DOI QR코드

DOI QR Code

GEOMETRY OF GENERALIZED BERGER-TYPE DEFORMED METRIC ON B-MANIFOLD

  • Received : 2023.03.03
  • Accepted : 2023.06.26
  • Published : 2023.10.31

Abstract

Let (M2m, 𝜑, g) be a B-manifold. In this paper, we introduce a new class of metric on (M2m, 𝜑, g), obtained by a non-conformal deformation of the metric g, called a generalized Berger-type deformed metric. First we investigate the Levi-Civita connection of this metric. Secondly we characterize the Riemannian curvature, the sectional curvature and the scalar curvature. Finally, we study the proper biharmonicity of the identity map and of a curve on M with respect to a generalized Berger-type deformed metric.

Keywords

Acknowledgement

The author expresses his gratitude to the referee for his valuable comments and suggestions towards the improvement of the paper. The author would also like to thank Prof. Ahmed Mohammed Cherif, University Mustapha Stambouli of Mascara for his helpful suggestions and valuable comments.

References

  1. M. Altunbas, R. Simsek, and A. Gezer, A study concerning Berger type deformed Sasaki metric on the tangent bundle, J. Math. Phys. Anal. Geom. 15 (2019), no. 4, 435-447. https://doi.org/10.15407/mag15.04.435 
  2. M. Altunbas, R. Simsek, and A. Gezer, Some harmonic problems on the tangent bundle with a Berger-type deformed Sasaki metric, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 82 (2020), no. 2, 37-42. 
  3. P. Baird, A. Fardoun, and S. Ouakkas, Conformal and semi-conformal biharmonic maps, Ann. Global Anal. Geom. 34 (2008), no. 4, 403-414. https://doi.org/10.1007/s10455-008-9118-8 
  4. P. Baird and D. Kamissoko, On constructing biharmonic maps and metrics, Ann. Global Anal. Geom. 23 (2003), no. 1, 65-75. https://doi.org/10.1023/A:1021213930520 
  5. P. Baird and J. C. Wood, Harmonic morphisms between Riemannian manifolds, London Mathematical Society Monographs. New Series, 29, The Clarendon Press, Oxford University Press, Oxford, 2003. https://doi.org/10.1093/acprof:oso/9780198503620.001.0001 
  6. A. Balmus, Biharmonic properties and conformal changes, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 50 (2004), no. 2, 361-372. 
  7. A. Benkartab and A. M. Cherif, New methods of construction for biharmonic maps, Kyungpook Math. J. 59 (2019), no. 1, 135-147. https://doi.org/10.5666/KMJ.2019. 59.1.135 
  8. A. Benkartab and A. M. Cherif, Deformations of metrics and biharmonic maps, Commun. Math. 28 (2020), no. 3, 263-275. https://doi.org/10.2478/cm-2020-0022 
  9. G. Chen, Y. Liu, and J. Wei, Nondegeneracy of harmonic maps from ℝ2 to 𝕊2, Discrete Contin. Dyn. Syst. 40 (2020), no. 6, 3215-3233. https://doi.org/10.3934/dcds.2019228 
  10. M. Crasmareanu, Killing potentials, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 45 (1999), no. 1, 169-176. 
  11. N. E. Djaa, F. Latti, and A. Zagane, Proper biharmonic maps on tangent bundle, Commun. Math. 31 (2023), no. 1, 137-154. https://doi.org/10.46298/cm.10305 
  12. N. E. Djaa and A. Zagane, Harmonicity of Mus-gradient metric, Int. J. Maps Math. 5 (2022), no. 1, 61-77. 
  13. N. E. Djaa and A. Zagane, Some results on the geometry of a non-conformal deformation of a metric, Commun. Korean Math. Soc. 37 (2022), no. 3, 865-879. https://doi.org/10.4134/CKMS.c210207 
  14. J. Eells and L. Lemaire, Another report on harmonic maps, Bull. London Math. Soc. 20 (1988), no. 5, 385-524. https://doi.org/10.1112/blms/20.5.385 
  15. J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109-160. https://doi.org/10.2307/2373037 
  16. J. J. Konderak, On harmonic vector fields, Publ. Mat. 36 (1992), no. 1, 217-228. https://doi.org/10.5565/PUBLMAT_36192_17 
  17. A. A. Salimov, M. Iscan, and F. Etayo, Paraholomorphic B-manifold and its properties, Topology Appl. 154 (2007), no. 4, 925-933. https://doi.org/10.1016/j.topol.2006.10.003 
  18. A. Yampolsky, On geodesics of tangent bundle with fiberwise deformed Sasaki metric over Kahlerian manifold, J. Math. Phys. Anal. Geom. 8 (2012), no. 2, 177-189. 
  19. A. Zagane, A study of harmonic sections of tangent bundles with vertically rescaled Berger-type deformed Sasaki metric, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb. 47 (2021), no. 2, 270-285. https://doi.org/10.30546/2409-4994.47.2.270 
  20. A. Zagane and N. E. Djaa, Notes about a harmonicity on the tangent bundle with vertical rescaled metric, Int. Electron. J. Geom. 15 (2022), no. 1, 83-95. https://doi.org/10.36890/iejg.1033998 
  21. A. Zagane and A. Gezer, Vertical rescaled Cheeger-Gromoll metric and harmonicity on the cotangent bundle, Adv. Stud. Euro-Tbil. Math. J. 15 (2022), no. 3, 11-29. https://doi.org/10.32513/asetmj/19322008221