On Semi-Riemannian Manifolds Satisfying Some Generalized Einstein Metric Conditions

Year 2023, , 539 - 576, 29.10.2023

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

https://doi.org/10.36890/iejg.1323352

Cited By: 2

Abstract

The derivation-commutator
$R \cdot C - C \cdot R$ of a
semi-Riemannian manifold $(M,g)$, $\dim M \geq 4$, formed by its
Riemann-Christoffel curvature tensor
$R$ and the Weyl conformal curvature tensor $C$,
under some assumptions,
can be expressed
as a linear combination of $(0,6)$-Tachibana tensors $Q(A,T)$,
where $A$ is a symmetric $(0,2)$-tensor and $T$
a generalized curvature tensor. These conditions
form a family of generalized Einstein metric conditions.
In this survey paper we present recent results
on manifolds and submanifolds, and in particular hypersurfaces,
satisfying such conditions.

Keywords

Warped product manifold, Einstein, quasi-Einstein, 2-quasi-Einstein and partially Einstein manifold, generalized Einstein metric condition, pseudosymmetry type curvature condition, hypersurface, Chen ideal submanifold

References

• [1] Abdalla, B. E., Dillen, F.: A Ricci-semi-symmetric hypersurface of Euclidean space which is not semi-symmetric. Proc. Amer. Math. Soc. 130 (6),1805-1808 (2002). DOI: https://doi.org/10.1090/S0002-9939-01-06220-7
• [2] Arslan, K., Deszcz, R., Ezenta¸s, R., Hotlo´s, M., Murathan, C.: On generalized Robertson-Walker spacetimes satisfying some curvature condition. Turkish J. Math. 38 (2), 353-373 (2014). https://doi.org/10.3906/mat-1304-3
• [3] Besse, A. L.: Einstein Manifolds. Ergeb. Math. Grenzgeb. (3) 10. Springer. Berlin (1987).
• [4] Bryant, R.: Some remarks on the geometry of austere manifolds. Bol. Soc. Brasil. Math. (N.S.). 21 (2), 133-157 (1991). https://doi.org/10.1007/BF01237361
• [5] Cecil, T. E., Ryan, P. J.: Geometry of Hypersurfaces. Springer Monographs in Mathematics. Springer. New York, Heidelberg, Dodrecht, London (2015).
• [6] 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
• [7] Chen, B.-Y.: A Riemannian invariant for submanifolds in space forms and its applications. In: Geometry and Topology of Submanifolds, VI. World Sci., River Edge, NJ, 58-81 (1996).
• [8] 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 Romˆane (2008).
• [9] Chen, B.-Y.: Pseudo-Riemannian Geometry, δ-Invariants and Applications. World Scientific (2011).
• [10] Chen, B.-Y.: Differential Geometry of Warped Product Manifolds and Submanifolds. World Scientific (2017).
• [11] Chen, B.-Y.: Recent developments in Wintgen inequality and Wintgen ideal submanifolds. International Electronic Journal Geometry 14 (1), 1-40 (2021). https://doi.org/10.36890/iejg.838446
• [12] Chen, B.-Y., Martin-Molina, V.: Optimal inequalities, contact δ-invariants and their applications. Bull. Malays. Math. Sci. Soc. (2) 36 (2), 263-276 (2013).
• [13] Chen, B.-Y., Verstraelen, L.: Codimension 2 submanifolds with a quasi-umbilical normal direction. J. Korean Math. Soc. 13 (1), 87-97 (1976).
• [14] Chen, B.-Y., Yıldırım, H.: Classification of ideal submanifolds of real space forms with type number ≤ 2. J. Geom. Phys. 92, 167-180 (2015). https://doi.org/10.1016/j.geomphys.2015.02.015
• [15] Chern, S. S.: Minimal submanifolds in a Riemannian manifold. Lect. Notes, Technical Rep. 19, Dept. Math. Univ. Kansas, (1968).
