On Chen Ideal Submanifolds Satisfying Some Conditions of Pseudo-symmetry Type

详细信息    查看全文

• 作者：Ryszard Deszcz ; Miroslava Petrović-Torgas̆ev…
• 关键词：Submanifold ; Condition of pseudo ; symmetry type ; Generalized Einstein metric condition ; Chen ideal submanifold ; Roter space ; Tachibana tensor ; Primary 53B20 ; 53B25 ; 53B30 ; 53B50 ; Secondary 53C25 ; 53C40
• 刊名：Bulletin of the Malaysian Mathematical Sciences Society
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：39
• 期：1
• 页码：103-131
• 全文大小：621 KB
• 参考文献：1.Arslan, K., Deszcz, R., Ezentaṣ, R., Hotloś, M., Murathan, C.: On generalized Robertson-Walker spacetimes satisfying some curvature condition. Turk. J. Math. 38, 353–373 (2014)
  2.Asperti, A.C., Lobos, G.A., Mercuri, F.: Pseudo-parallel submanifolds of a space form. Adv. Geom. 2, 57–71 (2002)MathSciNet MATH
  3.Besse, A.L.: Einstein Manifolds. Ergeb. Math. Grenzgeb., 3. Folge, Bd. 10. Springer, Berlin (1987)
  4.Blair, D.E., Verheyen, P., Verstraelen, L.: Hypersurfaces satisfaisant a \(R \cdot C = 0\) ou \(C \cdot R = 0\) . C. R. Acad. Bulg. Sci. 37, 459–462 (1984)MathSciNet
  5.Cartan, E.: Leçons sur la géométrie des espaces de Riemann, 2nd edn. Gauthier-Villars, Paris (1946)MATH
  6.Chen, B.-Y.: Some pinching and classification theorems for minimal submanifolds. Arch. Math. (Basel) 60, 568–578 (1993)MathSciNet CrossRef MATH
  7.Chen, B.-Y.: Riemannian submanifolds. In: Dillen, F., Verstraelen, L. (eds.) Handbook of Differential Geometry, pp. 187–418. North Holland, Amsterdam (2000)
  8.Chen, B.-Y.: \(\delta \) -invariants, inequalities of submanifolds and their applications, Chap. 2. Topics in Differential Geometry, pp. 29–155. Romanian Academy of Science, Bucharest (2008)
  9.Chen, B.-Y.: Pseudo-Riemannian geometry, \( \delta \) -Invariants and Applications. World Scientific, Hackensack (2011)MATH
  10.Chen, B.-Y.: A tour through \(\delta \) -invariants: From Nash embedding theorem to ideal immersions, best ways of living and beyond. Publ. Inst. Math. (Beograd) (N.S.) 94(108), 67–80 (2013)
  11.Chen, B.-Y.: \(\delta \) -Invariants, Inequalities of Submanifolds and Their Applications, arXiv:​1307.​1877V1 [math.DG] 7 (2013)
  12.Chen, B.-Y.: Ideal hypersurfaces of Euclidean 4-space, Arab. J. Math. Sci. 19, 129–144 (2013)
  13.Chen, B.-Y., Dillen, F.: Totally real bisectional curvature, Bochner-Kaehler and Einstein-Kaehler manifolds. Diff. Geom. Appl. 10, 145–154 (1999)MathSciNet CrossRef MATH
  14.Chen, B.-Y., Dillen, F., Verstraelen, L.: \(\delta \) -invariants and their applications to centroaffine geometry. Diff. Geom. Appl. 22, 341–354 (2005)MathSciNet CrossRef MATH
  15.Chen, B.-Y., Dillen, F., Verstraelen, L., Vrancken, L.: Characterizations of Riemannian space forms, Einstein spaces and conformally flat spaces. Proc. Am. Math. Soc. 128, 589–598 (2000)MathSciNet CrossRef MATH
  16.Chen, B.-Y., Martin-Molina, V.: Optimal inequalities, contact \(\delta \) -invariants and their applications. Bull. Malays. Math. Sci. Soc. (2) 36(2), 263–276 (2013)MathSciNet MATH
  17.Chern, S.S.: Minimal submanifolds in a Riemannian manifold. Lecture Notes in Technical Report 19, Department of Mathematics, University of Kansas (1968)
  18.Chojnacka-Dulas, J., Deszcz, R., Głogowska, M., Prvanović, M.: On warped products manifolds satisfying some curvature conditions. J. Geom. Phys. 74, 328–341 (2013)MathSciNet CrossRef MATH
  19.Dajczer, M., Florit, L.A.: On Chen’s basic equality. Ill. J. Math. 42, 97–106 (1998)MathSciNet MATH
  20.Derdziński, A.: Examples de métriques de Kaehler et d’Einstein autoduales sur le plan complexe. Géométrie riemannienne en dimension 4 (Séminaire Arthur Besse 1978/79), Cedic/Fernand Nathan Paris 1981, pp. 334–346
  21.Defever, F., Deszcz, R.: On semi-Riemannian manifolds satisfying the condition \(R \cdot R = Q(S,R)\) . Geometry and Topology of Submanifolds, vol. III, pp. 108–130. World Scientific, River Edge, NJ (1991)
