×

On some classes of \(N(k)\)-quasi Einstein manifolds. (English) Zbl 1325.53064

Summary: The object of the present paper is to study \(N(k)\)-quasi Einstein manifolds. We study an \(N(k)\)-quasi Einstein manifold satisfying the conditions \(S\cdot R=0\), \(R\cdot C=f\hat{Q}(g,C)\). Next we prove that the curvature condition \(C\cdot S=0\) holds in an \(N(k)\)-quasi Einstein manifold. Then we study an \(N(k)\)-quasi Einstein manifold satisfying the condition \(R\cdot R=f\hat{Q}(S,R)\). Finally, we construct some examples of \(N(k)\)-quasi Einstein manifolds.

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C35 Differential geometry of symmetric spaces
53D10 Contact manifolds (general theory)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Besse AL (1987) Einstein manifolds, Ergeb. Math. Grenzgeb., 3. Folge, Bd. 10. Springer, Berlin, Heidelberg, New York
[2] Tamassy L, Binh TQ (1993) On weakly symmetries of Einstein and Sasakin manifolds. Tensor N S 53:140–148
[3] Chaki MC, Maity RK (2000) On quasi Einstein manifolds. Publ Math Debrecen 57:297–306 · Zbl 0968.53030
[4] El Naschie MS (2006) Is Einstein’s general field equation more fundamental than quantum field theory and particle physics?. Chaos Solitons Fractals 30:525–531 · doi:10.1016/j.chaos.2005.04.123
[5] Einstein A (2002) Grundzuge der relativitats theory. Springer, Berlin
[6] El Naschie MS (2005) Gödel universe, dualities and high energy particle in E-infinity. Chaos Solitons Fractals 25:759–764 · Zbl 1073.83531 · doi:10.1016/j.chaos.2004.12.010
[7] Deszcz R, Hotlos M, Senturk Z (2001) On curvature properties of quasi-Einstein hypersurfaces in semi-Euclidean spaces. Soochow J Math 27:375–389
[8] De UC, Ghosh GC (2004) On quasi Einstein and special quasi Einstein manifolds, Proc. of the Int. Conf. of Mathematics and its applications, Kuwait University, April 5–7, 178–191
[9] Chaki MC (2001) On generalized quasi-Einstein manifolds. Publ Math Debrecen 58:683–691 · Zbl 1062.53035
[10] Guha SR (2003) On quasi-Einstein and generalized quasi-Einstein manifolds. Facta Universitatis 3:821–842 · Zbl 1056.53033
[11] De UC, Ghosh GC (2004) On quasi-Einstein manifolds. Period Math Hungar 48:223–231 · Zbl 1059.53030 · doi:10.1023/B:MAHU.0000038977.94711.ab
[12] De UC, Ghosh GC (2005) On conformally flat special quasi-Einstein manifolds. Publ Math Debrecen 66:129–136 · Zbl 1075.53039
[13] Özgür C (2009) On some classes of super quasi-Einstein manifolds. Chaos Solitons Fractals 40:1156–1161 · Zbl 1197.53059 · doi:10.1016/j.chaos.2007.08.070
[14] Özgür C (2006) On a class of generalized quasi-Einstein manifolds. Applied Sciences, Balkan Society of Geometers, Geometry Balkan Press, Bucharest, 8:138–141 · Zbl 1103.53023
[15] Tanno S (1988) Ricci curvatures of contact Riemannian manifolds. Tohoku Math J 40:441–448 · Zbl 0655.53035 · doi:10.2748/tmj/1178227985
[16] Özgür C (2008) N(k)-quasi Einstein manifolds satisfying certain conditions. Chaos Solitons Fractals 38:1373–1377 · Zbl 1154.53311 · doi:10.1016/j.chaos.2008.03.016
[17] Taleshian A, Hosseeinzadeh AA (2011) Investigation of some conditions on N(k)-quasi Einstein manifolds. Bull Malays Math Sci Soc 34:455–464 · Zbl 1232.53041
[18] Hosseeinzadeh AA, Taleshian A (2012) On conformal and quasi-conformal curvature tensors of an N(k)-quasi Einstein manifold. Commun Korean Math Soc 27:317–326 · Zbl 1252.53057 · doi:10.4134/CKMS.2012.27.2.317
[19] Nagaraja HG (2010) On N(k)-mixed quasi-Einstein manifolds. Eur J Pure Appl Math 3:16–25 · Zbl 1213.53043
[20] De A, De UC, Gazi AK (2011) On a class of N(k)-quasi Einstein manifolds. Commun Korean Math Soc 26:623–634 · Zbl 1227.53057 · doi:10.4134/CKMS.2011.26.4.623
[21] Singh RN, Pandey MK, Gautam D (2010) On N(k)-quasi-Einstein manifolds. Novi Sad J Math 40:23–28 · Zbl 1299.53111
[22] Yano K, Kon M (1984) Structures on manifolds, series in pure mathematics. World Scientific Publishing Co., Singapore · Zbl 0557.53001
[23] Deszcz R (1992) On pseudosymmetric spaces. Bull Soc Belg Math Ser A 44:1–34 · Zbl 0808.53012
[24] Okumura M (1962) Some remarks on space with a certain contact structure. Tohoku Math J 14:135–145 · Zbl 0119.37701 · doi:10.2748/tmj/1178244168
[25] Patterson EM (1952) Some theorems on Ricci-recurrent spaces. J Lond Math Soc 27:287–295 · Zbl 0048.15604 · doi:10.1112/jlms/s1-27.3.287
[26] Mantica CA, Suh YJ (2011) Conformally symmetric manifolds and quasi conformally recurrent riemann manifolds. Balkan J Geom Appl 16:66–77 · Zbl 1226.53007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.