• [16] Chojnacka-Dulas, J., Deszcz, R., Głogowska, M., Prvanovic, M.: On warped products manifolds satisfying some curvature conditions. J. Geom. Phys. 74, 328-341 (2013). https://doi.org/10.1016/j.geomphys.2013.08.007
• [17] Dajczer, M., Florit, L. A.: On Chen’s basic equality. Illinois J. Math. 42 (1), 97-106 (1998). DOI: 10.1215/ijm/1255985615
• [18] Decu, S., Deszcz, R., Haesen, S.: A classification of Roter type spacetimes. Int. J. Geom. Meth. Modern Phys. 18 (9), art. 2150147, 13 pp. (2021). https://doi.org/10.1142/S0219887821501474
• [19] Defever, F., Deszcz, R.: On semi-Riemannian manifolds satisfying the condition R · R = Q(S,R). In: Geometry and Topology of Submanifolds, III. World Sci., River Edge, NJ, 108-130 (1991).
• [20] Defever, F., Deszcz, R., Dhooghe, P., Verstraelen, L., Yaprak, ¸S.: On Ricci-pseudosymmetric hypersurfaces in spaces of constant curvature. Results Math. 27, 227-236 (1995). https://doi.org/10.1007/BF03322827
• [21] Defever, F., Deszcz, R., Prvanovic, M.: On warped product manifolds satisfying some curvature condition of pseudosymmetry type. Bull. Greek Math. Soc. 36, 43-62 (1994). http://eudml.org/doc/237192
• [22] 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
• [23] Derdzinski, A., Roter, W.: Some theorems on conformally symmetric manifolds. Tensor (N.S.). 32 (1), 11-23 (1978).
• [24] Derdzinski A., Roter, W.: Some properties of conformally symmetric manifolds which are not Ricci-recurrent. Tensor (N.S.). 34 (1), 11-20 (1980).
• [25] Derdzinski, A., Roter, W.: Projectively flat surfaces, null parallel distributions, and conformally symmetric manifolds. Tohoku Math. J. 59 (4), 565-602 (2007). https://doi.org/10.2748/tmj/1199649875
• [26] Derdzinski A., Roter, W.: Global properties of indefinite metrics with parallel Weyl tensor. In: Pure and Applied Differential Geometry - PADGE 2007. Berichte aus der Mathematik, Shaker Verlag, Aachen, 63-72 (2007).
• [27] Derdzinski, A., Roter,W.: On compact manifolds admitting indefinite metrics with parallelWeyl tensor. J. Geom. Phys. 58 (9), 1137-1147 (2008). https://doi.org/10.1016/j.geomphys.2008.03.011
• [28] Derdzinski, A., Roter, W.: The local structure of conformally symmetric manifolds. Bull. Belg. Math. Soc. Simon Stevin. 16 (1), 117-128 (2009). DOI:2010.36045/bbms/1235574196
• [29] Derdzinski, A., Roter, W.: Compact pseudo-Riemannian manifolds with parallel Weyl tensor. Ann. Global Anal. Geom. 37 (1), 73-90 (2010). https://doi.org/10.1007/s10455-009-9173-9
• [30] Derdzinski, A., Terek, I.: New examples of compact Weyl-parallel manifolds. Preprint arXiv: 2210.03660v1 (2022).
• [31] Derdzinski, A., Terek, I.: The topology of compact rank-one ECS manifolds. Preprint arXiv: 2210.09195v1 (2022).
• [32] Derdzinski, A., Terek, I.: Rank-one ECS manifolds of dilational type. Preprint arXiv: 2301.09558v1 (2023).
• [33] Derdzinski, A., Terek, I.: The metric structure of compact rank-one ECS manifolds. Preprint arXiv: 2304.10388v1 (2023).
• [34] Derdzinski, A., Terek, I.: Compact locally homogeneous manifolds with parallel Weyl tensor. Preprint arXiv: 2306.01600v1 (2023).
• [35] Deszcz, R.: On pseudosymmetric spaces. Bull. Soc. Math. Belg. Sér. A. 44 Fasc. 1, 1-34 (1992).
