Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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From the Eisenhart Problem to Ricci Solitons in Quaternion Space Forms

  • Received : 2016.06.19
  • Accepted : 2018.04.25
  • Published : 2018.06.23
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Abstract

In this paper we obtain the condition for the existence of Ricci solitons in nonflat quaternion space form by using Eisenhart problem. Also it is proved that if (g, V, ${\lambda}$) is Ricci soliton then V is solenoidal if and only if it is shrinking, steady and expanding depending upon the sign of scalar curvature. Further it is shown that Ricci soliton in semi-symmetric quaternion space form depends on quaternion sectional curvature c if V is solenoidal.

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References

  1. T. Adachi and S. Maeda, Some characterizations of quaternionic space forms, Proc. Japan Acad. Ser. A Math. Sci., 76(2000), 168-172. https://doi.org/10.3792/pjaa.76.168
  2. C. S. Bagewadi and G. Ingalahalli, Ricci solitons in Lorentzian ${\alpha}$-Sasakian manifolds, Acta Math. Acad. Paedagog. Nyhazi., 28(1)(2012), 59-68.
  3. C. Calin and M. Crasmareanu, From the Eisenhart problem to Ricci solitons in f-Kenmotsu manifolds, Bull. Malays. Math. Sci. Soc. (2), 33(3)(2010), 361-368.
  4. S. Debnath and A. Bhattacharyya, Second order parallel tensor in Trans-Sasakian manifolds and connection with ricci soliton, Lobachevskii J. Math., 33(4)(2012), 312-316. https://doi.org/10.1134/S1995080212040075
  5. R. Deszcz, On pseudosymmetric spaces, Bull. Soc. Math. Belg. Ser. A, 44(1992), 1-34.
  6. L. P. Eisenhart, Symmetric tensors of the second order whose first covariant derivatives are zero, Trans. Amer. Math, Soc., 25(2)(1923), 297-306. https://doi.org/10.1090/S0002-9947-1923-1501245-6
  7. R. S. Hamilton, The Ricci flow on surfaces, Mathematics and general relativity (Santa Cruz, CA, 1986), Contemp. Math., 71, Amer. Math. Soc., (1988), 237-262.
  8. G. Ingalahalli and C. S. Bagewadi, Ricci solitons in ${\alpha}$-Sasakian manifolds, ISRN Geometry, (2012), Article ID 421384, 1-13.
  9. S. Ishihara, Quaternion Kahlerian manifolds, J. Differential Geometry, 9(1974), 483-500. https://doi.org/10.4310/jdg/1214432544
  10. H. Levy, Symmetric tensors of the second order whose covariant derivatives vanish, Ann. Math. (2), 27(2)(1925), 91-98. https://doi.org/10.2307/1967964
  11. R. Sharma, Second order parallel tensor in real and complex space forms, Internat. J. Math. Math. Sci., 12(4)(1989), 787-790. https://doi.org/10.1155/S0161171289000967
  12. Z. I. Szabo, Structure theorems on Riemannian spaces satisfying R(X; Y ) ${\cdot}$ R = 0, J. Differential Geom., 17(1982), 531-582. https://doi.org/10.4310/jdg/1214437486
  13. K. Yano and M. Kon, Structures on manifolds, Series In Pure Mathematics 3, World Scientific Publishing Co., Singapore, 1984.