Yıl 2023,
Cilt: 16 Sayı: 2, 539 - 576, 29.10.2023
Ryszard Deszcz
,
Małgorzata Głogowska
,
Marian Hotloś
,
Miroslava Petrović-torgašev
,
Georges Zafindratafa
Kaynakça
- [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
On Semi-Riemannian Manifolds Satisfying Some Generalized Einstein Metric Conditions
Yıl 2023,
Cilt: 16 Sayı: 2, 539 - 576, 29.10.2023
Ryszard Deszcz
,
Małgorzata Głogowska
,
Marian Hotloś
,
Miroslava Petrović-torgašev
,
Georges Zafindratafa
Öz
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.
Kaynakça
- [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