On Wintgen Ideal Submanifolds Satisfying Some Pseudo-symmetry Type Curvature Conditions

Year 2024, , 72 - 96, 23.04.2024

Miroslava Petrović-torgašev , Ryszard Deszcz , Małgorzata Głogowska , Georges Zafindratafa

https://doi.org/10.36890/iejg.1404576

Abstract

Let M be a Wintgen ideal submanifold of dimension n in a real space form Rn+m(k) of dimension (n + m) and of constant curvaturek, n ≥ 4, m ≥ 1. Let g, R, Ricc, g ∧ Ricc and C be the metric tensor, the Riemann-Christoffel curvature tensor, the Ricci tensor, the Kulkarni-Nomizu product of g and Ricc, and the Weyl conformal curvature tensor of M, respectively. In this paper we study Wintgen ideal submanifolds M in real space forms Rn+m(k), n ≥ 4, m ≥ 1, satisfying the following pseudo-symmetry type curvature conditions:
(i) the tensors R · C and Q(g, R) (resp., Q(g, C), Q(g, g ∧ Ricc), Q(Ricc, R) or Q(Ricc, g ∧ Ricc)) are linearly dependent;
(ii) the tensors C · R and Q(g, R) (resp., Q(g, C), Q(g, g ∧ Ricc), Q(Ricc, R) or Q(Ricc, g ∧ Ricc)) are linearly dependent;
(iii) the tensors R·C -C ·R and Q(g, R) (resp., Q(g, C), Q(g, g∧Ricc), Q(Ricc, R) or Q(Ricc, g∧Ricc)) are linearly dependent.

Keywords

Tachibana tensor, pseudo-symmetry type curvature condition, generalized Einstein metric condition, submanifold, Wintgen ideal submanifold, DDVV conjecture

