Abstract
In this paper, we consider a new notion of Reeb parallel shape operator for real hypersurfaces \(M\) in complex two-plane Grassmannians \(G_2({\mathbb C}^{m+2})\). When \(M\) has Reeb parallel shape operator and non-vanishing geodesic Reeb flow, it becomes a real hypersurface of Type \((A)\) with exactly four distinct constant principal curvatures. Moreover, if \(M\) has vanishing geodesic Reeb flow and Reeb parallel shape operator, then \(M\) is model space of Type \((A)\) with the radius \(r = \frac{\pi }{4\sqrt{2}}\).
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References
Alekseevskii, D.V.: Compact quaternion spaces. Funct. Anal. Appl. 2, 106–114 (1968)
Berndt, J.: Riemannian geometry of complex two-plane Grassmannian. Rend. Sem. Mat. Univ. Politec. Torino 55, 19–83 (1997)
Berndt, J., Suh, Y.J.: Real hypersurfaces in complex two-plane Grassmannians. Monatshefte für Math. 127, 1–14 (1999)
Berndt, J., Suh, Y.J.: Isometric flows on real hypersurfaces in complex two-plane Grassmannians. Monatshefte für Math. 137, 87–98 (2002)
Choi, Y.S., Suh, Y.J.: Real hypersurfaces with \(\eta \)-parallel shape operator in complex two-plane Grassmannians. Bull. Austral. Math. Soc. 75, 1–16 (2007)
Kim, S., Lee, H., Yang, H.: Real hypersurfaces in complex two-plane Grassmannians with recurrent shape operator. Bull. Malays. Math. Sci. Soc. 34(2), 295–305 (2011)
Jeong, I., Machado, C.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. World Scientific, Singapore 22(5), 655–673 (2011)
Lee, H., Kim, S.: Hopf hypersurfaces with \(\eta \)-parallel shape operator in complex two-plane Grassmannians, Bull. Malays. Math. Sci. Soc. (2) 36(4), 937–948 (2013)
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)
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)
Suh, Y.J.: Real hypersurfaces in complex two-plane Grassmannians with parallel shape operator. Bull. Austral. Math. Soc. 68, 493–502 (2003)
Suh, Y.J.: Real hypersurfaces in complex two-plane Grassmannians with parallel shape operator II. J. Korean Math. Soc. 41, 535–565 (2004)
Suh, Y.J.: Real hypersurfaces in complex two-plane Grassmannians with \(\xi \)-invariant Ricci tensor. J. Geom. Phys. 61, 808–814 (2011)
Acknowledgments
The authors would like to express their hearty thanks to Professors Juan de Dios Pérez and Young Jin Suh for their valuable suggestions and continuous encouragement to develop our works in the first version. This work was supported by Grant Project No. NRF-220-2011-1-C00002 from National Research Foundation of Korea. The first author by NRF Grants No. 2012-R1A1A3002031 and No. 2011-0030044 (SRC-GAIA) and the third by Grant No. NRF-2013-Fostering Core Leaders of Future Basic Science Program.
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Communicated by Suh Young Jin.
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Lee, H., Choi, Y.S. & Woo, C. Hopf Hypersurfaces in Complex Two-Plane Grassmannians with Reeb Parallel Shape Operator. Bull. Malays. Math. Sci. Soc. 38, 617–634 (2015). https://doi.org/10.1007/s40840-014-0039-3
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DOI: https://doi.org/10.1007/s40840-014-0039-3
Keywords
- Complex two-plane Grassmannians
- Hopf hypersurface
- Shape operator
- Parallel shape operator
- \(\mathfrak F\)-parallel shape operator
- Reeb parallel shape operator