# zbMATH — the first resource for mathematics

Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster parallel normal Jacobi operator. (English) Zbl 1363.53049
Summary: We study the classifying problem of immersed submanifolds in Hermitian symmetric spaces. Typically in this paper, we deal with real hypersurfaces in a complex two-plane Grassmannian $$G_2(\mathbb{C}^{m+2})$$ which has a remarkable geometric structure as a Hermitian symmetric space of rank 2. In relation to the generalized Tanaka-Webster connection, we consider a new concept of the parallel normal Jacobi operator for real hypersurfaces in $$G_2(\mathbb{C}^{m+2})$$ and prove non-existence of real hypersurfaces in $$G_2(\mathbb{C}^{m+2})$$ with generalized Tanaka-Webster parallel normal Jacobi operator.

##### MSC:
 53C40 Global submanifolds 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
Full Text:
##### References:
 [1] D. V. Alekseevskij: Compact quaternion spaces. Funkts. Anal. Prilozh. 2 (1968), 11–20. (In Russian.) [2] J. Berndt, Y. J. Suh: Real hypersurfaces with isometric Reeb flow in complex two-plane Grassmannians. Monatsh. Math. 137 (2002), 87–98. · Zbl 1015.53034 [3] J. Berndt, Y. J. Suh: Real hypersurfaces in complex two-plane Grassmannians. Monatsh. Math. 127 (1999), 1–14. · Zbl 0920.53016 [4] J. T. Cho: Levi-parallel hypersurfaces in a complex space form. Tsukuba J. Math. 30 (2006), 329–343. · Zbl 1131.53025 [5] J. T. Cho: CR structures on real hypersurfaces of a complex space form. Publ. Math. 54 (1999), 473–487. · Zbl 0929.53029 [6] J. de Dios Pérez, I. Jeong, Y. J. Suh: Real hypersurfaces in complex two-plane Grassmannians with commuting normal Jacobi operator. Acta Math. Hung. 117 (2007), 201–217. · Zbl 1220.53070 [7] J. de Dios Pérez, Y. J. Suh: Real hypersurfaces of quaternionic projective space satisfying $${\nabla _{{U_i}}}R = 0$$ . Differ. Geom. Appl. 7 (1997), 211–217. · Zbl 0901.53011 [8] I. Jeong, H. J. Kim, Y. J. Suh: Real hypersurfaces in complex two-plane Grassmannians with parallel normal Jacobi operator. Publ. Math. 76 (2010), 203–218. · Zbl 1274.53080 [9] I. Jeong, M. Kimura, H. Lee, Y. J. Suh: Real hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster Reeb parallel shape operator. Monatsh. Math. 171 (2013), 357–376. · Zbl 1277.53049 [10] I. Jeong, Y. J. Suh: Real hypersurfaces in complex two-plane Grassmannians with F-parallel normal Jacobi operator. Kyungpook Math. J. 51 (2011), 395–410. · Zbl 1243.53099 [11] H. Lee, Y. J. Suh: Real hypersurfaces of type B in complex two-plane Grassmannians related to the Reeb vector. Bull. Korean Math. Soc. 47 (2010), 551–561. · Zbl 1206.53064 [12] N. Tanaka: On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections. Jap. J. Math., new Ser. 2 (1976), 131–190. · Zbl 0346.32010 [13] S. Tanno: Variational problems on contact Riemannian manifolds. Trans. Am. Math. Soc. 314 (1989), 349–379. · Zbl 0677.53043 [14] S. M. Webster: Pseudo-Hermitian structures on a real hypersurface. J. Differ. Geom. 13 (1978), 25–41. · Zbl 0379.53016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.