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Year 2020, Volume: 8 Issue: 2, 79 - 85, 15.10.2020

Krishnendu De

https://doi.org/10.36753/mathenot.727083

Abstract

References

  • [1] Blaga, A.M.: $\eta$-Ricci solitons on Lorentzian para-Sasakian manifolds, Filomat 30 , 489–496(2016).
  • [2] Blaga, A.M.: $\eta$-Ricci solitons on para-Kenmotsu manifolds, Balkan J. Geom. Appl. 20 , 1–13(2015).
  • [3] Blair, D.E.: Contact manifold in Riemannian geometry. Lecture Notes on Mathematics, Springer, Berlin, 509,(1976).
  • [4] Blair, D.E.: Riemannian geometry on contact and symplectic manifolds, Progr. Math., 203, Birkhäuser, (2010).
  • [5] Blair, D.E.: Koufogiorgos, T., Papantoniou, B.J., Contact metric manifolds satisfying a nullity condition, Israel. J. Math. 91, 189–214(1995).
  • [6] Dai, X., Zhao,Y., De, U.C.: $\eta$-Ricci soliton on $(k; \mu)'$-almost Kenmotsu manifolds, Open Math. 17 , 874-882(2019).
  • [7] Dileo, G., Pastore, A.M.: Almost Kenmotsu manifolds and nullity distributions, J. Geom. 93, 46–61(2009).
  • [8] Duggal, K. L.: Almost Ricci Solitons and Physical Applications, Int. El. J. Geom. 2 , 1–10(2017).
  • [9] Ghosh, A., Patra, D.S.: $\eta$-Ricci Soliton within the framework of Sasakian and (k; )-contact manifold, Int. J. Geom. Methods Mod. Phys. 15 (7) 1850120 (2018).
  • [10] Gray, A.: Spaces of constancy of curvature operators, Proc. Amer. Math. Soc., 17, 897–902(1966).
  • [11] Hamada, T.: Real Hypersurfaces of Complex Space Forms in Terms of Ricci $\eta$-Tensor, Tokyo J. Math. 25 , 473– 483(2002).
  • [12] Hamilton, R. S.: The Ricci flow on surfaces, Mathematics and general relativity (Santa Cruz, CA, 1986), 237–262, Contemp. Math. 71, American Math. Soc., (1988).
  • [13] Kaimakamis, G., Panagiotidou, K.: $\eta$-Ricci solitons of real hypersurfaces in non-flat complex space forms, J. Geom. and Phys. 86 , 408–413(2014).
  • [14] Majhi, P., De, U. C., Suh, Y. J.: $\eta$-Ricci solitons and Sasakian 3-manifolds, Publ. Math. Debrecen 93 , 241–252(2018).
  • [15] Petersen, P., Wylie, W.: Rigidity of gradient Ricci solitons, Pacific J. Math. 241, 329–345(2009).
  • [16] Petersen, P., Wylie,W.: On gradient Ricci solitons with symmetry, Proc. Amer. Math. Soc. 137, 2085–2092(2009).
  • [17] Pigola, S., Rigoli, M. Rimoldi,M., Setti, A.: Ricci almost solitons, Ann. Sc. Norm. Super. Pisa Cl. Sci. 10, 757–799(2011).
  • [18] Prakasha, D.G., Veeresha, P.: Para-Sasakian manifolds and $\eta$-Ricci solitons, arXiv:1801.01727v1.
  • [19] Tachibana, S.: On almost-analytic vectors in almost-Kählerian manifolds, Tohoku Math. J. 11 , 247–265(1959).
  • [20] Tanno, S.: Some differential equations on Riemannian manifolds, J. Math. Soc. Japan, 30, 509–531(1978).

A Note on Gradient $\ast$-Ricci Solitons

Year 2020, Volume: 8 Issue: 2, 79 - 85, 15.10.2020

Krishnendu De

https://doi.org/10.36753/mathenot.727083

Abstract

In the offering exposition we characterize $(k,\mu)'$- almost Kenmotsu $3$-manifolds admitting gradient $\ast$-Ricci soliton. It is shown that a $(k,\mu)'$- almost Kenmotsu manifold with $k<-1$ is admitting a gradient $\ast$-Ricci soliton, either the soliton is steady or the manifold is locally isometric to a rigid gradient Ricci soliton $\mathbb{H}^{2}(-4)\times \mathbb{R}$.                                                                                                                                 .                                                                      

Keywords

$(k \mu)'$- almost Kenmotsu manifolds $\ast$-Ricci solitons gradient $\ast$-Ricci solitons

