# Article

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Keywords:
complex two-plane Grassmannians; Hopf hypersurface; $\mathfrak D^{\bot }$-invariant hypersurface; commuting shape operator; Reeb vector field
Summary:
Lee, Kim and Suh (2012) gave a characterization for real hypersurfaces $M$ of Type ${\rm (A)}$ in complex two plane Grassmannians $G_2({\mathbb C}^{m+2})$ with a commuting condition between the shape operator $A$ and the structure tensors $\phi$ and $\phi _{1}$ for $M$ in $G_2({\mathbb C}^{m+2})$. Motivated by this geometrical notion, in this paper we consider a new commuting condition in relation to the shape operator $A$ and a new operator $\phi \phi _{1}$ induced by two structure tensors $\phi$ and $\phi _{1}$. That is, this commuting shape operator is given by $\phi \phi _{1} A = A \phi \phi _{1}$. Using this condition, we prove that $M$ is locally congruent to a tube of radius $r$ over a totally geodesic $G_2({\mathbb C}^{m+1})$ in $G_2({\mathbb C}^{m+2})$.
References:
[1] Alekseevskij, D. V.: Compact quaternion spaces. Funkts. Anal. Prilozh. 2 (1968), 11-20 Russian. MR 0231314 | Zbl 0175.19001
[2] Berndt, J.: Riemannian geometry of complex two-plane Grassmannians. Rend. Semin. Mat., Torino 55 (1997), 19-83. MR 1626089 | Zbl 0909.53038
[3] Berndt, J., Suh, Y. J.: Real hypersurfaces in complex two-plane Grassmannians. Monatsh. Math. 127 (1999), 1-14. DOI 10.1007/s006050050018 | MR 1666307 | Zbl 0920.53016
[4] Berndt, J., Suh, Y. J.: Real hypersurfaces with isometric Reeb flow in complex two-plane Grassmannians. Monatsh. Math. 137 (2002), 87-98. DOI 10.1007/s00605-001-0494-4 | MR 1937621 | Zbl 1015.53034
[5] Jeong, I., Lee, H. J., Suh, Y. J.: Anti-commuting real hypersurfaces in complex two-plane Grassmannians. Bull. Aust. Math. Soc. 78 (2008), 199-210. DOI 10.1017/S0004972708000609 | MR 2466859 | Zbl 1154.53031
[6] Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry I. Interscience Publishers, a division of John Wiley and Sons New York (1963). MR 0152974 | Zbl 0119.37502
[7] Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry Vol. II. Interscience Tracts in Pure and Applied Mathematics No. 15, Vol. II Interscience Publishers, a division of John Wiley and Sons, New York (1969). MR 0238225 | Zbl 0175.48504
[8] Lee, H., Kim, S., Suh, Y. J.: Real hypersurfaces in complex two-plane Grassmannians with certain commuting condition. Czech. Math. J. 62 (2012), 849-861. DOI 10.1007/s10587-012-0049-y | MR 2984638 | Zbl 1260.53097
[9] 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 (2010), 551-561. DOI 10.4134/BKMS.2010.47.3.551 | MR 2666376 | Zbl 1206.53064
[10] Pérez, J. D., Jeong, I., Suh, Y. J.: Real hypersurfaces in complex two-plane Grassmannians with commuting normal Jacobi operator. Acta Math. Hung. 117 (2007), 201-217. DOI 10.1007/s10474-007-6091-9 | MR 2361601 | Zbl 1220.53070
[11] Pérez, J. D., Suh, Y. J.: The Ricci tensor of real hypersurfaces in complex two-plane Grassmannians. J. Korean Math. Soc. 44 (2007), 211-235. DOI 10.4134/JKMS.2007.44.1.211 | MR 2283469 | Zbl 1156.53034
[12] Pérez, J. D., Suh, Y. J., Watanabe, Y.: Generalized Einstein real hypersurfaces in complex two-plane Grassmannians. J. Geom. Phys. 60 (2010), 1806-1818. DOI 10.1016/j.geomphys.2010.06.017 | MR 2679423 | Zbl 1197.53071
[13] Suh, Y. J.: Real hypersurfaces in complex two-plane Grassmannians with commuting shape operator. Bull. Aust. Math. Soc. 68 (2003), 379-393. DOI 10.1017/S0004972700037795 | MR 2027682 | Zbl 1058.53046
[14] Suh, Y. J.: Real hypersurfaces in complex two-plane Grassmannians with harmonic curvature. J. Math. Pures Appl. 100 (2013), 16-33. DOI 10.1016/j.matpur.2012.10.010 | MR 3057300 | Zbl 1279.53052
[15] Suh, Y. J.: Real hypersurfaces in complex two-plane Grassmannians with parallel Ricci tensor. Proc. Roy. Soc. Edinburgh Sect. A 142 (2012), 1309-1324. MR 3002598 | Zbl 1293.53071
[16] Suh, Y. J.: Real hypersurfaces in complex two-plane Grassmannians with $\xi$-invariant Ricci tensor. J. Geom. Phys. 61 (2011), 808-814. DOI 10.1016/j.geomphys.2010.12.010 | MR 2765405 | Zbl 1209.53046
[17] Suh, Y. J.: Real hypersurfaces in complex two-plane Grassmannians with Reeb parallel Ricci tensor. J. Geom. Phys. 64 (2013), 1-11. DOI 10.1016/j.geomphys.2012.10.005 | MR 3004010 | Zbl 1259.53052
[18] Suh, Y. J.: Real hypersurfaces of Type $B$ in complex two-plane Grassmannians. Monatsh. Math. 147 (2006), 337-355. DOI 10.1007/s00605-005-0329-9 | MR 2215841 | Zbl 1094.53050

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