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Generalized Sasakian-space-forms and Ricci almost solitons with a conformal killing vector field

Year 2016, Volume: 4 Issue: 3, 263 - 269, 30.09.2016

Abstract



The present paper deals with the study of generalized
Sasakian-space-forms whose metric is Ricci almost soliton with a conformal
killing vector field. We obtain sufficient conditions of such type of Ricci
almost solitons to be expanding, steady and shrinking respectively.




References

  • Alegre, P., Blair, D. E. and Carriazo, A., Generalized Sasakian-space-forms, Israel J. Math., 14 (2004), 157–183.
  • Alegre, P. and Carriazo, A., Submanifolds of generalized Sasakian-space-forms, Taiwanese J. Math., 13 (2009), 923–941.
  • Alegre, P. and Carriazo, A., Structures on generalized Sasakian-space-forms, Diff. Geo. and its Application, 26 (2008), 656–666.
  • Alegre, P. and Carriazo, A., Generalized Sasakian-space-forms and conformal changes of the metric, Results in Math., 59 (2011), 485–493.
  • Ashoka, S. R., Bagewadi, C. S. and Ingalahalli, G., Certain results on Ricci Solitons in α-Sasakian manifolds, Hindawi Publ. Corporation, Geometry, Vol.(2013), Article ID 573925, 4 Pages.
  • Ashoka, S. R., Bagewadi, C. S. and Ingalahalli, G., A geometry on Ricci solitons in (LCS)_n-manifolds, Diff. Geom.-Dynamical Systems, 16 (2014), 50–62.
  • Bagewadi, C. S. and Ingalahalli,G., Ricci solitons in Lorentzian-Sasakian manifolds, Acta Math. Acad. Paeda. Nyire., 28 (2012), 59-68.
  • Bejan, C. L. and Crasmareanu, M., Ricci Solitons in manifolds with quasi-contact curvature, Publ. Math. Debrecen, 78/1 (2011), 235-243.
  • Belkhelfa, M., Deszcz, R. and Verstraelen, L., Symmetry properties of generalized Sasakian-space-forms, Soochow J. Math., 31 (2005), 611–616.
  • Blaga, A. M., η-Ricci solitons on para-kenmotsu manifolds, Balkan J. Geom. Appl., 20 (2015), 1–13.
  • Blair, D. E., Contact manifolds in Riemannian geometry, Lecture Notes in Math., 509, Springer-Verlag, 1976.
  • Carriazo, A., On generalized Sasakian-space-forms, Proceedings of the Ninth International Workshop on Diff. Geom., 9 (2005), 31–39.
  • Chandra, S., Hui, S. K. and Shaikh, A. A., Second order parallel tensors and ricci solitons on (LCS)_n-manifolds, Commun. Korean Math. Soc., 30 (2015), 123–130.
  • Chen, B. Y. and Deshmukh, S., Geometry of compact shrinking Ricci solitons, Balkan J. Geom. Appl., 19 (2014), 13–21.
  • Deshmukh, S., Al-Sodais, H. and Alodan, H., A note on Ricci solitons, Balkan J. Geom. Appl., 16 (2011), 48–55.
  • Ghefari, R. A., Al-Solamy, F. R. and Shahid, M. H., CR-submanifolds of generalized Sasakian-space-forms, JP J. Geom. and Topology, 6 (2006), 151–166.
  • Gherib, F. and Belkhelfa, M., Second order parallel tensors on generaized Sasakian-space-forms and semi parallel hypersurfaces in Sasakian-space-forms, Beitrage Zur Algebra and Geom., 51 (2010), 1–7.
  • Gherib, F., Gorine, M. and Belkhelfa, M., Parallel and semi symmetry of some tensors in generalized Sasakian-space-forms, Bull. Trans. Univ. Brasov, Series III: Mathematics, Informatics, Physics, 1(50) (2008), 139–148.
  • Hamilton, R. S., Three-manifolds with positive Ricci curvature, J. Diff. Geom., 17 (1982), 255–306.
  • Hamilton, R. S., The Ricci flow on surfaces, Mathematics and general relativity, Contemp. Math., 71, American Math. Soc., 1988, 237–262.
