J., Woo C.: Real hypersurfaces in complex two-plane Grassmannians with commuting Jacobi operators. Houst. J. Math. 40(3), 751–766 (2014)
Pérez J.D., Santos F.G., Suh Y.J.: Real hypersurfaces in complex projective space whose structure Jacobi operator is \({{{\mathfrak{D} }}}\)-parallel. Bull. Belg. Math. Soc. Simon Stevin 13, 459–469 (2006)
Pérez J.D., Suh Y.J.: Real hypersurfaces of quaternionic projective space satisfying \({\nabla _{U_{i}} R=0}\). Differ. Geom. Appl. 7, 211–217 (1997)
Pérez J.D., Suh Y.J.: The Ricci tensor of real hypersurfaces in complex two-plane Grassmannians. J. Korean Math. Soc. 44, 211–235 (2007)
Tanaka N.: On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections. Jpn. J. Math. 20, 131–190 (1976)
Tanno S.: Variational problems on contact Riemannian manifolds. Trans. Am. Math. Soc. 314, 349–379 (1989)
Webster S.M.: Pseudo-Hermitian structures on a real hypersurface. J. Differ. Geom. 13, 25–41 (1978)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by Grant Proj. No. NRP-2012-R1A2A2A-01043023.
Rights and permissions
About this article
Cite this article
Pak, E., Kim, G.J. & Suh, Y.J. Real Hypersurfaces in Complex Two-Plane Grassmannians with GTW Reeb Lie Derivative Structure Jacobi Operator. Mediterr. J. Math. 13, 1263–1272 (2016). https://doi.org/10.1007/s00009-015-0535-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00009-015-0535-1