  22.Deprez, J., Roter, W., Verstraelen, L.: Conditions on the projective curvature tensor of conformally flat Riemannian manifolds. Kyungpook Math. J. 29, 153–165 (1989)MathSciNet MATH
  23.Deszcz, R.: On pseudo-symmetric spaces. Bull. Belg. Math. Soc. Ser. A 44, 1–34 (1992)MathSciNet MATH
  24.Deszcz, R., Głogowska, M., Hashiguchi, H., Hotloś, M., Yawata, M.: On semi-Riemannian manifolds satisfying some conformally invariant curvature condition. Colloq. Math. 131, 149–170 (2013)MathSciNet CrossRef MATH
  25.Deszcz, R., Głogowska, M., Hotloś, M.: Some identities on hypersurfaces in conformally flat spaces. In: Proceedings of the International Conference XVI Geometrical Seminar, Vrnjačka banja, 20–25 Sept 2010, Faculty of Science and Mathematics, University of Niš, Serbia, pp. 34–39 (2011)
  26.Deszcz, R., Głogowska, M., Hotloś, M., Sawicz, K.: A survey on generalized Einstein metric conditions. AMS/IP Stud. Adv. Math. 49, 27–45 (2011)
  27.Deszcz, R., Głogowska, M., Hotloś, M., Ṣentürk, Z.: On certain quasi-Einstein semi-symmetric hypersurfaces. Ann. Univ. Sci. Bp. Eötvös Sect. Math. 41, 151–164 (1998)MATH
  28.Deszcz, R., Głogowska, M., Hotloś, M., Verstraelen, L.: On some generalized Einstein metric conditions on hypersurfaces in semi-Riemannian space forms. Colloq. Math. 96, 149–166 (2003)MathSciNet CrossRef MATH
  29.Deszcz, R., Głogowska, M., Hotloś, M., Zafindratafa, G.: On some curvature conditions of pseudo-symmetry type. Period. Math. Hung. 70, 153–170 (2015)CrossRef
  30.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)
  31.Deszcz, R., Głogowska, M., Jełowicki, J., Zafindratafa, G.: Curvature properties of some class of warped product manifolds. Int. J. Geom. Meth. Modern Phys. doi:10.​1142/​SO21988781550135​2
  32.Deszcz, R., Głogowska, M., Petrović-Torgašev, M., Verstraelen, L.: On the Roter type of Chen ideal submanifolds. Results Math. 59, 401–413 (2011)MathSciNet CrossRef MATH
  33.Deszcz, R., Głogowska, M., Plaue, M., Sawicz, K., Scherfner, M.: On hypersurfaces in space forms satisfying particular curvature conditions of Tachibana type. Kragujev. J. Math. 35, 223–247 (2011)MATH
  34.Deszcz, R., Grycak, W.: On some class of warped product manifolds. Bull. Inst. Math. Acad. Sinica 15, 311–322 (1987)MathSciNet MATH
  35.Deszcz, R., Haesen, S., Verstraelen, L.: On natural symmetries, Chap. 6. In: Topics in differential geometry, pp. 249–308. Romanian Academy of Science, Bucharest (2008)
  36.Deszcz, R., Hotloś, M.: On a certain extension of class of semi-symmetric manifolds. Publ. Inst. Math. (Beograd) (N.S.) 63(77), 115–130 (1998)MathSciNet
  37.Deszcz, R., Hotloś, M.: On certain subclass of pseudo-symmetric manifolds. Publ. Math. Debr. 53, 29–48 (1998)MATH
  38.Deszcz, R., Hotloś, M.: On some pseudo-symmetry type curvature condition. Tsukuba J. Math. 27, 13–30 (2003)MathSciNet MATH
  39.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)MATH
  40.Deszcz, R., Hotloś, M., Ṣentürk, Z.: On a certain application of Patterson’s curvature identity. Publ. Math. Debr. 58, 93–107 (2001)MATH
  41.Deszcz, R., Hotloś, M., Ṣentürk, Z.: On some family of generalized Einstein metric conditions. Demonstr. Math. 34, 943–954 (2001)MATH
  42.Deszcz, R., Hotloś, M., Ṣentürk, Z.: On curvature properties of certain quasi-Einstein hypersurfaces. Int. J. Math. 23, 1250073 (17 pp) (2012)CrossRef
  43.Deszcz, R., Petrović-Torgašev, M., Verstraelen, L., Zafindratafa, G.: On the intrinsic symmetries of Chen ideal submanifolds. Bull. Transilvania Univ. Braşov 1(50), series III 99–109 (2008)
  44.Deszcz, R., Verstraelen, L.: Hypersurfaces of semi-Riemannian conformally flat manifolds. Geometry and Topology of Submanifolds, V, pp. 109–131. World Scientific, River Edge, NJ (1993)
  45.Deszcz, R., Verstraelen, L., Vrancken, L.: The symmetry of warped product space-times. Gen. Relat. Gravit. 23, 671–681 (1991)MathSciNet CrossRef MATH
