A Note on Some Generalized Curvature Tensor

Year 2023, Volume: 16 Issue: 1, 379 - 397, 30.04.2023

Ryszard Deszcz Małgorzata Głogowska Marian Hotloś Miroslava Petrović-torgašev Georges Zafındratafa

https://doi.org/10.36890/iejg.1273631

Cited By: 2

Abstract

For any semi-Riemannian manifold (M, g) we define some generalized curvature tensor E as a linear combination of Kulkarni-Nomizu products formed by the metric tensor, the Ricci tensor and its square of given manifold. That tensor is closely related to quasi-Einstein spaces, Roter spaces and some Roter type spaces.

Keywords

warped product manifold, Einstein manifold, quasi-Einstein manifold, partially Einstein manifold, generalized Einstein metric condition, pseudosymmetry type curvature condition, Roter space, generalized Roter space, hypersurface, principal curvature, quasi-umbilical hypersurface

References

• [1] Arslan, K., Deszcz, R., Ezentaş, R., Hotloś, 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
• [2] Besse, A. L.: Einstein Manifolds. Ergeb. Math. Grenzgeb. (3) 10. Springer. Berlin (1987).
• [3] Cecil, T. E., Ryan, P. J.: Geometry of Hypersurfaces. Springer Monographs in Mathematics. Springer. New York, Heidelberg, Dodrecht, London (2015).
• [4] Chen, B. Y.: Pseudo-Riemannian Geometry, δ-Invariants and Applications. World Scientific (2011).
• [5] Chen, B.Y.: Differential Geometry of Warped Product Manifolds and Submanifolds. World Scientific (2017).
• [6] Chen, B.-Y.: Recent developments inWintgen inequality andWintgen ideal submanifolds. International Electronic Journal Geometry 14 (1) , 1-40 (2021). https://doi.org/10.36890/iejg.838446
• [7] Chojnacka-Dulas, J., Deszcz, R., Głogowska, M., Prvanović, 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
• [8] 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
• [9] Decu, S., Petrović-Torgašev, M., Šebeković, A., Verstraelen, L.: On the Roter type of Wintgen ideal submanifolds. Rev. Roumaine Math. Pures Appl. 57 (1) , 75-90 (2012).
• [10] Defever, F., Deszcz, R., Prvanović, M.: On warped product manifolds satisfying some curvature condition of pseudosymmetry type. Bull. Greek Math. Soc. 36 , 43-67 (1994).
• [11] Derdziński, A., Roter, W.: Some theorems on conformally symmetric manifolds. Tensor (N.S.). 32 (1), 11-23 (1978).
• [12] Derdziński A., Roter, W.: Some properties of conformally symmetric manifolds which are not Ricci-recurrent. Tensor (N.S.). 34 (1), 11-20 (1980).
• [13] Derdziński, 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
• [14] Derdziński A., Roter,W.: Global properties of indefinite metrics with parallelWeyl tensor. In: Pure and Applied Differential Geometry - PADGE 2007. Shaker Verlag, Aachen, 63-72 (2007).
• [15] Derdziński, A., Roter, W.: On compact manifolds admitting indefinite metrics with parallel Weyl tensor. J. Geom. Phys. 58 (9), 1137-1147 (2008). https://doi.org/10.1016/j.geomphys.2008.03.011
• [16] Derdziński, 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
• [17] Derdziński, A., Roter, W.: Compact pseudo-Riemannian manifolds with parallel Weyl tensor. Ann. Global Anal. Geom. 37, 73-90 (2010). https://doi.org/10.1007/s10455-009-9173-9
• [18] Derdzinski, A., Terek, I.: New examples of compact Weyl-parallel manifolds. Preprint arXiv: 2210.03660v1 (2022).
• [19] Derdzinski, A., Terek, I.: The topology of compact rank-one ECS manifolds. Preprint arXiv: 2210.09195v1 (2022).
• [20] Deszcz, R.: On conformally flat Riemannian manifolds satisfying certain curvature conditions. Tensor (N.S.). 49 (2), 134-145 (1990).
• [21] Deszcz, R.: On four-dimensional warped product manifolds satisfying certain pseudo-symmetry curvature conditions. Colloq. Math. 62 (1), 103-120 (1991). DOI: 10.4064/cm-62-1-103-120
• [22] Deszcz, R.: On some Akivis-Goldberg type metrics. Publ. Inst. Math. (Beograd) (N.S.). 74 (88) , 71-83 (2003). DOI: 10.2298/PIM0374071D
• [23] 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