• [36] Deszcz, R.: On some Akivis-Goldberg type metrics. Publ. Inst. Math. (Beograd) (N.S.). 74 (88), 71-83 (2003). DOI: 10.2298/PIM0374071D
• [37] Deszcz, R., Dillen, F., Verstraelen, L., Vrancken, L.: Quasi-Einstein totally real submanifolds of the nearly Kähler 6-sphere. Tôhoku Math. J. 51 (4), 461-478 (1999). https://doi.org/10.2748/tmj/1178224715
• [38] Deszcz, R., Głogowska, M.: Some nonsemisymmetric Ricci-semisymmetric warped product hypersurfaces. Publ. Inst. Math. (Beograd) (N.S.). 72 (86) 81-93 (2002).
• [39] Deszcz, R., Głogowska, M.: On some generalized Einstein metric conditions. International Geometry Symposium in Memory of Prof. Erdoğan Esin. 9-10 February 2023. Abstract Book, Ankara, 4-6 (2023). https://igsm-erdoganesin.gazi.edu.tr/view/page/292264/abstract-book
• [40] Deszcz, R., Głogowska, M., Hashiguchi, H., Hotlos, M., Yawata, M.: On semi-Riemannian manifolds satisfying some conformally invariantcurvature condition. Colloq. Math. 131 (2), 149-170 (2013). DOI: 10.4064/cm131-2-1
• [41] Deszcz, R., Głogowska, M., Hotlos, M.: On hypersurfaces satisfying conditions determined by the Opozda-Verstraelen affine curvature tensor.Ann. Polon. Math. 126 (3), 215-240 (2021). DOI: 10.4064/ap200715-6-5
• [42] Deszcz, R., Głogowska, M., Hotlos, M., Petrovic-Torgašev, M., Zafindratafa, G.: A note on some generalized curvature tensor. Int. Electron. J.Geom. 17 (1), 379-397 (2023). https://doi.org/10.36890/iejg.1273631
• [43] Deszcz, R., Głogowska, M., Hotlos, 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).
• [44] Deszcz, R., Głogowska, M., Hotlos, M., Sawicz, K.: Hypersurfaces in space forms satisfying a particular Roter type equation. Preprint arXiv:2211.06700v2 (2022).
• [45] Deszcz, R., Głogowska, M., Hotlos, M., ¸Sentürk, Z.: On certain quasi-Einstein semisymmetric hypersurfaces. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 41, 151-164 (1998).
• [46] Deszcz, R., Głogowska, M., Hotlos, 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
• [47] Deszcz, R., Głogowska, M., Hotlos, M., Verstraelen, L.: On some generalized Einstein metric conditions on hypersurfaces in semi-Riemannian space forms. Colloq. Math. 96 (2), 149-166 (2003). DOI: 10.4064/cm96-2-1
• [48] Deszcz, R., Głogowska, M., Hotlos, M., Zafindratafa, G.: On some curvature conditions of pseudosymmetry type. Period. Math. Hung. 70 (2), 153-170 (2015). DOI 10.1007/s10998-014-0081-9
• [49] Deszcz, R., Głogowska, M., Hotlos, M., Zafindratafa, G.: Hypersurfaces in space forms satisfying some curvature conditions. J. Geom. Phys. 99, 218-231 (2016). https://doi.org/10.1016/j.geomphys.2015.10.010
• [50] Deszcz, R., Głogowska, M., Jełowicki, J., Petrovic-Torgašev, M., Zafindratafa, G.: On Riemann andWeyl compatible tensors. Publ. Inst. Math. (Beograd) (N.S.). 94 (108), 111-124 (2013). DOI: 10.2298/PIM1308111D
• [51] Deszcz, R., Głogowska, M., Jełowicki, J., Zafindratafa, G.: Curvature properties of some class of warped product manifolds. Int. J. Geom. Methods Modern Phys. 13 (1), art. 1550135, 36 pp. (2016). https://doi.org/10.1142/S0219887815501352
• [52] Deszcz, R., Głogowska, M., Petrovic-Torgašev, M., Verstraelen, L.: On the Roter type of Chen ideal submanifolds. Results Math. 59, 401-413 (2011). https://doi.org/10.1007/s00025-011-0109-x
• [53] 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
• [54] 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).
