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Jeong, Imsoon; Kim, Gyu Jong; Woo, Changhwa

Semi-symmetric structure Jacobi operator for real hypersurfaces in the complex quadric. (English) Zbl 1528.53059

Bull. Korean Math. Soc. 60, No. 4, 849-861 (2023).
Summary: In this paper, we introduce the notion of semi-symmetric structure Jacobi operator for Hopf real hypersufaces in the complex quadric \(Q^m = SO_{m+2}/SO_m SO_2\). Next we prove that there does not exist any Hopf real hypersurface in the complex quadric \(Q^m = SO_{m+2}/SO_m SO_2\) with semi-symmetric structure Jacobi operator. As a corollary, we also get a non-existence property of Hopf real hypersurfaces in the complex quadric \(Q^m\) with either symmetric (parallel), or recurrent structure Jacobi operator.

Cited in 1 Document

MSC:

53C40 Global submanifolds
53C55 Global differential geometry of Hermitian and Kählerian manifolds
32S25 Complex surface and hypersurface singularities

Keywords:

complex quadric; real hypersurface; Reeb flow; Hopf hypersurface; Jacobi operator
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References:

[1] J. T. Cho and M. Kimura, Curvature of Hopf hypersurfaces in a complex space form, Re-sults Math. 61 (2012), no. 1-2, 127-135. https://doi.org/10.1007/s00025-010-0080-y · Zbl 1251.53007 · doi:10.1007/s00025-010-0080-y
[2] A. De and T.-H. Loo, Pseudo parallel CR-submanifolds in a non-flat complex space form, Taiwanese J. Math. 18 (2014), no. 5, 1549-1562. https://doi.org/10.11650/ tjm.18.2014.4034 · Zbl 1357.53057 · doi:10.11650/tjm.18.2014.4034
[3] A. De, T.-H. Loo, and C. Woo, Real hypersurfaces in complex Grassmannians of rank two with semi-parallel structure Jacobi operator, Rev. Un. Mat. Argentina 60 (2019), no. 2, 505-515. https://doi.org/10.33044/revuma.v60n2a15 · Zbl 1431.53054 · doi:10.33044/revuma.v60n2a15
[4] U.-H. Ki, J. D. Pérez, F. G. Santos, and Y. J. Suh, Real hypersurfaces in complex space forms with ξ-parallel Ricci tensor and structure Jacobi operator, J. Korean Math. Soc. 44 (2007), no. 2, 307-326. https://doi.org/10.4134/JKMS.2007.44.2.307 · Zbl 1144.53069 · doi:10.4134/JKMS.2007.44.2.307
[5] S. Klein, Totally geodesic submanifolds of the complex and the quaternionic 2-Grassmannians, Trans. Amer. Math. Soc. 361 (2009), no. 9, 4927-4967. https://doi. org/10.1090/S0002-9947-09-04699-6 · Zbl 1176.53054 · doi:10.1090/S0002-9947-09-04699-6
[6] S. Klein and Y. J. Suh, Contact real hypersurfaces in the complex hyperbolic quadric, Ann. Mat. Pura Appl. (4) 198 (2019), no. 4, 1481-1494. https://doi.org/10.1007/ s10231-019-00827-y · Zbl 1420.53064 · doi:10.1007/s10231-019-00827-y
[7] H. Lee, J. D. Pérez, and Y. J. Suh, Derivatives of normal Jacobi operator on real hypersurfaces in the complex quadric, Bull. Lond. Math. Soc. 52 (2020), no. 6, 1122-1133. https://doi.org/10.1112/blms.12386 · Zbl 1462.53052 · doi:10.1112/blms.12386
[8] H. Lee and Y. J. Suh, Commuting Jacobi operators on real hypersurfaces of type B in the complex quadric, Math. Phys. Anal. Geom. 23 (2020), no. 4, Paper No. 44, 21 pp. https://doi.org/10.1007/s11040-020-09370-2 · Zbl 1456.53044 · doi:10.1007/s11040-020-09370-2
[9] H. Lee, Y. J. Suh, and C. Woo, A classification of Ricci semi-symmetric real hyper-surfaces in the complex quadric, J. Geom. Phys. 164 (2021), Paper No. 104177, 15 pp. https://doi.org/10.1016/j.geomphys.2021.104177 · Zbl 1464.53077 · doi:10.1016/j.geomphys.2021.104177
[10] R. Niebergall and P. J. Ryan, Semi-parallel and semi-symmetric real hypersurfaces in complex space forms, Kyungpook Math. J. 38 (1998), no. 1, 227-234. · Zbl 0906.53015
[11] M. Okumura, On some real hypersurfaces of a complex projective space, Trans. Amer. Math. Soc. 212 (1975), 355-364. https://doi.org/10.2307/1998631 · Zbl 0288.53043 · doi:10.2307/1998631
