×

zbMATH — the first resource for mathematics

A new classifcation of contact hypersurfaces in the complex hyperbolic two-plane Grassmannian. (English) Zbl 1452.53052
Summary: We give a complete classification of contact hypersurfaces in the complex hyperbolic two-plane Grassmannian \(G_2^\ast(\mathbb{C}^{m+2})\), \(m \geq 2\). In fact, they are locally congruent to a tube over a totally real totally geodesic \(\mathbb{H} H^n, m = 2 n\), in the complex hyperbolic two-plane Grassmannian \(G_2^\ast(\mathbb{C}^{m+2})\), a horosphere whose center at the infinity is singular, or another exceptional case.

MSC:
53C40 Global submanifolds
53C30 Differential geometry of homogeneous manifolds
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] D. V. Alekseevskii, “Compact quaternion spaces”, Funkcional. Anal. i Priložen 2:2 (1968), 11-20. In Russian; translated in Func. Anal. Appl. 2:2 (1968), 106-114. Mathematical Reviews (MathSciNet): MR0231314
Zentralblatt MATH: 0175.19001
· Zbl 0175.19001
[2] J. Berndt, “Riemannian geometry of complex two-plane Grassmannians”, Rend. Sem. Mat. Univ. Politec. Torino 55:1 (1997), 19-83. Mathematical Reviews (MathSciNet): MR1626089
Zentralblatt MATH: 0909.53038
· Zbl 0909.53038
[3] J. Berndt and Y. J. Suh, “Real hypersurfaces in complex two-plane Grassmannians”, Monatsh. Math. 127:1 (1999), 1-14. Mathematical Reviews (MathSciNet): MR1666307
Zentralblatt MATH: 0920.53016
Digital Object Identifier: doi:10.1007/s006050050018
· Zbl 0920.53016
[4] J. Berndt and Y. J. Suh, “Hypersurfaces in noncompact complex Grassmannians of rank two”, Internat. J. Math. 23:10 (2012), 1250103, 35. Mathematical Reviews (MathSciNet): MR2999048
Zentralblatt MATH: 1262.53046
Digital Object Identifier: doi:10.1142/S0129167X12501030
· Zbl 1262.53046
[5] J. Berndt and Y. J. Suh, “Real hypersurfaces in Hermitian symmetric spaces”, pp. 37-46 in Geometry of submanifolds in symmetric spaces and related problems (Daegu, 2019), to appear in series Advances in Analysis and Geometry, de Gruyter, 2021. · Zbl 1455.53003
[6] J. Berndt, H. Lee, and Y. J. Suh, “Contact hypersurfaces in noncompact complex Grassmannians of rank two”, Internat. J. Math. 24:11 (2013), 1350089, 11. Mathematical Reviews (MathSciNet): MR3143606
Zentralblatt MATH: 1286.53062
Digital Object Identifier: doi:10.1142/S0129167X13500894
· Zbl 1286.53062
[7] I. Jeong and Y. J. Suh, “Real hypersurfaces of type (A) in complex two-plane Grassmannians related to the commuting shape operator”, Forum Math. 25:1 (2013), 179-192. Mathematical Reviews (MathSciNet): MR3010853
Zentralblatt MATH: 1260.53094
Digital Object Identifier: doi:10.1515/forum-2012-0029
· Zbl 1260.53094
[8] J. D. Pérez and Y. J. Suh, “Real hypersurfaces of quaternionic projective space satisfying \(\nabla_{U_i}R=0\)”, Differential Geom. Appl. 7:3 (1997), 211-217. Mathematical Reviews (MathSciNet): MR1480534
Zentralblatt MATH: 0901.53011
Digital Object Identifier: doi:10.1016/S0926-2245(97)00003-X
· Zbl 0901.53011
[9] Y. J. Suh, “Real hypersurfaces of type \(B\) in complex two-plane Grassmannians”, Monatsh. Math. 147:4 (2006), 337-355. Mathematical Reviews (MathSciNet): MR2215841
Zentralblatt MATH: 1094.53050
Digital Object Identifier: doi:10.1007/s00605-005-0329-9
· Zbl 1094.53050
[10] Y. J. Suh, “Real hypersurfaces in complex two-plane Grassmannians with commuting Ricci tensor”, J. Geom. Phys. 60:11 (2010), 1792-1805. Mathematical Reviews (MathSciNet): MR2679422
Zentralblatt MATH: 1197.53070
Digital Object Identifier: doi:10.1016/j.geomphys.2010.06.007
· Zbl 1197.53070
[11] Y. J. Suh, “Real hypersurfaces in complex two-plane Grassmannians with \(\xi \)-invariant Ricci tensor”, J. Geom. Phys. 61:4 (2011), 808-814. Mathematical Reviews (MathSciNet): MR2765405
Zentralblatt MATH: 1209.53046
Digital Object Identifier: doi:10.1016/j.geomphys.2010.12.010
· Zbl 1209.53046
[12] Y. J. Suh, “Hypersurfaces with isometric Reeb flow in complex hyperbolic two-plane Grassmannians”, Adv. in Appl. Math. 50:4 (2013), 645-659. Mathematical Reviews (MathSciNet): MR3032310
Zentralblatt MATH: 1279.53051
Digital Object Identifier: doi:10.1016/j.aam.2013.01.001
· Zbl 1279.53051
[13] Y. J. Suh, “Real hypersurfaces in complex hyperbolic two-plane Grassmannians with Reeb vector field”, Adv. in Appl. Math. 55 (2014), 131-145. Mathematical Reviews (MathSciNet): MR3176718
Zentralblatt MATH: 1296.53123
Digital Object Identifier: doi:10.1016/j.aam.2014.01.005
· Zbl 1296.53123
[14] Y. J. Suh, “Pseudo-anti commuting Ricci tensor and Ricci soliton real hypersurfaces in the complex quadric”, J. Math. Pures Appl. \((9) 107\):4 (2017), 429-450. Mathematical Reviews (MathSciNet): MR3623639
Digital Object Identifier: doi:10.1016/j.matpur.2016.07.005
[15] Y. J. Suh, “Real hypersurfaces in the complex hyperbolic quadric and related topics”, pp. 15-36 in Proceedings of the 22nd international workshop on differential geometry of submanifolds in symmetric spaces & the 17th RIRCM-OCAMI joint differential geometry workshop, edited by Y. J. Suh et al., 2019. Mathematical Reviews (MathSciNet): MR3971844
· Zbl 1466.53065
[16] Y. · Zbl 1307.53043
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.