• [55] Deszcz, R., Głogowska, M., Zafindratafa, G.: Hypersurfaces in space forms satisfying some generalized Einstein metric condition. J. Geom. Phys. 148, 103562 20 pp. (2020). https://doi.org/10.1016/j.geomphys.2019.103562
• [56] 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).
• [57] Deszcz, R., Hotlos, M.: On a certain subclass of pseudosymmetric manifolds. Publ. Math. Debrecen. 53 (1-2), 29-48 (1998). DOI: 10.5486/PMD
• [58] Deszcz, R., Hotlos, M.: On hypersurfaces with type number two in spaces of constant curvature. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 46, 19-34 (2003).
• [59] Deszcz, R., Hotlos, M.: On some pseudosymmetry type curvature conditions. Tsukuba J. Math. 27 (1), 13-30 (2003). DOI: 10.21099/tkbjm/1496164557
• [60] Deszcz, R., Hotlos, M., Jełowicki, J., Kundu, H., Shaikh, A. A.: Curvature properties of Gödel metric. Int. J. Geom. Meth. Modern Phys. 11(3), 1450025, 20 pp. (2014). https://doi.org/10.1142/S021988781450025X
• [61] Deszcz, R., Hotlos, M., Şentürk Z.: On the equivalence of the Ricci-pseudosymmetry and pseudosymmetry. Colloq. Math. 79 (2), 211-227 (1999). DOI: 10.4064/cm-79-2-211-227
• [62] Deszcz, R., Hotlos, M., Şentürk Z.: On curvature properties of quasi-Einstein hypersurfaces in semi-Euclidean spaces. Soochow J. Math. 27 (4), 375-389 (2001).
• [63] Deszcz, R., Hotlos, M., Şentürk Z.: On some family of generalized Einstein metric conditions. Demonstr. Math. 34 (4), 943-954 (2001). https://doi.org/10.1515/dema-2001-0422
• [64] Deszcz, R., Hotlos, M., Şentürk, Z.: On curvature properties of certain quasi-Einstein hypersurfaces. Int. J. Math. 23 (7), 1250073 17 pp. (2012). https://doi.org/10.1142/S0129167X12500735
• [65] Deszcz, R., Kowalczyk, D.: On some class of pseudosymmetric warped products. Colloq. Math. 97 (1), 7-22 (2003). DOI: 10.4064/cm97-1-2
• [66] Deszcz, R., Kucharski, M.,: On curvature properties of certain generalized Robertson-Walker spacetimes. Tsukuba J. Math. 23 (1), 113-130 (1999). https://www.jstor.org/stable/43686121
• [67] Deszcz, R., Petrovic-Torgašev, M., Verstraelen, L., Zafindratafa, G.: On the intrinsic symmetries of Chen ideal submanifolds. Bull. Transilvania Univ. Brasov, Ser. III, Math., Inform., Phys. 1 (50), 99-108 (2008).
• [68] Deszcz, R., Petrovic-Torgašev, M., Verstraelen, L., Zafindratafa, G.: On Chen ideal submanifolds satisfying some conditions of pseudo-symmetry type. Bull. Malaysian Math. Sci. Soc. 39 (1), 103-131 (2016). https://doi.org/10.1007/s40840-015-0164-7
• [69] Deszcz, R., Plaue, M., Scherfner, M.: On Roter type warped products with 1-dimensional fibres. J. Geom. Phys. 69, 1-11 (2013). https://dx.doi.org/10.1016/j.geomphys.2013.02.006
• [70] Deszcz, R., Scherfner, M.: On a particular class of warped products with fibres locally isometric to generalized Cartan hypersurfaces. Colloq. Math. 109 (1), 13-29 (2007). DOI: 10.4064/cm109-1-3
• [71] Deszcz, R., Verstraelen, L.: Hypersurfaces of semi-Riemannian conformally flat manifolds. In: Geometry and Topology of Submanifolds, III. World Sci., River Edge, NJ, 131-147 (1991).
