Lie Derivatives and Ricci Tensor on Real Hypersurfaces in Complex Two-plane Grassmannians

Jeong, Imsoon ; Pérez, Juan de Dios ; Suh, Young Jin ; Woo, Changhwa

Canadian mathematical bulletin, Tome 61 (2018) no. 3, pp. 543-552

Résumé

On a real hypersurface $M$ in a complex two-plane Grassmannian ${{G}_{2}}\left( {{\mathbb{C}}^{m+2}} \right)$ we have the Lie derivation $\mathcal{L}$ and a differential operator of order one associated with the generalized Tanaka–Webster connection ${{\widehat{\mathcal{L}}}^{\left( k \right)}}$ . We give a classification of real hypersurfaces $M$ on ${{G}_{2}}\left( {{\mathbb{C}}^{m+2}} \right)$ satisfying $\widehat{\mathcal{L}}_{\xi }^{\left( k \right)}\,S\,=\,{{\mathcal{L}}_{\xi }}S$ , where $\xi$ is the Reeb vector field on $M$ and $s$ the Ricci tensor of $M$ .

Détail

DOI : 10.4153/CMB-2017-049-5

Mots-clés : 53C40, 53C15, real hypersurface, complex two-plane Grassmannian, Hopf hypersurface, shape operator, Ricci tensor, Lie derivation

Bibliographie

[1] [1] Cho, J. T., CR structures on real hypersurfaces of a complex Space form. Publ. Math. Debrecen 54(1999), no. 3-4, 473–487. Google Scholar

[2] [2] Cho, J. T., Levi parallel hypersurfaces in a complex Space form. Tsukuba J. Math. 30 (2006), no. 2, 329–343. http://dx.doi.Org/10.21099/tkbjm/1496165066 Google Scholar

[3] [3] Hörn, R. A. and Johnson, C. R., Matrix analysis. Cambridge University Press, Cambridge, 1985. http://dx.doi.Org/10.1017/CBO9780511810817 Google Scholar

[4] [4] Jeong, I., Machado, C., Perez, J. D., and Suh, Y. J., Real hypersurfaces in complex two-plane Grassmannians with D1-parallel structure facobi Operator. Internat. J. Math. 22 (2011), no. 5, 655–673. http://dx.doi.org/!0.1142/S0129167X11 006957 Google Scholar

[5] [5] Lee, H., Choi, Y. S., and Woo, C., Hopf hypersurfaces in complex two-plane Grassmannians with Reeb parallel shape Operator. Bull. Malays. Math. Sei. Soc. 38 (2015), no. 2, 617–634. http://dx.doi.Org/10.1007/s40840-014-0039-3 Google Scholar

[6] [6] Lee, H. and Suh, Y. J., Real hypersurfaces oftype B in complex two-plane Grassmannians related to the Reeb vector. Bull. Korean Math. Soc. 47 (2010), no. 3, 551–561. http://dx.doi.Org/10.4134/BKMS.2010.47.3.551 Google Scholar

[7] [7] Loo, T. H., Semi-parallel real hypersurfaces in complex two-plane Grassmannians. Differential Geom. Appl. 34(2014), 87–102. http://dx.doi.Org/10.1016/j.difgeo.2014.03.011 Google Scholar

[8] [8] Pak, E., Suh, Y. J., and Woo, C., Restricted Ricci conditions for real hypersurfaces in complex two-plane Grassmannians. Houston J. Math. 41 (2015), no. 3, 767–783. Google Scholar

[9] [9] Perez, J. D., Jeong, I., and Suh, Y. J., Real hypersurfaces in complex two-plane Grassmannians with commuting normal Jacobi Operator. ActaMath. Hungar. 117(2007), 201–217. http://dx.doi.org/10.1007/s10474-007-6091-9 Google Scholar

[10] [10] Perez, J. D. and Suh, Y. J., The Ricci tensor ofreal hypersurfaces in complex two-plane Grassmannians. J. Korean Math. Soc. 44(2007), 211–235. http://dx.doi.Org/10.4134/JKMS.2007.44.1.211 Google Scholar

[11] [11] Perez, J. D., Suh, Y. J., and Watanabe, Y., Generalized Einstein real hypersurfaces in complex two-plane Grassmannians. J. Geom. Phys. 60(2010), 1806–1818. http://dx.doi.Org/10.1016/j.geomphys.2010.06.01 7 Google Scholar

[12] [12] Suh, Y. J., Real hypersurfaces in complex two-plane Grassmannians with commuting Ricci tensor. J. Geom. Phys. 60(2010), 1792–1805. http://dx.doi.Org/10.1016/j.geomphys.2010.06.007 Google Scholar

[13] [13] Suh, Y. J., Real hypersurfaces in complex two-plane Grassmannians with parallel Ricci tensor. Proc. Roy. Soc. Edinburgh Sect. A. 142(2012), 1309–1324. http://dx.doi.org/10.1017/S0308210510001472 Google Scholar

[14] [14] Suh, Y. J., Real hypersurfaces in the complex quadric with parallel Ricci tensor. Adv. Math. 281(2015), 886–905. http://dx.doi.Org/10.1016/j.aim.2015.05.012 Google Scholar

[15] [15] Suh, Y. J., Real hypersurfaces in the complex quadric with harmonic curvature. J. Math. Pures Appl. 106(2016), 393–410. http://dx.doi.Org/10.1016/j.matpur.2016.02.015 Google Scholar

[16] [16] Suh, Y. J. and Yang, H. Y., Real hypersurfaces in complex two-plane Grassmannians with commuting strueture Jacobi Operator. Bull. Korean Math. Soc. 45 (2008), no. 3, 495–507. http://dx.doi.Org/10.4134/BKMS.2008.45.3.495 Google Scholar

[17] [17] Tanaka, N., On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections. Japan. J. Math. (N.S.) 2(1976), 131–190. Google Scholar

[18] [18] Tanno, S., Variationalproblems on contact Riemannian manifolds. Trans. Amer. Math. Soc. 314(1989), 349–379. http://dx.doi.org/10.1090/S0002-9947-1989-1000553-9 Google Scholar

[19] [19] Webster, S. M., Peudo-Hermitian struetures on a real hypersurface. J. Differential Geom. 13(1978), 25–41. http://dx.doi.Org/10.4310/jdg/1214434345 Google Scholar