References

• [1] Cartan, E.: Leçons sur la géométrie des espaces de Riemann. 2nd ed., Paris: Gauthier-Villars (1946).
• [2] Chen, B.-Y.: Some pinching and classification theorems for minimal submanifolds. Arch. Math. (Basel). 60 (6), 568-578 (1993). https://doi.org/10.1007/BF01236084
• [3] Chen, B.-Y.: Complex extensors and Lagrangian submanifolds in complex Euclidean spaces. Tôhoku Math. J. 49 (2), 277-297 (1997). DOI: 10.2748/tmj/1178225151
• [4] Chen, B.-Y.: δ-invariants, inequalities of submanifolds and their applications. Topics in Differential Geometry, Ch. 2, Editors A. Mihai, I. Mihai and R. Miron. Editura Academiei Romane (2008).
• [5] Chen, B.-Y.: Classification of Wintgen ideal surfaces in Euclidean 4-space with equal Gauss and normal curvatures. Ann. Glob Anal. Geom. 38 (2), 145-160 (2010). https://doi.org/10.1007/s10455-010-9205-5
• [6] Chen, B.-Y.: AWintgen type inequality for surfaces in 4D neutral neutral pseudo-Riemannian space forms and its applications to minimal immersions. JMI Int. J. Math. Sci. 1, 1-12 (2010).
• [7] Chen, B.-Y.: Pseudo-Riemannian Geometry, δ-Invariants and Applications. World Scientific (2011).
• [8] Chen, B.-Y.: Wintgen ideal surfaces in four-dimensional neutral indefinite space form R42
• (c). Results Math. 61 (3-4), 329-345 (2012). https://doi.org/10.1007/s00025-011-0119-8
• [9] Chen, B.-Y.: Differential Geometry of Warped Product Manifolds and Submanifolds. World Scientific (2017).
• [10] Chen, B.-Y.: Recent developments in Wintgen inequality and Wintgen ideal submanifolds. Int. Electron. J. Geom. 14 (1), 1-40 (2021). https://doi.org/10.36890/iejg.838446
• [11] Choi, T., Lu, Z.: On the DDVV conjecture and the comass in calibrated geometry (I). Math. Z. 260 (2), 409-429 (2008). https://doi.org/10.1007/s00209-007-0281-6
• [12] Decu, S., Petrovic-Torgašev, M., Šebekovic, A., Verstraelen, L.: On the intrinsic Deszcz symmetries and the extrinsic Chen character of Wintgen ideal submanifolds. Tamkang J. Math. 41 (2), 109-116 (2010).
• [13] Decu, S., Petrovic-Torgašev, M., Šebekovi´c, A., Verstraelen, L.: On the Roter type of Wintgen ideal submanifolds. Rev. Roum. Math. Pures Appl. 57 (1), 75-90 (2012).
• [14] Deprez, J., Deszcz, R., Verstraelen, L.: Examples of pseudosymmetric conformally flat warped products. Chinese J. Math. 17 (1), 51-65 (1989). https://www.jstor.org/stable/43836355
• [15] De Smet, P.J., Dillen, F., Verstraelen, L., Vrancken, L.: A pointwise inequality in submanifold theory. Arch. Math. (Brno). 35 (2), 115-128 (1999).
• [16] Deszcz, R.: On pseudosymmetric spaces. Bull. Soc. Math. Belg. Sér. A. 44 Fasc. 1, 1-34 (1992).
• [17] Deszcz, R., Głogowska, M., Hashiguchi, H., Hotlo´s, M., Yawata, M.: On semi-Riemannian manifolds satisfying some conformally invariant curvature condition. Colloq. Math. 131 (2), 149-170 (2013). DOI: 10.4064/cm131-2-1
• [18] Deszcz, R., Głogowska, M., Hotlo´s, M., Petrovi´c-Torgašev, M., Zafindratafa, G.: A note on some generalized curvature tensor. Int. Electron. J. Geom. 16 (1), 379-397 (2023). https://doi.org/10.36890/iejg.1273631
• [19] Deszcz, R., Głogowska, M., Hotlo´s, M., Petrovi´c-Torgašev, M., Zafindratafa, G.: On semi-Riemannian manifolds satisfying some generalized Einstein metric conditions, Int. Electron. J. Geom. 16 (2), 539-576 (2023). https://doi.org/10.36890/iejg.1323352
• [20] Deszcz, R., Głogowska, M., Hotlo´s, M., Sawicz, K.: A Survey on Generalized Einstein Metric Conditions. In: Advances in Lorentzian Geometry, Proceedings of the Lorentzian Geometry Conference in Berlin, AMS/IP Studies in Advanced Mathematics. 49, S.-T. Yau (series ed.), M. Plaue, A.D. Rendall and M. Scherfner (eds.), 27-46 (2011).
• [21] Deszcz, R., Głogowska, M., Hotlo´s, M., ¸Sentürk, Z.: On some quasi-Einstein and 2-quasi-Einstein manifolds. AIP Conference Proceedings 2483, 100001 (2022). https://doi.org/10.1063/5.0118057
• [22] Deszcz, R., Głogowska, M., Petrovic-Torgašev, M., Verstraelen, L.: On the Roter type of Chen ideal submanifolds. Results Math. 59 (3-4), 401-413 (2011). https://doi.org/10.1007/s00025-011-0109-x
• [23] Deszcz, R., Głogowska, M., Petrovic-Torgašev, M., Verstraelen, L.: Curvature properties of some class of minimal hypersurfaces in Euclidean spaces. Filomat. 29 (3), 479-492 (2015). DOI 10.2298/FIL1503479D
• [24] Deszcz, R., Głogowska, M., Plaue, M., Sawicz, K., Scherfner, M.: On hypersurfaces in space forms satisfying particular curvature conditions of Tachibana type. Kragujevac J. Math. 35 (2), 223-247 (2011).
• [25] Deszcz, R., Haesen, S., Verstraelen L.: On natural symmetries. Topics in Differential Geometry, Ch. 6. Editors A. Mihai, I. Mihai and R. Miron. Editura Academiei Romˆane (2008).
• [26] Deszcz, R., Hotlo´s, M., ¸Sentürk, Z.: On some family of generalized Einstein metric conditions, Demonstr. Math. 34 (4), 943–954 (2001) . https://doi.org/10.1515/dema-2001-0422
• [27] Deszcz, R., Petrovic-Torgašev, M., Verstraelen, L., Zafindratafa, G.: On the intrinsic symmetries of Chen ideal submanifolds. Bull. Transilv. Univ. Braşov, Ser. III, Math., Inform., Phys. 1 (50), 99-108 (2008).