• [72] Deszcz, R., Verstraelen, L., Vrancken, L.: The symmetry of warped product spacetimes. Gen. Relativ. Gravit. 23 (6), 671-681 (1991). https://doi.org/10.1007/BF00756772
• [73] Deszcz, R., Verstraelen, L., Yaprak, Ş.: Warped products realizing a certain condition of pseudosymmetry type imposed on the Weyl curvature tensor. Chinese J. Math. 22 (2), 139-157 (1994). https://www.jstor.org/stable/43836548
• [74] Deszcz, R., Verstraelen, L., Yaprak, Ş.: 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
• [75] Deszcz, R., Verstraelen, L., Yaprak, Ş.: Pseudosymmetric hypersurfaces in 4-dimensional spaces of constant curvature. Bull. Inst. Math. Acad. Sinica 22 (1), 167-179 (1994).
• [76] Deszcz, R., Yaprak, Ş.: Curvature properties of Cartan hypersurfaces, Colloq. Math. 67 (1), 91-98 (1994). DOI: 10.4064/cm-67-1-91-98
• [77] Deszcz, R., Yaprak, Ş.: Curvature properties of certain pseudosymmetric manifolds. Publ. Math. Debrecen. 45 (3-4), 333-345 (1994). DOI: 10.5486/PMD
• [78] Dillen, F., Petrovic-Torgašev, M., Verstraelen, L.: Einstein, conformally flat and semi-symmetric sumbmanifolds satisfying Chen’s equality. Israel J. Math. 100, 163-169 (1997). https://doi.org/10.1007/BF02773638
• [79] Fu, Y.: Biharmonic hypersurfaces with three distinct principal curvatures in Euclidean 5-space. J. Geom. Phys. 75 (2014), 113-119. https://doi.org/10.1016/j.geomphys.2013.09.004
• [80] Głogowska, M.: Semi-Riemannian manifolds whoseWeyl tensor is a Kulkarni-Nomizu square. Publ. Inst. Math. (Beograd) (N.S.). 72 (86), 95-106 (2002). DOI: 10.2298/PIM0272095G
• [81] Głogowska, M.: On a curvature characterization of Ricci-pseudosymmetric hypersurfaces. Acta Math. Scientia, 24 B (3), 361-375 (2004). https://doi.org/10.1016/S0252-9602(17)30160-1
• [82] Głogowska, M.: Curvature conditions on hypersurfaces with two distinct principal curvatures. In: Banach Center Publ., Inst. Math. Polish Acad. Sci. 69, 133-143 (2005). DOI: 10.4064/bc69-0-8
• [83] Głogowska, M.: On Roter type manifolds. In: Pure and Applied Differential Geometry - PADGE 2007. Berichte aus der Mathematik, Shaker Verlag, Aachen. 114-122 (2007).
• [84] Głogowska, M.: On quasi-Einstein Cartan type hypersurfaces. J. Geom. Phys. 58 (5), 599-614 (2008). doi:10.1016/j.geomphys.2007.12.012
• [85] Gödel, K.: An example of a new type of cosmological solutions of Einstein’s field equations of gravitation. Rev. Mod. Phys. 21 (3), 447-450 (1949). https://doi.org/10.1103/RevModPhys.21.447
• [86] Haesen, S., Verstraelen, L.: Properties of a scalar curvature invariant depending on two planes. Manuscripta Math. 122, 59-72 (2007). https://doi.org/10.1007/s00229-006-0056-0
• [87] Haesen, S., Verstraelen, L.: Natural intrinsic geometrical symmetries. SIGMA. 5, 086, 15 pp. (2009). https://doi.org/10.3842/SIGMA.2009.086
• [88] Hájková, V., Kowalski, O., Sekizawa, M.: On three-dimensional hypersurfaces with type number two in H4 and S4 treated in intrinsic way. In: Jan Slovák and Martin Cˇ adek (eds.): The proceedings of the 23rd winter school "Geometry and Physics". Srní, Czech Republic, January 18-25, 2003. Palermo: Circ. Mat. di Palermo. Suppl. Rend. Circ. Mat. Palermo, Sér. II, Suppl. 72, 107-126, (2004).