References

  • [1] Blaga, A.M.: $\eta$-Ricci solitons on Lorentzian para-Sasakian manifolds, Filomat 30 , 489–496(2016).
  • [2] Blaga, A.M.: $\eta$-Ricci solitons on para-Kenmotsu manifolds, Balkan J. Geom. Appl. 20 , 1–13(2015).
  • [3] Blair, D.E.: Contact manifold in Riemannian geometry. Lecture Notes on Mathematics, Springer, Berlin, 509,(1976).
  • [4] Blair, D.E.: Riemannian geometry on contact and symplectic manifolds, Progr. Math., 203, Birkhäuser, (2010).
  • [5] Blair, D.E.: Koufogiorgos, T., Papantoniou, B.J., Contact metric manifolds satisfying a nullity condition, Israel. J. Math. 91, 189–214(1995).
  • [6] Dai, X., Zhao,Y., De, U.C.: $\eta$-Ricci soliton on $(k; \mu)'$-almost Kenmotsu manifolds, Open Math. 17 , 874-882(2019).
  • [7] Dileo, G., Pastore, A.M.: Almost Kenmotsu manifolds and nullity distributions, J. Geom. 93, 46–61(2009).
  • [8] Duggal, K. L.: Almost Ricci Solitons and Physical Applications, Int. El. J. Geom. 2 , 1–10(2017).
  • [9] Ghosh, A., Patra, D.S.: $\eta$-Ricci Soliton within the framework of Sasakian and (k; )-contact manifold, Int. J. Geom. Methods Mod. Phys. 15 (7) 1850120 (2018).
  • [10] Gray, A.: Spaces of constancy of curvature operators, Proc. Amer. Math. Soc., 17, 897–902(1966).
  • [11] Hamada, T.: Real Hypersurfaces of Complex Space Forms in Terms of Ricci $\eta$-Tensor, Tokyo J. Math. 25 , 473– 483(2002).
  • [12] Hamilton, R. S.: The Ricci flow on surfaces, Mathematics and general relativity (Santa Cruz, CA, 1986), 237–262, Contemp. Math. 71, American Math. Soc., (1988).
  • [13] Kaimakamis, G., Panagiotidou, K.: $\eta$-Ricci solitons of real hypersurfaces in non-flat complex space forms, J. Geom. and Phys. 86 , 408–413(2014).
  • [14] Majhi, P., De, U. C., Suh, Y. J.: $\eta$-Ricci solitons and Sasakian 3-manifolds, Publ. Math. Debrecen 93 , 241–252(2018).
  • [15] Petersen, P., Wylie, W.: Rigidity of gradient Ricci solitons, Pacific J. Math. 241, 329–345(2009).
  • [16] Petersen, P., Wylie,W.: On gradient Ricci solitons with symmetry, Proc. Amer. Math. Soc. 137, 2085–2092(2009).
  • [17] Pigola, S., Rigoli, M. Rimoldi,M., Setti, A.: Ricci almost solitons, Ann. Sc. Norm. Super. Pisa Cl. Sci. 10, 757–799(2011).
  • [18] Prakasha, D.G., Veeresha, P.: Para-Sasakian manifolds and $\eta$-Ricci solitons, arXiv:1801.01727v1.
  • [19] Tachibana, S.: On almost-analytic vectors in almost-Kählerian manifolds, Tohoku Math. J. 11 , 247–265(1959).
  • [20] Tanno, S.: Some differential equations on Riemannian manifolds, J. Math. Soc. Japan, 30, 509–531(1978).
There are 20 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Krishnendu De 0000-0001-5264-5861

Publication Date October 15, 2020
Submission Date April 26, 2020
Acceptance Date October 10, 2020
Published in Issue Year 2020 Volume: 8 Issue: 2

Cite

APA De, K. (2020). A Note on Gradient $\ast$-Ricci Solitons. Mathematical Sciences and Applications E-Notes, 8(2), 79-85. https://doi.org/10.36753/mathenot.727083
AMA De K. A Note on Gradient $\ast$-Ricci Solitons. Math. Sci. Appl. E-Notes. October 2020;8(2):79-85. doi:10.36753/mathenot.727083
Chicago De, Krishnendu. “A Note on Gradient $\ast$-Ricci Solitons”. Mathematical Sciences and Applications E-Notes 8, no. 2 (October 2020): 79-85. https://doi.org/10.36753/mathenot.727083.
EndNote De K (October 1, 2020) A Note on Gradient $\ast$-Ricci Solitons. Mathematical Sciences and Applications E-Notes 8 2 79–85.
IEEE K. De, “A Note on Gradient $\ast$-Ricci Solitons”, Math. Sci. Appl. E-Notes, vol. 8, no. 2, pp. 79–85, 2020, doi: 10.36753/mathenot.727083.
ISNAD De, Krishnendu. “A Note on Gradient $\ast$-Ricci Solitons”. Mathematical Sciences and Applications E-Notes 8/2 (October 2020), 79-85. https://doi.org/10.36753/mathenot.727083.
JAMA De K. A Note on Gradient $\ast$-Ricci Solitons. Math. Sci. Appl. E-Notes. 2020;8:79–85.
MLA De, Krishnendu. “A Note on Gradient $\ast$-Ricci Solitons”. Mathematical Sciences and Applications E-Notes, vol. 8, no. 2, 2020, pp. 79-85, doi:10.36753/mathenot.727083.
Vancouver De K. A Note on Gradient $\ast$-Ricci Solitons. Math. Sci. Appl. E-Notes. 2020;8(2):79-85.
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