• [24] Deszcz, R., Głogowska, M., Hashiguchi, H., Hotloś, 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
• [25] Deszcz, R., Głogowska, M., Hotloś, 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
• [26] Deszcz, R., Głogowska, M., Hotloś, M., Jełowicki, J., Zafindratafa, G.: Curvature properties of some warped product manifolds. Poster, Conf. "Differential Geometry", Banach Conf. Center at B˛edlewo, June 19 to June 24 (2017).
• [27] Deszcz, R., Głogowska, M., Hotloś, 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).
• [28] Deszcz, R., Głogowska, M., Hotlośs, M., Sawicz, K.: Hypersurfaces in space forms satisfying a particular Roter type equation. Preprint arXiv: 2211.06700v2 (2022).
• [29] Deszcz, R., Głogowska, M., Hotloś, M., ¸Sentürk, Z.: On certain quasi-Einstein semisymmetric hypersurfaces. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 41 , 151-164 (1998).
• [30] Deszcz, R., Głogowska, M., Hotloś, 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
• [31] Deszcz, R., Głogowska, M., Hotloś, 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
• [32] Deszcz, R., Głogowska, M., Hotloś, 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
• [33] Deszcz, R., Głogowska, M., Jełowicki, J., Petrović-Torgašev, M., Zafindratafa, G.: On Riemann and Weyl compatible tensors. Publ. Inst. Math. (Beograd) (N.S.). 94 (108), 111-124 (2013). DOI: 10.2298/PIM1308111D
• [34] 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
• [35] Deszcz, R., Głogowska, M., Petrović-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
• [36] 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).
• [37] 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
• [38] 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).
• [39] Deszcz, R., Hotloś, M.: On a certain subclass of pseudosymmetric manifolds. Publ. Math. Debrecen. 53 (1-2), 29-48 (1998). DOI: 10.5486/PMD
• [40] Deszcz, R., Hotloś,M.: On hypersurfaces with type number two in spaces of constant curvature. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 46, 19-34 (2003).
• [41] Deszcz, R., Hotloś, M.: On geodesic mappings in a particular class of Roter spaces. Colloq. Math. 166 (2), 267-290 (2021). DOI: 10.4064/cm7797- 1-2021
• [42] Deszcz, R., Hotloś, 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
• [43] Deszcz, R., Hotloś, 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
• [44] Deszcz, R., Kowalczyk, D.: On some class of pseudosymmetric warped products. Colloq. Math. 97 (1), 7-22 (2003). DOI: 10.4064/cm97-1-2
• [45] Deszcz, R., Petrović-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
• [46] 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
• [47] 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
• [48] 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).
• [49] 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
• [50] 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/43836548395 dergipark
• [51] Deszcz, R., Verstraelen, L., Yaprak, ¸S.: On hypersurfaces with pseudosymmetric Weyl tensor. In: Geometry and Topology of Submanifolds, VIII, World Sci., River Edge, NJ, 111-120 (1996).
• [52] Deszcz, R., Verstraelen, L., Yaprak, ¸S.: On 2-quasi-umbilical hypersurfaces in conformally flat spaces, Acta Math. Hungarica. 78 (1-2), 45-57 (1998). https://doi.org/10.1023/A:1006566319359
• [53] Deszcz, R., Verstraelen, L., Yaprak, ¸S.: Hypersurfaces with pseudosymmetric Weyl tensor in conformally flat spaces. In: Geometry and Topology of Submanifolds, IX. World Sci., River Edge, NJ, 108-117 (1999).
• [54] Deszcz, R., Yaprak, ¸S.: Curvature properties of certain pseudosymmetric manifolds. Publ. Math. Debrecen. 45 (3-4), 333-345 (1994). DOI: 10.5486/PMD
• [55] Duggal, K. L., Sharma, R.: Hypersurfaces in a conformally flat space with curvature collineation. Internat. J. Math. and Math. Sci. 14 (3), 595-604 (1991). https://doi.org/10.1155/S0161171291000807