  • He, C. and Zhu, M., Ricci solitons on Sasakian manifolds, arxiv: 1109.4407V2, [Math DG], (2011).
  • Hui, S. K. and Chakraborty, D., Some types of Ricci solitons on (LCS)n-manifolds, J. Math. Sci. Advances and Applications, 37 (2016), 1–17.
  • Hui, S. K., Lemence, R. S. and Chakraborty, D., Ricci solitons on three dimensional generalized Sasakian-space-forms, Tensor Society, N. S., 76 (2015).
  • Hui, S. K. and Prakasha, D. G., On the C-Bochner curvature tensor of generalized Sasakian-space-forms, Proc. Natl. Acad. Sci., India, Sec. A, Phys. Sci., Springer, 85(3) (2015), 401–405.
  • Hui, S. K. and Sarkar, A., On the W_2-curvature tensor of generalized Sasakian-space-forms, Math. Pannonica, 23 (2012), 1-12.
  • Ingalahalli, G. and Bagewadi, C. S., Ricci solitons in α-Sasakian manifolds, ISRN Geometry, Vol.(2012), Article ID 421384, 13 Pages.
  • Kim, U. K., Conformally flat generalized Sasakian-space-forms and locally symmetric generalized Sasakian-space-forms, Note di Matematica, 26 (2006), 55–67.
  • Maralabhavi, Y. B. and Shivaprasanna, G. S., Second order parallel tensors on generalized Sasakian-space-forms, Int. J. Math. Eng. and Sci., 1 (2012), 11–21.
  • Nagaraja, H.G. and Premalatta, C.R., Ricci solitons in Kenmotsu manifolds, J. Math. Analysis, 3(2) (2012), 18–24.
  • Narain, D., Yadav, S. and Dwivedi, P. K., On generalized Sasakian-space-forms satisfying certain conditions, Int. J. Math. and Analysis, 3 (2011), 1–12.
  • Olteanu, A., Legendrian warped product submanifolds in generalized Sasakian-space-forms, Acta Mathematica Academiae Paedagogice Nyiregyhaziensis, 25 (2009), 137–144.
  • Patterson, E. M., Some theorems on Ricci-recurrent spaces, J. London Math. Soc., 27 (1952), 287–295.
  • Perelman, G., The entropy formula for the Ricci flow and its geometric applications, http://arXiv.org/abs/math/0211159, 2002, 1–39.
  • Perelman, G., Ricci flow with surgery on three manifolds, http://arXiv.org/abs/math/0303109, 2003, 1–22.
  • Pigola, S., Rigoli, M., Rimoldi, M. and Setti, A. G., Ricci almost solitons, Ann. Sc. Norm. Super. Pisa Cl. Sci, 10 (2011), 757–799.
  • Pokhariyal, G. P. and Mishra, R. S., The curvature tensor and their relativistic significance, Yokohoma Math. J., 18 (1970), 105–108.
  • Shaikh, A. A. and Hui, S. K., Some global properties of pseudo cyclic Ricci symmetric manifolds, Applied Sciences, Balkan Soc. of Geom., 13 (2011), 97–101.
  • Shaikh, A. A. and Hui, S. K., On ϕ-symmetric generalized Sasakian-space-forms admitting semi-symmetric metric connection, Tensor, N. S., 74 (2013), 265–274.
  • Sharma, R., Certain results on k-contact and (k,μ)-contact manifolds, J. Geom., 89 (2008),138–147.
  • Shukla, S. S. and Chaubey, P. K., On invariant submanifolds in generalized Sasakian-space-forms, J. Dynamical Systems and Geometric Theories, 8 (2010), 173–188.
  • Szabo', Z. I., Structure theorems on Riemannian spaces satisfying R(X,Y)⋅R=0, The local version, J. Diff. Geom., 17 (1982), 531–582.
  • Tripathi, M. M., Ricci solitons in contact metric manifolds, arxiv: 0801,4221 V1, [Math DG], (2008).
  • Yadav, S., Suthar, D. L. and Srivastava, A. K., Some results on M(f_1,f_2,f_3 )_(2n+1)-manifolds, Int. J. Pure and Appl. Math., 70 (2011), 415–423.
Year 2016, Volume: 4 Issue: 3, 263 - 269, 30.09.2016