• [89] Harvey, R., Lawson, H. B. Jr., Calibrated geometries. Acta Math. 148, 47-157 (1982). DOI: 10.1007/BF02392726
• [90] Hashimoto, N.; Sekizawa, M.: Three-dimensional conformally flat pseudo-symmetric spaces of constant type. Arch. Math. (Brno) 36 (4), 279-286 (2000). http://dml.cz/dmlcz/107742
• [91] Hotlos, M.: On conformally symmetric warped products. Ann. Acad. Paedagog. Crac. 23. Studia Math. 4, 75-85 (2004).
• [92] Kowalczyk, D.: On some class of semisymmetric manifolds. Soochow J. Math. 27 (4), 445-461 (2001).
• [93] Kowalczyk, D.: On the Reissner-Nordström-de Sitter type spacetimes. Tsukuba J. Math. 30 (2), 363-381 (2006). DOI: 10.21099/tkbjm/ 1496165068
• [94] Kowalski, O.: Classification of generalized symmetric Riemannian spaces of dimension n ≤ 5. Rozpr. Cˇ esk. Akad. Veˇd, Rˇ adaMat. Prˇir. Veˇd, 85 (8), 1-61, (1975).
• [95] Kowalski, O.: Generalized Symmetric Spaces. Lecture Notes in Mathematics 865, Berlin - Heidelberg - New York, Springer Verlag, 1980.
• [96] Kowalski, O.: Generalized Symmetric Spaces. MIR, Moscow, 1984 (in Russian).
• [97] Kowalski, O.: A classification of Riemannian 3-manifolds with constant principal Ricci curvatures ρ1 = ρ2 ̸= ρ3. Nagoya Math. J. 132, 1-36 (1993). https://doi.org/10.1017/S002776300000461X
• [98] Kowalski, O.: An explicit classification of 3-dimensional Riemannian spaces satisfying R(X; Y ) · R = 0. Czechoslovak Math. J. 46 (3), 427-474 (1996). http://dml.cz/dmlcz/127308
• [99] Kowalski, O., Sekizawa, M.: Three-dimensional Riemannian manifolds of c-conullity two, Chapter 11. Riemannian Manifolds of Conullity Two. World Sci., Singapore, 1966.
• [100] Kowalski, O., Sekizawa, M.: Local isometry classes of Riemannian 3-manifolds with constant Ricci eigenvalues ρ1 = ρ2 ̸= ρ3. Arch. Math. (Brno) 32 (2), 137-145 (1996). http://dml.cz/dmlcz/107568
• [101] Kowalski, O., Sekizawa, M.: Pseudo-symmetric spaces of constant type in dimension three - elliptic spaces. Rend. Mat. Appl., Ser. VII, 17 (3), 477-512 (1997).
• [102] Kowalski, O., Sekizawa, M.: Riemannian 3-manifolds with c-conullity two. Boll. Unione Mat. Ital., Ser. VII, B 11, No. 2, Suppl., 161-184 (1997).
• [103] Kowalski, O., Sekizawa, M.: Pseudo-symmetric spaces of constant type in dimension three - non-elliptic spaces. Bull. Tokyo Gakugei Univ. Sect. IV, Math. Nat. Sci. 50, 1-28 (1998).
• [104] Kowalski, O., Sekizawa, M.: Pseudo-symmetric spaces of constant type in dimension three. Personal Note, Charles University - Tokyo Gakugei University, Prague - Tokyo, 1-56 (1998).
