Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Integrability of the Metallic Structures on the Frame Bundle

  • Received : 2021.01.16
  • Accepted : 2021.06.21
  • Published : 2021.12.31
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Abstract

Earlier investigators have made detailed studies of geometric properties such as integrability, partial integrability, and invariants, such as the fundamental 2-form, of some canonical f-structures, such as f3 ± f = 0, on the frame bundle FM. Our aim is to study metallic structures on the frame bundle: polynomial structures of degree 2 satisfying F2 = pF +qI where p, q are positive integers. We introduce a tensor field Fα, α = 1, 2…, n on FM show that it is a metallic structure. Theorems on Nijenhuis tensor and integrability of metallic structure Fα on FM are also proved. Furthermore, the diagonal lifts gD and the fundamental 2-form Ωα of a metallic structure Fα on FM are established. Moreover, the integrability condition for horizontal lift FαH of a metallic structure Fα on FM is determined as an application. Finally, the golden structure that is a particular case of a metallic structure on FM is discussed as an example.

Keywords

Acknowledgement

The author would like to thank the referee for his valuable remarks and suggestions which helped to improve the manuscript.

References

  1. S. Azami, Natural metallic structure on tangent bundle, Iran. J. Sci. Technol. Trans. A Sci., 42(1)(2018), 81-88. https://doi.org/10.1007/s40995-018-0488-x
  2. L. Bilen, S. Turanli and A. Gezer, On Kahler-Norden-Codazzi golden structures on pseudo-Riemannian manifolds, Int. J. Geom. Methods Mod. Phys., 15(5)(2018), 1850080, 10pp. https://doi.org/10.1142/S0219887818500809
  3. A. Bonome, R. Castro and L. M. Hervella, Almost complex structure in the frame bundle of an almost contact metric manifold, Math. Z., 193(1986), 431-440. https://doi.org/10.1007/BF01229810
  4. L. A. Cordero, C. T. Dodson and M. D. Leon, Differential Geometry of Frame Bundles, Springer Netherlands, Kluwer Academic Publishers, 1989.
  5. L. A. Cordero and D. M. Len, Prolongation of G-structures to the frame bundle, Ann. Mat. Pura. Appl., 143(1986), 123-141. https://doi.org/10.1007/BF01769212
  6. M. Crasmareanu and C. E. Hretcanu, Golden differential geometry, Chaos Solitons Fractals, 38(2008), 1229-1238. https://doi.org/10.1016/j.chaos.2008.04.007
  7. S. I. Goldberg and K. Yano, Polynomial structures on manifolds, Kodai Math. Sem. Rep., 22(1970), 199-218. https://doi.org/10.2996/kmj/1138846118
  8. S. I. Goldberg and N. C. Petridis, Differentiable solutions of algebraic equations on manifolds, Kodai Math. Sem. Rep., 25(1973), 111-128. https://doi.org/10.2996/kmj/1138846727
  9. S. Gonul, I. Kupeli Erken, A. Yazla and C. Murathan, Neutral relation between metallic structure and almost quadratic ø-structure, Turkish J. Math., 43(1)(2019), 268-278. https://doi.org/10.3906/mat-1807-72
  10. C. E. Hretcanu and M. Crasmareanu, Metallic structures on Riemannian manifolds, Rev. Un. Mat. Argentina, 54(2)(2013), 15-27.
  11. C. E. Hretcanu and A. M. Blaga, Submanifolds in metallic Riemannian manifolds, Differ. Geom. Dyn. Syst., 20(2018), 83-97.
  12. C. E. Hretcanu and A. M. Blaga, Slant and semi-slant submanifolds in metallic Riemannian manifolds, J. Funct. Spaces, 2018 ID 2864263 (2018), 1-13.
  13. C. E. Hretcanu and A. M. Blaga, Hemi-slant submanifolds in metallic Riemannian manifolds, Carpathian J. Math., 35(1)(2019), 59-68. https://doi.org/10.37193/CJM.2019.01.07
  14. M. N. I. Khan, Complete and horizontal lifts of metallic structures, Int. J. Math. Comput. Sci., 15(4)(2020), 983-992.
  15. M. N. I. Khan, Tangent bundle endowed with quarter-symmetric non-metric connection on an almost Hermitian manifold, Facta Univ. Ser. Math. Inform., 35(1)(2020), 167-178.
  16. M. N. I. Khan, Novel theorems for the frame bundle endowed with metallic structures on an almost contact metric manifold, Chaos Solitons Fractals, 146(2021), 110872. https://doi.org/10.1016/j.chaos.2021.110872.
  17. O. Kowalski and M. Sekizawa, Curvatures of the diagonal lift from an affine manifold to the linear frame bundle, Cent. Eur. J. Math., 10(3)(2012), 837-843. https://doi.org/10.2478/s11533-012-0033-7
  18. M. Lachieze-Rey, Connections and frame bundle Reductions, arXiv:2002.01410 [mathph], 4 Feb, 2020.
  19. K. Niedzialomski, On the frame bundle adapted to a submanifold, Math. Nachr., 288(5-6)(2015), 648-664. https://doi.org/10.1002/mana.201400041
  20. M. Ozkan and F. Yilmaz, Metallic structures on differentiable manifolds, Journal of Science and Arts, 44(2018), 645-660.
  21. M. Ozkan, Prolongations of golden structures to tangent bundles, Differ. Geom. Dyn. Syst., 16(2014), 227-239.
  22. M. Ozkan and B. Peltek, A new structure on manifolds: silver structure, Int. Electron. J. Geom., 9(2)(2016), 59-69. https://doi.org/10.36890/iejg.584592
  23. A. Gezer, C. Karaman, On metallic Riemannian structures, Turkish J. Math., 39(6)(2015), 954-962. https://doi.org/10.3906/mat-1504-50
  24. V. W. de Spinadel, The metallic means family and multifractal spectra, Nonlinear Anal., 36(1999), 721-745. https://doi.org/10.1016/S0362-546X(98)00123-0
  25. V. W. de. Spinadel, On characterization of the onset to chaos, Chaos Solitons Fractals, 8(10)(1997), 1631-1643. https://doi.org/10.1016/S0960-0779(97)00001-5
  26. A. Stakhov, The generalized golden proportions, a new theory of real numbers, and ternary mirror-symmetrical arithmetic, Chaos Solitons and Fractals, 33(2007), 315-334. https://doi.org/10.1016/j.chaos.2006.01.028
  27. S. Tuanli, A. Gezer and H. Cakicioglu, Metallic Kahler and nearly Kahler manifolds, arXiv:1907.00726 [math.GM], 25 June, 2019.
  28. K. Yano and S. Ishihara, Tangent and cotangent bundles, Marcel Dekker, Inc., New York, 1973.
  29. K. Yano and M. Kon, Structures on manifolds, Series in Pure Mathematics. Singapore: World Scientific, 1984.