  46.Deszcz, R., Verstraelen, L., Yaprak, S.: Pseudo-symmetric hypersurfaces in 4-dimensional spaces of constant curvature. Bull. Inst. Math. Acad. Sinica 22, 167–179 (1994)MathSciNet MATH
  47.Deszcz, R., Yaprak, S.: Curvature properties of certain pseudosymmetric manifolds. Publ. Math. Debr. 45, 333–345 (1994)MathSciNet MATH
  48.Dillen, F., Petrović-Torgašev, M., Verstraelen, L.: Einstein, conformally flat and semi-symmetric sumbmanifolds satisfying Chen’s equality. Israel J. Math. 100, 163–169 (1997)MathSciNet CrossRef MATH
  49.Głogowska, M.: Semi-Riemannian manifolds whose Weyl tensor is a Kulkarni-Nomizu square. Publ. Inst. Math. (Beograd) (N.S.) 72(86), 95–106 (2002)MathSciNet CrossRef
  50.Głogowska, M.: On a curvature characterization of Ricci-pseudosymmetric hypersurfaces. Acta Math. Sci. 24 B, 361–375 (2004)
  51.Głogowska, M.: On quasi-Einstein Cartan type hypersurfaces. J. Geom. Phys. 58, 599–614 (2008)MathSciNet CrossRef MATH
  52.Grycak, W.: Riemannian manifolds with a symmetry condition imposed on the 2-end derivative of the conformal curvature tensor. Tensor (N.S.) 46, 287–290 (1987)MATH
  53.Haesen, S., Verstraelen, L.: Classification of the pseudo-symmetric space-times. J. Math. Phys. 45, 2343–2346 (2004)MathSciNet CrossRef MATH
  54.Haesen, S., Verstraelen, L.: Properties of a scalar curvature invariant depending on two planes. Manuscr. Math. 122, 59–72 (2007)MathSciNet CrossRef MATH
  55.Haesen, S., Verstraelen, L.: Natural intrinsic geometrical symmetries. SIGMA 5, 086, 15 pages (2009)MathSciNet
  56.Hinić, A.: Some curvature conditions of the type \(2 \times 4\) on the submanifolds satisfying Chen’s equality. Mat. Vesnik 59, 189–196 (2007)MathSciNet MATH
  57.Hotloś, M.: On certain curvature condition of pseudo-symmetry type (to appear)
  58.Jahanara, B., Haesen, S., Petrović-Torgašev, M., Verstraelen, L.: On the Weyl curvature of Deszcz. Publ. Math. Debr. 74, 417–431 (2009)MATH
  59.Jamali, M., Jun, J.-B., Shahid, M.H., Ahmad. M.: A remark on the axiom of spheres with lightlike submanifolds and application of Jacobi equation, Bull. Malays. Math. Sci. Soc. (2) 37, 565–574 (2014)
  60.Kowalczyk, D.: On some subclass of semi-symmetric manifolds. Soochow J. Math. 27, 445–461 (2001)MathSciNet MATH
  61.Leung, D.S., Nomizu, K.: The axiom of spheres in Riemannian Geometry. J. Differ. Geom. 5, 487–489 (1971)MathSciNet MATH
  62.Petrović-Torgašev, M., Hinić, A.: Some curvature conditions of the type \( 4 \times 2 \) on the submanifolds satisfying Chen’s equality. Math. Vesnik 59, 143–150 (2007)MATH
  63.Sawicz, K.: On some class of hypersurfaces with three distinct principal curvatures, Banach Center Publications 69. Inst. Math. Polish Acad. Sci. 145–156 (2005)
  64.Szabó, Z.I.: Structure theorems on Riemannian spaces satisfying \(R(X, Y)\cdot R = 0\) . I. The local version. J. Differ. Geom. 17, 531–582 (1982)MATH
  65.Taleshian, A., Hosseinzadeh, A.A.: Investigation of some conditions on \(N(k)\) -quasi Einstein manifolds. Bull. Malays. Math. Sci. Soc. (2) 34(3), 455–464 (2011)MathSciNet MATH
  66.Taṣtan, H.M.: The axiom of hemi-slant \(3\) -spheres in almost Hermitian geometry. Bull. Malays. Math. Sci. Soc. (2) 37, 555–564 (2014)
  67.van Lindt, D., Verstraelen, L.: Some axioms of Einsteinian and conformally fiat hypersurfaces. J. Differ. Geom. 16, 205–212 (1981)MATH
  68.van Lindt, D., Verstraelen, L.: A survey on axioms of submanifolds in Riemannian and Kaehlerian geometry. Colloq. Math. 54, 193–213 (1987)MathSciNet MATH
  69.Verstraelen, L.: Some comments on the \( \delta \) -curvatures of Bang-Yen Chen. Rend. Sem. Matem. Messina, 1998 International Congress in honour of R. Calapso, 331–337
  70.Yildiz, A., De, U.C., C̣etinkaya, A.C.: \(N(k)\) -quasi Einstein manifolds satisfying certain curvature conditions. Bull. Malays. Math. Sci. Soc. (2) 36, 1139–1149 (2013)
  71.Zafindratafa, G.K.: Sous-variétés soumises a des courbure, Doctoral Thesis. K.U, Leuven (1991)
  