Abstract

References

  • Alegre, P., Blair, D. E. and Carriazo, A., Generalized Sasakian-space-forms, Israel J. Math., 14 (2004), 157–183.
  • Alegre, P. and Carriazo, A., Submanifolds of generalized Sasakian-space-forms, Taiwanese J. Math., 13 (2009), 923–941.
  • Alegre, P. and Carriazo, A., Structures on generalized Sasakian-space-forms, Diff. Geo. and its Application, 26 (2008), 656–666.
  • Alegre, P. and Carriazo, A., Generalized Sasakian-space-forms and conformal changes of the metric, Results in Math., 59 (2011), 485–493.
  • Ashoka, S. R., Bagewadi, C. S. and Ingalahalli, G., Certain results on Ricci Solitons in α-Sasakian manifolds, Hindawi Publ. Corporation, Geometry, Vol.(2013), Article ID 573925, 4 Pages.
  • Ashoka, S. R., Bagewadi, C. S. and Ingalahalli, G., A geometry on Ricci solitons in (LCS)_n-manifolds, Diff. Geom.-Dynamical Systems, 16 (2014), 50–62.
  • Bagewadi, C. S. and Ingalahalli,G., Ricci solitons in Lorentzian-Sasakian manifolds, Acta Math. Acad. Paeda. Nyire., 28 (2012), 59-68.
  • Bejan, C. L. and Crasmareanu, M., Ricci Solitons in manifolds with quasi-contact curvature, Publ. Math. Debrecen, 78/1 (2011), 235-243.
  • Belkhelfa, M., Deszcz, R. and Verstraelen, L., Symmetry properties of generalized Sasakian-space-forms, Soochow J. Math., 31 (2005), 611–616.
  • Blaga, A. M., η-Ricci solitons on para-kenmotsu manifolds, Balkan J. Geom. Appl., 20 (2015), 1–13.
  • Blair, D. E., Contact manifolds in Riemannian geometry, Lecture Notes in Math., 509, Springer-Verlag, 1976.
  • Carriazo, A., On generalized Sasakian-space-forms, Proceedings of the Ninth International Workshop on Diff. Geom., 9 (2005), 31–39.
  • Chandra, S., Hui, S. K. and Shaikh, A. A., Second order parallel tensors and ricci solitons on (LCS)_n-manifolds, Commun. Korean Math. Soc., 30 (2015), 123–130.
  • Chen, B. Y. and Deshmukh, S., Geometry of compact shrinking Ricci solitons, Balkan J. Geom. Appl., 19 (2014), 13–21.
  • Deshmukh, S., Al-Sodais, H. and Alodan, H., A note on Ricci solitons, Balkan J. Geom. Appl., 16 (2011), 48–55.
  • Ghefari, R. A., Al-Solamy, F. R. and Shahid, M. H., CR-submanifolds of generalized Sasakian-space-forms, JP J. Geom. and Topology, 6 (2006), 151–166.
  • Gherib, F. and Belkhelfa, M., Second order parallel tensors on generaized Sasakian-space-forms and semi parallel hypersurfaces in Sasakian-space-forms, Beitrage Zur Algebra and Geom., 51 (2010), 1–7.
  • Gherib, F., Gorine, M. and Belkhelfa, M., Parallel and semi symmetry of some tensors in generalized Sasakian-space-forms, Bull. Trans. Univ. Brasov, Series III: Mathematics, Informatics, Physics, 1(50) (2008), 139–148.
  • Hamilton, R. S., Three-manifolds with positive Ricci curvature, J. Diff. Geom., 17 (1982), 255–306.
  • Hamilton, R. S., The Ricci flow on surfaces, Mathematics and general relativity, Contemp. Math., 71, American Math. Soc., 1988, 237–262.
  • He, C. and Zhu, M., Ricci solitons on Sasakian manifolds, arxiv: 1109.4407V2, [Math DG], (2011).
  • Hui, S. K. and Chakraborty, D., Some types of Ricci solitons on (LCS)n-manifolds, J. Math. Sci. Advances and Applications, 37 (2016), 1–17.
  • Hui, S. K., Lemence, R. S. and Chakraborty, D., Ricci solitons on three dimensional generalized Sasakian-space-forms, Tensor Society, N. S., 76 (2015).
  • Hui, S. K. and Prakasha, D. G., On the C-Bochner curvature tensor of generalized Sasakian-space-forms, Proc. Natl. Acad. Sci., India, Sec. A, Phys. Sci., Springer, 85(3) (2015), 401–405.
  • Hui, S. K. and Sarkar, A., On the W_2-curvature tensor of generalized Sasakian-space-forms, Math. Pannonica, 23 (2012), 1-12.
  • Ingalahalli, G. and Bagewadi, C. S., Ricci solitons in α-Sasakian manifolds, ISRN Geometry, Vol.(2012), Article ID 421384, 13 Pages.
  • Kim, U. K., Conformally flat generalized Sasakian-space-forms and locally symmetric generalized Sasakian-space-forms, Note di Matematica, 26 (2006), 55–67.
  • Maralabhavi, Y. B. and Shivaprasanna, G. S., Second order parallel tensors on generalized Sasakian-space-forms, Int. J. Math. Eng. and Sci., 1 (2012), 11–21.
  • Nagaraja, H.G. and Premalatta, C.R., Ricci solitons in Kenmotsu manifolds, J. Math. Analysis, 3(2) (2012), 18–24.
  • Narain, D., Yadav, S. and Dwivedi, P. K., On generalized Sasakian-space-forms satisfying certain conditions, Int. J. Math. and Analysis, 3 (2011), 1–12.
  • Olteanu, A., Legendrian warped product submanifolds in generalized Sasakian-space-forms, Acta Mathematica Academiae Paedagogice Nyiregyhaziensis, 25 (2009), 137–144.
  • Patterson, E. M., Some theorems on Ricci-recurrent spaces, J. London Math. Soc., 27 (1952), 287–295.
  • Perelman, G., The entropy formula for the Ricci flow and its geometric applications, http://arXiv.org/abs/math/0211159, 2002, 1–39.
  • Perelman, G., Ricci flow with surgery on three manifolds, http://arXiv.org/abs/math/0303109, 2003, 1–22.
  • Pigola, S., Rigoli, M., Rimoldi, M. and Setti, A. G., Ricci almost solitons, Ann. Sc. Norm. Super. Pisa Cl. Sci, 10 (2011), 757–799.
  • Pokhariyal, G. P. and Mishra, R. S., The curvature tensor and their relativistic significance, Yokohoma Math. J., 18 (1970), 105–108.
  • Shaikh, A. A. and Hui, S. K., Some global properties of pseudo cyclic Ricci symmetric manifolds, Applied Sciences, Balkan Soc. of Geom., 13 (2011), 97–101.
  • Shaikh, A. A. and Hui, S. K., On ϕ-symmetric generalized Sasakian-space-forms admitting semi-symmetric metric connection, Tensor, N. S., 74 (2013), 265–274.
  • Sharma, R., Certain results on k-contact and (k,μ)-contact manifolds, J. Geom., 89 (2008),138–147.
  • Shukla, S. S. and Chaubey, P. K., On invariant submanifolds in generalized Sasakian-space-forms, J. Dynamical Systems and Geometric Theories, 8 (2010), 173–188.
  • Szabo', Z. I., Structure theorems on Riemannian spaces satisfying R(X,Y)⋅R=0, The local version, J. Diff. Geom., 17 (1982), 531–582.
  • Tripathi, M. M., Ricci solitons in contact metric manifolds, arxiv: 0801,4221 V1, [Math DG], (2008).
  • Yadav, S., Suthar, D. L. and Srivastava, A. K., Some results on M(f_1,f_2,f_3 )_(2n+1)-manifolds, Int. J. Pure and Appl. Math., 70 (2011), 415–423.
There are 43 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Shyamal Kumar Hui This is me