• [105] Kowalski, O., Sekizawa, M.: Hypersurfaces of type number two in the hyperbolic four-space and their etensions to Riemannian Geometry. In: Non-Euclidean Geometries: János Bolyai Memorial Volume, Springer, 407-426 (2006).
• [106] Kruchkovich, G. I.: On some class of Riemannian spaces. Trudy sem. po vekt. i tenz. analizu, 11, 103-128 (1961) (in Russian).
• [107] Lumiste, Ü.: Semiparallel Submanifolds in Space Forms. Springer Science + Business Media, New York, LLC (2009).
• [108] Murathan, C., Arslan, K., Deszcz, R., Ezenta¸s, R., Özgür, C.: On a certain class of hypersurfaces of semi-Euclidean spaces. Publ. Math. Debrecen 58 (4), 587-604 (2001). DOI: 10.5486/PMD.2001.2367
• [109] O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Academic Press. London (1983).
• [110] Ou, Y.-L., Chen, B.-Y.: Biharmonic Submanifolds and biharmonic maps in Riemannian Geometry. World Sci., 2020.
• [111] Sawicz, K.: Hypersurfaces in spaces of constant curvature satisfying some Ricci-type equations. Colloq. Math. 101 (2), 183-201 (2004). DOI: 10.4064/cm101-2-4
• [112] Sawicz, K.: On some class of hypersurfaces with three distinct principal curvatures. In: Banach Center Publ., Inst. Math. Polish Acad. Sci. 69, 145-156 (2005). DOI: 10.4064/bc69-0-9
• [113] Sawicz, K.: On curvature characterization of some hypersurfaces in spaces of constant curvature. Publ. Inst. Math. (Beograd) (N.S.). 79 (93), 95-107 (2006). DOI: 10.2298/PIM0693095S
• [114] Sawicz, K.: Curvature identities on hypersurfaces in semi-Riemannian space forms. In: Pure and Applied Differential Geometry - PADGE 2007. Berichte aus der Mathematik, Shaker Verlag, Aachen, 252-260 (2007).
• [115] Sawicz, K.: Curvature properties of some class of hypersurfaces in Euclidean spaces. Publ. Inst. Math. (Beograd) (N.S.). 98 (112) , 165-177 (2015). DOI: 10.2298/PIM141025001S
• [116] Shaikh, A. A., Deszcz, R., Hotlos, M., Jełowicki, J., Kundu, H.: On pseudosymmetric manifolds. Publ. Math. Debrecen 86 (3-4), 433-456 (2015). DOI: 10.5486/PMD.2015.7057
• [117] Shaikh, A. A., Kundu, H.: On warped product generalized Roter type manifolds. Balkan J. Geom. Appl. 21 (2), 82-94 (2016).
• [118] Shaikh, A. A., Kundu, H.: On generalized Roter type manifolds. Kragujevac J. Math. 43 (3), 471-493 (2019).
• [119] 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
• [120] Szabó, Z. I.: Structure theorems on Riemannian manifolds satisfying R(X, Y ) · R = 0. II. Global version. Geom. Dedicata 19, 65-108 (1985). https://doi.org/10.1007/BF00233102
• [121] Szabó, Z. I.: Classification and construction of complete hypersurfaces satisfying R(X, Y ) · R = 0. Acta Sci. Math. (Szeged). 47 (3-4), 321-348 (1984).
• [122] Verstraelen, L.: Some comments on the δ-curvatures of Bang-Yen Chen, Rend. Sem. Matem. Messina, International Congress in honour of R. Calapso, 331-337 (1998).
• [123] 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).
• [124] Verstraelen, L.: A coincise mini history of Geometry. Kragujevac J. Math. 38 (1), 5-21 (2014).
• [125] 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
• [126] Verstraelen, L.: Foreword, In: B.-Y. Chen, Differential Geometry ofWarped Product Manifolds and Submanifolds.World Scientific, vii-xxi (2017).
• [127] 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