Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

RICCI 𝜌-SOLITONS ON 3-DIMENSIONAL 𝜂-EINSTEIN ALMOST KENMOTSU MANIFOLDS

  • Received : 2019.03.04
  • Accepted : 2019.07.05
  • Published : 2020.04.30
Download PDF
⟨ Previous Next ⟩

Abstract

The notion of quasi-Einstein metric in theoretical physics and in relation with string theory is equivalent to the notion of Ricci soliton in differential geometry. Quasi-Einstein metrics or Ricci solitons serve also as solution to Ricci flow equation, which is an evolution equation for Riemannian metrics on a Riemannian manifold. Quasi-Einstein metrics are subject of great interest in both mathematics and theoretical physics. In this paper the notion of Ricci 𝜌-soliton as a generalization of Ricci soliton is defined. We are motivated by the Ricci-Bourguignon flow to define this concept. We show that if a 3-dimensional almost Kenmotsu Einstein manifold M is a 𝜌-soliton, then M is a Kenmotsu manifold of constant sectional curvature -1 and the 𝜌-soliton is expanding with λ = 2.

Keywords

References

  1. P. Alegre and A. Carriazo, Generalized Sasakian space forms and conformal changes of the metric, Results Math. 59 (2011), no. 3-4, 485-493. https://doi.org/10.1007/s00025-011-0115-z
  2. M. D. Atkinson, B. E. Sagan, and V. Vatter, Counting (3 + 1)-avoiding permutations, European J. Combin. 33 (2012), no. 1, 49-61. https://doi.org/10.1016/j.ejc.2011.06.006
  3. P. Baird and L. Danielo, Three-dimensional Ricci solitons which project to surfaces, J. Reine Angew. Math. 608 (2007), 65-91.
  4. D. E. Blair, Riemannian geometry of contact and symplectic manifolds, second edition, Progress in Mathematics, 203, Birkhauser Boston, Inc., Boston, MA, 2010. https://doi.org/10.1007/978-0-8176-4959-3
  5. J.-P. Bourguignon, Ricci curvature and Einstein metrics, in Global differential geometry and global analysis (Berlin, 1979), 42-63, Lecture Notes in Math., 838, Springer, Berlin, 1981.
  6. G. Catino, L. Cremaschi, Z. Djadli, C. Mantegazza, and L. Mazzieri, The Ricci-Bourguignon flow, Pacific J. Math. 287 (2017), no. 2, 337-370. https://doi.org/10.2140/pjm.2017.287.337
  7. J. T. Cho, Almost contact 3-manifolds and Ricci solitons, Int. J. Geom. Methods Mod. Phys. 10 (2013), no. 1, 1220022, 7 pp. https://doi.org/10.1142/S0219887812200228
  8. U. C. De, M. Turan, A. Yildiz, and A. De, Ricci solitons and gradient Ricci solitons on 3-dimensional normal almost contact metric manifolds, Publ. Math. Debrecen 80 (2012), no. 1-2, 127-142.
  9. G. Dileo and A. M. Pastore, Almost Kenmotsu manifolds and local symmetry, Bull. Belg. Math. Soc. Simon Stevin 14 (2007), no. 2, 343-354. http://projecteuclid.org/euclid.bbms/1179839227
  10. A. Ghosh, Kenmotsu 3-metric as a Ricci soliton, Chaos Solitons Fractals 44 (2011), no. 8, 647-650. https://doi.org/10.1016/j.chaos.2011.05.015
  11. K. Kenmotsu, A class of almost contact Riemannian manifolds, Tohoku Math. J. (2) 24 (1972), 93-103. https://doi.org/10.2748/tmj/1178241594
  12. A. M. Pastore and V. Saltarelli, Generalized nullity distributions on almost Kenmotsu manifolds, Int. Electron. J. Geom. 4 (2011), no. 2, 168-183.
  13. V. Saltarelli, Three-dimensional almost Kenmotsu manifolds satisfying certain nullity conditions, Bull. Malays. Math. Sci. Soc. 38 (2015), no. 2, 437-459. https://doi.org/10.1007/s40840-014-0029-5
  14. M. Turan, U. C. De, and A. Yildiz, Ricci solitons and gradient Ricci solitons in three-dimensional trans-Sasakian manifolds, Filomat 26 (2012), no. 2, 363-370. https://doi.org/10.2298/FIL1202363T
  15. Y. Wang and X. Liu, Ricci solitons on three-dimensional ${\eta}$-Einstein almost Kenmotsu manifolds, Taiwanese J. Math. 19 (2015), no. 1, 91-100. https://doi.org/10.11650/tjm.19.2015.4094
  16. K. Yano, Integral formulas in Riemannian geometry, Pure and Applied Mathematics, No. 1, Marcel Dekker, Inc., New York, 1970.