Debabrata Chakraborty This is me

Publication Date September 30, 2016
Published in Issue Year 2016 Volume: 4 Issue: 3

Cite

APA Kumar Hui, S., & Chakraborty, D. (2016). Generalized Sasakian-space-forms and Ricci almost solitons with a conformal killing vector field. New Trends in Mathematical Sciences, 4(3), 263-269.
AMA Kumar Hui S, Chakraborty D. Generalized Sasakian-space-forms and Ricci almost solitons with a conformal killing vector field. New Trends in Mathematical Sciences. September 2016;4(3):263-269.
Chicago Kumar Hui, Shyamal, and Debabrata Chakraborty. “Generalized Sasakian-Space-Forms and Ricci Almost Solitons With a Conformal Killing Vector Field”. New Trends in Mathematical Sciences 4, no. 3 (September 2016): 263-69.
EndNote Kumar Hui S, Chakraborty D (September 1, 2016) Generalized Sasakian-space-forms and Ricci almost solitons with a conformal killing vector field. New Trends in Mathematical Sciences 4 3 263–269.
IEEE S. Kumar Hui and D. Chakraborty, “Generalized Sasakian-space-forms and Ricci almost solitons with a conformal killing vector field”, New Trends in Mathematical Sciences, vol. 4, no. 3, pp. 263–269, 2016.
ISNAD Kumar Hui, Shyamal - Chakraborty, Debabrata. “Generalized Sasakian-Space-Forms and Ricci Almost Solitons With a Conformal Killing Vector Field”. New Trends in Mathematical Sciences 4/3 (September 2016), 263-269.
JAMA Kumar Hui S, Chakraborty D. Generalized Sasakian-space-forms and Ricci almost solitons with a conformal killing vector field. New Trends in Mathematical Sciences. 2016;4:263–269.
MLA Kumar Hui, Shyamal and Debabrata Chakraborty. “Generalized Sasakian-Space-Forms and Ricci Almost Solitons With a Conformal Killing Vector Field”. New Trends in Mathematical Sciences, vol. 4, no. 3, 2016, pp. 263-9.
Vancouver Kumar Hui S, Chakraborty D. Generalized Sasakian-space-forms and Ricci almost solitons with a conformal killing vector field. New Trends in Mathematical Sciences. 2016;4(3):263-9.