• [28] Deszcz, R., Petrovicc-Torgašev, M., Verstraelen, L., Zafindratafa, G.: On Chen ideal submanifolds satisfying some conditions of pseudo-symmetry type. Bull. Malaysian Math. Sci. Soc. (2) 39 (1), 103-131 (2016). https://doi.org/10.1007/s40840-015-0164-7
• [29] Deszcz, R., Petrovicc-Torgašev, M., ¸Sentürk, Z., Verstraelen, L.: Characterization of the pseudo-symmetries of Wintgen ideal submanifolds of dimension 3. Publ. Inst. Math. (Beograd) (N.S.). 88 (102), 53-65 (2010). https://doi.org/10.2298/PIM1002053D
• [30] Deszcz, R., Verheyen, P., Verstraelen, L.: On some generalized Einstein metric conditions. Publ. Inst. Math. (Beograd) (N.S.). 60 (74), 108-120 (1996).
• [31] Deszcz, R., Verstraelen, L., Yaprak, ¸S.: Pseudosymmetric hypersurfaces in 4-dimensional spaces of constant curvature. Bull. Inst. Math. Acad. Sinica. 22 (1), 167-179 (1994).
• [32] Deszcz, R., Verstraelen, L., Yaprak, ¸S.: On 2-quasi-umbilical hypersurfaces in conformally flat spaces. Acta Math. Hung. 78 (1-2), 45-57 (1998). https://doi.org/10.1023/A:1006566319359
• [33] Deszcz, R., Yaprak, ¸S.: Curvature properties of Cartan hypersurfaces. Colloq. Math. 67 (1), 91-98 (1994). DOI: 10.4064/cm-67-1-91-98
• [34] Ge, J., Tang, Z.: A proof of the DDVV conjecture and its equality case. Pacific J. Math. 237 (1), 87-95 (2008). DOI: 10.2140/pjm.2008.237.87
• [35] Ge, J., Tang, Z.: A survey on the DDVV conjecture. Preprint arXiv: 1006.5326 (2010).
• [36] Gudalupe, I.V., Rodriguez, I.: Normal curvature of surfaces in space forms. Pacific J. Math. 106 (1), 95-103 (1983). DOI: 10.2140/pjm.1983.106.95
• [37] Haesen, S., Verstraelen, L.: Classification of the pseudo-symmetric space-times. J. Math. Phys. 45 (6), 2343-2346 (2004). https://doi.org/10.1063/1.1745129
• [38] Haesen, S., Verstraelen, L.: Properties of a scalar curvature invariant depending on two planes. Manuscr. Math. 122 (1), 59-72 (2007). https://doi.org/10.1007/s00229-006-0056-0
• [39] Haesen, S., Verstraelen, L.: Natural intrinsic geometrical symmetries. SIGMA. 5, 086, 15 pp. (2009). https://doi.org/10.3842/SIGMA.2009.086
• [40] Lu, Z.: Normal scalar curvature conjecture and its applications. J. Funct. Anal. 261 (5), 1284-1308 (2011). https://doi.org/10.1016/j.jfa.2011.05.002
• [41] Petrovic-Torgašev, M., Pantic, A.: Pseudo-symmetries of generalized Wintgen ideal Lagrangian submanifolds. Publ. Inst. Math. (Beograd) (N.S.). 103 (117), 181-190 (2018). https://doi.org/10.2298/PIM1817181P
• [42] Petrovi´c-Torgašev, M., Verstraelen, L.: On Deszcz symmetries of Wintgen ideal submanifolds. Arch. Math. (Brno). 44 (1), 57-67 (2008).
• [43] Rouxel, B.: Sur une famille de A-surfaces d’un espace euclidien E4. Proc. 10 Österreichischer Mathematiker Kongress, Innsbruck, 185, 1981.
• [44] Šebekovic, A.: Symmetries of Wintgen ideal submanifolds. Bull. Transilv. Univ. Bra¸sov. ser. III. Math., Inform. Phys. 1 (50), 333-342 (2008).
• [45] Šebekovi´c, A., Petrovi´c-Torgašev, M., Panti´c, A.: Pseudosymmetry properties of generalized Wintgen ideal Legendrian submanifolds. Filomat. 33 (4), 1209-1215 (2019). https://doi.org/10.2298/FIL1904209S
• [46] Şentürk, Z.: Characterisation of the Deszcz symmetric ideal Wintgen submanifolds. An. ¸Stiin¸t Univ. Al. I. Cuza Iaşi, Ser. Noua. Mat. 53, Suppl., 53, 309-316 (2007).
• [47] Szabó, Z. I.: Structure theorems on Riemannian spaces satisfying R(X, Y ) · R = 0. I. The local version. J. Differential Geom. 17 (4), 531-582 (1982). DOI: 10.4310/jdg/1214437486
• [48] Verstraelen, L.: Comments on the pseudo-symmetry in the sense of Ryszard Deszcz. In: Geometry and Topology of Submanifolds, VI. World Sci., Singapore, 119-209 (1994).
• [49] Verstraelen, L.: A coincise mini history of Geometry. Kragujevac J. Math. 38 (1), 5-21 (2014).
• [50] Verstraelen, L.: Natural extrinsic geometrical symmetries – an introduction. In: Recent advances in the geometry of submanifolds: dedicated to the memory of Franki Dillen (1963-2013). AMS special session on geometry of submanifolds, San Francisco State University, San Francisco, CA, USA, October 25-26, 2014, and the AMS special session on recent advances in the geometry of submanifolds: dedicated to the memory of Franki Dillen (1963–2013), Michigan State University, East Lansing, Ml, USA, March 14-15, 2015. Proceedings. Providence. Suceavˇa, Bogdan D. (ed.) et al. Contemporary Math. 674, 5-16 (2016). DOI: http://dx.doi.org/10.1090/conm/674
• [51] Verstraelen, L.: Foreword, In: B.-Y. Chen, Differential Geometry ofWarped Product Manifolds and Submanifolds.World Scientific, vii-xxi (2017).
• [52] Verstraelen, L.: Submanifolds theory – a contemplation of submanifolds. In: Geometry of Submanifolds. AMS special session in honor of Bang- Yen Chen’s 75th birthday, University of Michigan, Ann Arbor, Michigan, October 20-21, 2018. Providence, RI: American Mathematical Society (AMS). J. Van der Veken (ed) et al. Contemporary Math. 756. 21-56 (2020). DOI: https://doi.org/10.1090/conm/756
• [53] Vilcu, G.-E.: Curvature Inequalities for Slant Submanifolds in Pointwise Kenmotsu Space Forms. in: Contact Geometry of Slant Submanifolds, B.-Y. Chen, M.H. Shahid, F. Al-Solamy (eds.), Singapore: Springer. 13-37 (2022).
• [54] Wintgen, P.: Sur l’inégalité de Chen-Willmore. C. R. Acad. Sci. Paris Sér. A. 288, 993-995 (1979).