• [56] Eisenhart, L.P.: Riemannian Geometry. Princeton Univ. Press. Princeton (1966).
• [57] Eyasmin, S.: Hypersurfaces in a conformally flat space. Int. J. Geom. Methods Modern Phys. 18 (5), art. 2150067, pp. 7 (2021). https://doi.org/10.1142/S0219887821500675
• [58] Głogowska, M.: Semi-Riemannian manifolds whose Weyl tensor is a Kulkarni-Nomizu square. Publ. Inst. Math. (Beograd) (N.S.). 72 (86), 95-106 (2002). DOI: 10.2298/PIM0272095G
• [59] 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
• [60] Głogowska, M.: On Roter type manifolds. In: Pure and Applied Differential Geometry - PADGE 2007. Shaker Verlag, Aachen. 114-122 (2007).
• [61] Głogowska, M.: On quasi-Einstein Cartan type hypersurfaces, J. Geom. Phys. 58 (5), 599-614 (2008). doi:10.1016/j.geomphys.2007.12.012
• [62] Griffiths, J.B.: Podolský, J.: Exact Space-Times in Einstein’s General Relativity. Cambridge Univ. Press (2009).
• [63] 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
• [64] Haesen, S., Verstraelen, L.: Natural intrinsic geometrical symmetries. SIGMA. 5, 086, 15 pp. (2009). https://doi.org/10.3842/SIGMA.2009.086
• [65] Hotloś, M.: On conformally symmetric warped products. Ann. Acad. Paedagog. Crac. 23. Studia Math. 4, 75-85 (2004).
• [66] Kon, M.: Pseudo-Einstein real hypersurfaces in complex space forms. J. Differential Geom. 14 (3) , 339-354 (1979). DOI: 10.4310/jdg/1214435100
• [67] Kowalczyk, D.: On some class of semisymmetric manifolds. Soochow J. Math. 27 (4), 445-461 (2001).
• [68] Kowalczyk, D.: On the Reissner-Nordström-de Sitter type spacetimes. Tsukuba J. Math. 30 (2), 363-381 (2006). DOI: 10.21099/tkbjm/ 1496165068
• [69] Kundu, H., Mandal, J. K., Baishya, K. K.: On hypersurfaces with semisymmetric and pseudosymmetric Weyl tensor embedded in a conformally flat manifold. Preprint, Research Square, 17 pp. (2022).
• [70] Lumiste, Ü.: Semiparallel Submanifolds in Space Forms. Springer Science + Business Media, New York, LLC (2009).
• [71] Maeda, Y.: On real hypersurfaces of a complex projective space. J. Math. Soc. Japan. 28 (3), 529-540 (1976). DOI: 10.2969/jmsj/02830529
• [72] 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
• [73] 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
• [74] 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
• [75] Shaikh, A. A., Deszcz, R., Hotlo´s, M., Jełowicki, J., Kundu, H.: On pseudosymmetric manifolds. Publ. Math. Debrecen 86 (3-4) , 433-456 (2015). DOI: 10.5486/PMD.2015.7057
• [76] Shaikh, A. A., Kundu, H.: On warped product generalized Roter type manifolds. Balkan J. Geom. Appl. 21 (2), 82-94 (2016).
• [77] Shaikh, A. A., Kundu, H.: On generalized Roter type manifolds. Kragujevac J. Math. 43 (3), 471-493 (2019).
• [78] Sharma, R.: Cauchy-Riemann (CR) - submanifolds of semi-Riemannian manifolds with applications to relativity and hydrodynamics. Electronic Theses and Dissertations, 1374. Windsor, Ontario, Canada. Ph.D. thesis, University of Windsor (1986).https://scholar.uwindsor.ca/etd/1374
• [79] Stuchlik, Z., Hledik, S.: Properties of the Reissner-Nordström spacetimes with a nonzero cosmological constant. Acta Phys. Slovaca. 52 (5), 363-407 (2002).
• [80] 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
• [81] 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).
• [82] Verstraelen, L.: A coincise mini history of Geometry. Kragujevac J. Math. 38 (1), 5-21 (2014).
• [83] 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
• [84] Verstraelen, L.: Foreword, In: B.-Y. Chen, Differential Geometry ofWarped Product Manifolds and Submanifolds.World Scientific, vii-xxi (2017).
• [85] 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
• [86] Yaprak, Ş.: Intrinsic and extrinsic differential geometry concerning curvature conditions of pseudosymmetry type. Ph.D. thesis, Katholieke Universiteit Leuven (1993).
• [87] Zafindratafa, G.: Sous-variétés soumises a des conditions de courbure (in French). Ph.D. thesis, Katholieke Universiteit Leuven (1991).