Abstract
The Eisenhart problem of finding parallel and symmetric tensors is considered in the framework ofN (k)-quasi Einstein manifolds and the result is connected with Ricci solitons. If the generator of the manifold provides a Ricci soliton then this is: i) shrinking on a class of conformally flat perfect fluid space-times and on quasi-umbilical hypersurfaces, in particular unit spheres; ii) expanding if the generator is of torse-forming type.
Similar content being viewed by others
References
H.-D. Cao, Recent progress on Ricci solitons, in Recent advances in geometric analysis, 1–38, Adv. Lect. Math. (ALM), 11, Int. Press, Somerville, MA, 2010, MR2648937 (2011d:53061).
C. Čalin and M. Crasmareanu, From the Eisenhart problem to Ricci solitons in f-Kenmotsu manifolds, Bull. Malays. Math. Sci. Soc., 33(3) (2010), 361–368, MR2732157 (2011k:53113).
B.-y. Chen and K. Yano, Hypersurfaces of a conformally flat space, Tensor, 26 (1972), 318–322, MR0331283 (48 #9617).
B. Chow, P. Lu and L. Ni, Hamilton’s Ricci flow, Graduate Studies in Mathematics, 77, American Mathematical Society, Providence, RI; Science Press, New York (2006), MR2274812 (2008a:53068).
S. Dragomir and Renata Grimaldi, On the topology of Riemann spaces of quasiconstant curvature, Publ. Inst. Math. (Beograd), 46(60) (1989), 183–187, MR1060072 (91f:53071).
L. P. Eisenhart, Symmetric tensors of the second order whose first covariant derivatives are zero, Trans. Amer. Math. Soc., 25(2) (1923), 297–306, MR1501245.
H. Levy, Symmetric tensors of the second order whose covariant derivatives vanish, Ann. of Math., (2) 27(2) (1925), 91–98, MR1502714.
C. Oniciuc, Nonlinear connections on tangent bundle and harmonicity, Ital. J. Pure Appl. Math., 6 (1999), 109–122 (2000), MR1758536 (2001e:53026).
C. Özgür, N (k)-quasi Einstein manifolds satisfying certain conditions, Chaos Solitons Fractals, 38(5) (2008), 1373–1377, MR2456527 (2009k:53099).
C. Özgür and S. Sular, On N (k)-quasi Einstein manifolds satisfying certain conditions, Balkan J. Geom. Appl., 13(2) (2008), 74–79, MR2395278 (2009b:53067).
C. Özgür and M. M. Tripathi, On the concircular curvature tensor of an N (k)-quasi Einstein manifold, Math. Pannon., 18(1) (2007), 95–100, MR2321959 (2008d:53055).
L. Rachünek and J Mikeš, On tensor fields semiconjugated with torse-forming vector fields, Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math., 44 (2005), 151–160, MR2218574 (2007b:53038).
A. A. Shaikh, D. W. Yoon and S. K. Hui, On quasi-Einstein spacetimes, Tsukuba J. Math., 33(2) (2009), 305–326, MR2605858 (2011b:53108).
R. Sharma, Second order parallel tensor in real and complex space forms, Internat. J. Math. Math. Sci., 12(4) (1989), 787–790, MR1024982 (91f:53035).
R. Sharma, Second order parallel tensors on contact manifolds. I, Algebras Groups Geom., 7(2) (1990), 145–152, MR1109567 (92b:53041).
R. Sharma, Second order parallel tensors on contact manifolds. II, C. R. Math. Rep. Acad. Sci. Canada, 13(6) (1991), 259–264, MR1145119 (93b:53026).
R. Sharma, On the curvature of contact metric manifolds, J. Geom., 53(1–2) (1995), 179–190, MR1337435 (96d:53031).
R. N. Singh, M. K. Pandey and D. Gautam, On N (k)-quasi Einstein manifold, Novi Sad J. Math., 40(2) (2010), 23–28, MR2827654.
A. Taleshian and A. A. Hosseinzadeh, Investigation of Some Conditions on N (k)-Quasi Einstein Manifolds, Bull. Malays. Math. Sci. Soc., 34(3) (2011), 455–464.
M. M. Tripathi and J.-S. Kim, On N (k)-quasi Einstein manifolds, Commun. Korean Math. Soc., 22(3) (2007), 411–417, MR2340568 (2008e:53073).
Y.-c. Wong, Existence of linear connections with respect to which given tensor fields are parallel or recurrent, Nagoya Math. J., 24 (1964), 67–108, MR0174015 (30 #4222).
H. Wu, Holonomy groups of indefinite metrics, Pacific J. Math., 20 (1967), 351–392, MR0212740 (35 #3606).
G. Zhao, Symmetric covariant tensor fields of order 2 on pseudo-Riemannian manifolds, Viena Preprint ESI 479 (1997). Available at http://www.esi.ac.at/preprints/esi479.ps.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Crasmareanu, M. Parallel tensors and Ricci solitons in N (k)-quasi Einstein manifolds. Indian J Pure Appl Math 43, 359–369 (2012). https://doi.org/10.1007/s13226-012-0022-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13226-012-0022-3