[12] J. D. Pérez, Lie derivatives and structure Jacobi operator on real hypersurfaces in com-plex projective spaces, Differential Geom. Appl. 50 (2017), 1-10. https://doi.org/10. 1016/j.difgeo.2016.10.004 · Zbl 1355.53016 · doi:10.1016/j.difgeo.2016.10.004
[13] J. D. Pérez, Commutativity of torsion and normal Jacobi operators on real hypersurfaces in the complex quadric, Publ. Math. Debrecen 95 (2019), no. 1-2, 157-168. https: //doi.org/10.5486/pmd.2019.8424 · Zbl 1449.53028 · doi:10.5486/pmd.2019.8424
[14] J. D. Pérez, Some real hypersurfaces in complex and complex hyperbolic quadrics, Bull. Malays. Math. Sci. Soc. 43 (2020), no. 2, 1709-1718. https://doi.org/10.1007/s40840-019-00769-x · Zbl 1434.53017 · doi:10.1007/s40840-019-00769-x
[15] J. D. Pérez, D. H. Hwang, and Y. J. Suh, Hypersurfaces in complex Grassmannians whose geodesics are circles and straight lines, Houston J. Math. 46 (2020), no. 3, 681-693. · Zbl 1468.53052
[16] J. D. Pérez and F. G. Santos, Real hypersurfaces in complex projective space whose structure Jacobi operator is cyclic-Ryan parallel, Kyungpook Math. J. 49 (2009), no. 2, 211-219. https://doi.org/10.5666/KMJ.2009.49.2.211 · Zbl 1178.53022 · doi:10.5666/KMJ.2009.49.2.211
[17] J. D. Pérez and Y. J. Suh, Derivatives of the shape operator of real hypersurfaces in the complex quadric, Results Math. 73 (2018), no. 3, Paper No. 126, 10 pp. https: //doi.org/10.1007/s00025-018-0888-4 · Zbl 1400.53019 · doi:10.1007/s00025-018-0888-4
[18] H. Reckziegel, On the geometry of the complex quadric, in Geometry and topology of submanifolds, VIII (Brussels, 1995/Nordfjordeid, 1995), 302-315, World Sci. Publ., River Edge, NJ, 1996. · Zbl 0936.53032
[19] B. Smyth, Differential geometry of complex hypersurfaces, Ann. of Math. (2) 85 (1967), 246-266. https://doi.org/10.2307/1970441 · Zbl 0168.19601 · doi:10.2307/1970441
[20] Y. J. Suh, Real hypersurfaces in the complex quadric with Reeb parallel shape opera-tor, Internat. J. Math. 25 (2014), no. 6, 1450059, 17 pp. https://doi.org/10.1142/ S0129167X14500591 · Zbl 1295.53058 · doi:10.1142/S0129167X14500591
[21] Y. J. Suh, Real hypersurfaces in the complex quadric with Reeb invariant shape operator, Differential Geom. Appl. 38 (2015), 10-21. https://doi.org/10.1016/j.difgeo.2014. 11.003 · Zbl 1306.32016 · doi:10.1016/j.difgeo.2014.11.003
[22] Y. J. Suh, Real hypersurfaces in the complex quadric with parallel Ricci tensor, Adv. Math. 281 (2015), 886-905. https://doi.org/10.1016/j.aim.2015.05.012 · Zbl 1405.53080 · doi:10.1016/j.aim.2015.05.012
[23] Y. J. Suh, Real hypersurfaces in the complex quadric with parallel normal Jacobi op-erator, Math. Nachr. 290 (2017), no. 2-3, 442-451. https://doi.org/10.1002/mana. 201500428 · Zbl 1379.53078 · doi:10.1002/mana.201500428
[24] Y. J. Suh, Real hypersurfaces in the complex quadric with parallel structure Jacobi oper-ator, Differential Geom. Appl. 51 (2017), 33-48. https://doi.org/10.1016/j.difgeo. 2017.01.001 · Zbl 1425.53072 · doi:10.1016/j.difgeo.2017.01.001
[25] Y. J. Suh, Real hypersurfaces in the complex hyperbolic quadrics with isometric Reeb flow, Commun. Contemp. Math. 20 (2018), no. 2, 1750031, 20 pp. https://doi.org/ 10.1142/S0219199717500316 · Zbl 1378.53068 · doi:10.1142/S0219199717500316
[26] Y. J. Suh, Real hypersurfaces in the complex quadric whose structure Jacobi operator is of Codazzi type, Math. Nachr. 292 (2019), no. 6, 1375-1391. https://doi.org/10. 1002/mana.201800184 · Zbl 1425.53071 · doi:10.1002/mana.201800184
[27] Y. J. Suh and G. J. Kim, Real hypersurfaces in the complex quadric with Lie invariant structure Jacobi operator, Canad. Math. Bull. 63 (2020), no. 1, 204-221. https://doi. org/10.4153/s0008439519000080 · Zbl 1433.53088 · doi:10.4153/s0008439519000080
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