• 作者单位：Ryszard Deszcz (1)
  Miroslava Petrović-Torgas̆ev (2)
  Leopold Verstraelen (3)
  Georges Zafindratafa (4)
  
  1. Department of Mathematics, Wrocław University of Environmental and Life Sciences, Grunwaldzka 53, 50-357, Wrocław, Poland
  2. Department of Mathematics, Faculty of Science, University of Kragujevac, Radoja Domanovića 12, Kragujevac, 34000, Serbia
  3. Departement Wiskunde, Fakulteit Wetenschappen, Katholieke Universiteit Leuven, Celestijnenlaan 200B, 3001, Heverlee, Belgium
  4. Laboratoire LAMAV - ISTV2, Université de Valenciennes et du Hainaut-Cambrésis, 59313, Valenciennes Cedex 9, France
  
• 刊物类别：Mathematics, general; Applications of Mathematics;
• 刊物主题：Mathematics, general; Applications of Mathematics;
• 出版者：Springer Singapore
• ISSN：2180-4206

文摘

In this paper, we study Chen ideal submanifolds \( M^n \) of dimension n in Euclidean spaces \( \mathbb {E}^{n+m} \) (\( n \ge 4 \), \( m \ge 1 \)) satisfying curvature conditions of pseudo-symmetry type of the form: the difference tensor \(R \cdot C - C \cdot R\) is expressed by some Tachibana tensors. Precisely, we consider one of the following three conditions: \( R \cdot C - C \cdot R \) is expressed as a linear combination of Q(g , R) and Q(S , R) , \( R \cdot C - C \cdot R \) is expressed as a linear combination of Q(g , C) and Q(S , C) and \( R \cdot C - C \cdot R \) is expressed as a linear combination of \( Q(g , g \wedge S) \) and \( Q(S , g \wedge S) \). We then characterize Chen ideal submanifolds \( M^n \) of dimension n in Euclidean spaces \( \mathbb {E}^{n+m} \) (\( n \ge 4\), \( m \ge 1 \)) which satisfy one of the following six conditions of pseudo-symmetry type: \( R \cdot C - C \cdot R \) and Q(g , R) are linearly dependent, \( R \cdot C - C \cdot R \) and Q(S , R) are linearly dependent, \( R \cdot C - C \cdot R \) and Q(g , C) are linearly dependent, \( R \cdot C - C \cdot R \) and Q(S , C) are linearly dependent, \( R \cdot C - C \cdot R \) and \( Q(g , g \wedge S) \) are linearly dependent and \( R \cdot C - C \cdot R \) and \( Q(S , g \wedge S) \) are linearly dependent. We also prove that the tensors \(R \cdot R - Q(S,R)\) and Q(g, C) are linearly dependent at every point of \( M^n \) at which its Weyl tensor C is non-zero. Keywords Submanifold Condition of pseudo-symmetry type Generalized Einstein metric condition Chen ideal submanifold Roter space Tachibana tensor