Research article

Liftings of metallic structures to tangent bundles of order $ r $

  • Received: 31 October 2021 Revised: 31 January 2022 Accepted: 07 February 2022 Published: 17 February 2022
  • MSC : 53C15, 58A30

  • It is well known that the prolongation of an almost complex structure from a manifold $ M $ to the tangent bundle of order $ r $ on $ M $ is also an almost complex structure if it is integrable. The general quadratic structure $ F^2 = \alpha F+\beta I $ is a generalization of an almost complex structure where $ \alpha = 0, \; \beta = -1. $ The purpose of this paper is to characterize a metallic structure defined by the general quadratic structure $ F^2 = \alpha F+\beta I, \; \alpha, \beta\in \mathbb{N} $, where $ \mathbb{N} $ is the set of natural numbers. We show that the $ r $-lift of the metallic structure $ F $ in the tangent bundle of order $ r $ is also a metallic structure. Furthermore, we deduce a theorem on the projection tensor in the tangent bundle of order $ r $. Moreover, prolongations of $ G $-structures immersed in the metallic structure to the tangent bundle of order $ r $ and 2 are discussed. Finally, we construct examples of metallic structures that admit an almost para contact structure on the tangent bundle of order 3 and 4.

    Citation: Mohammad Nazrul Islam Khan, Uday Chand De. Liftings of metallic structures to tangent bundles of order $ r $[J]. AIMS Mathematics, 2022, 7(5): 7888-7897. doi: 10.3934/math.2022441

    Related Papers:

  • It is well known that the prolongation of an almost complex structure from a manifold $ M $ to the tangent bundle of order $ r $ on $ M $ is also an almost complex structure if it is integrable. The general quadratic structure $ F^2 = \alpha F+\beta I $ is a generalization of an almost complex structure where $ \alpha = 0, \; \beta = -1. $ The purpose of this paper is to characterize a metallic structure defined by the general quadratic structure $ F^2 = \alpha F+\beta I, \; \alpha, \beta\in \mathbb{N} $, where $ \mathbb{N} $ is the set of natural numbers. We show that the $ r $-lift of the metallic structure $ F $ in the tangent bundle of order $ r $ is also a metallic structure. Furthermore, we deduce a theorem on the projection tensor in the tangent bundle of order $ r $. Moreover, prolongations of $ G $-structures immersed in the metallic structure to the tangent bundle of order $ r $ and 2 are discussed. Finally, we construct examples of metallic structures that admit an almost para contact structure on the tangent bundle of order 3 and 4.



    加载中


    [1] S. Azami, Metallic structures on the tangent bundle of a $P$-Sasakian manifold, arXiv Preprint, 2019. Available from: https://arXiv.org/abs/1904.12637.
    [2] S. Azami, General natural metallic structure on tangent bundle, Iran. J. Sci. Technol. Trans. A Sci., 42 (2018), 81–88. https://doi.org/10.1007/s40995-018-0488-x doi: 10.1007/s40995-018-0488-x
    [3] S. Azami, M. G. Moghadam, Metallic structure on tangent bundles of order $ r$, In: Proceedings of the first national congress on mathematics and statistics, Gonbad Kavous, 2018,139–143. Available from: http://profs.gonbad.ac.ir/Uploads/Doc/file_169_mscgku1en.pdf.
    [4] A. Gezer, A. Magden, Geometry of the second-order tangent bundles of Riemannian manifolds, Chin. Ann. Math. Ser. B, 38 (2017), 985–998. https://doi.org/10.1007/s11401-017-1107-4 doi: 10.1007/s11401-017-1107-4
    [5] S. I. Goldberg, 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 doi: 10.2996/kmj/1138846727
    [6] S. I. Goldberg, K. Yano, Polynomial structures on manifolds, Kodai Math. Sem. Rep., 22 (1970), 199–218. https://doi.org/10.2996/kmj/1138846118 doi: 10.2996/kmj/1138846118
    [7] C. E. Hretcanu, M. Crasmareanu, Metallic structures on Riemannian manifolds, Rev. Union Mat. Argent., 54 (2013), 15–27.
    [8] M. N. I. Khan, Complete and horizontal lifts of metallic structures, Int. J. Math. Comput. Sci., 15 (2020), 983–992.
    [9] M. N. I. Khan, Tangent bundle endowed with quarter-symmetric non-metric connection on an almost Hermitian manifold, Facta Univ. Ser. Math. Inform., 35 (2020), 167–178. https://doi.org/10.22190/FUMI2001167K doi: 10.22190/FUMI2001167K
    [10] M. N. I. Khan, Novel theorems for the frame bundle endowed with metallic structures on an almost contact metric manifold, Chaos Soliton. Fract., 146 (2021), 110872. https://doi.org/10.1016/j.chaos.2021.110872 doi: 10.1016/j.chaos.2021.110872
    [11] M. N. I. Khan, J. B. Jun, Prolongations of $G$-structures immersed in generalized almost $r$-contact structure to tangent bundle of order 2, J. Chungcheong Math. Soc., 31 (2018), 421–427. https://doi.org/10.14403/jcms.2018.31.1.421 doi: 10.14403/jcms.2018.31.1.421
    [12] A. J. Ladger, K. Yano, Almost complex structures on tensor bundles, J. Differ. Geom., 1 (1967), 355–368.
    [13] A. Morimoto, Lifting of tensor fields and connections to tangent bundles of higher order, Nagoya Math. J., 40 (1970), 99–120. https://doi.org/10.1017/S002776300001388X doi: 10.1017/S002776300001388X
    [14] A. Morimoto, Prolongations of G-structures to tangent bundles, Nagoya Math. J., 32 (1968), 67–108. https://doi.org/10.1017/S002776300002660X doi: 10.1017/S002776300002660X
    [15] A. Mağden, A. Gezer, K. Karaca, Some problems concerning with Sasaki metric on the second-order tangent bundles, Int. Electron. J. Geom., 13 (2020), 75–86. https://doi.org/10.36890/iejg.750905 doi: 10.36890/iejg.750905
    [16] M. Özkan, F. Yilmaz, Prolongations of golden structures to tangent bundles of order $r$, Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat., 65 (2016), 35–48. https://doi.org/10.1501/Commua1_0000000742 doi: 10.1501/Commua1_0000000742
    [17] V. W. de Spinadel, The metallic means family and multifractal spectra, Nonlinear Anal., 36 (1999), 721–745.
    [18] V. W. de Spinadel, The metallic means family and forbidden symmetries, Int. Math. J., 2 (2002), 279–288.
    [19] V. W. de Spinadel, The metallic means family and renormalization group techniques, Trudy Inst. Mat. i Mekh. UrO RAN, 6 (2000), 173–189.
    [20] V. W. de Spinadel, A new family of irrational numbers with curious properties, Humanist. Math. Netw. J., 1 (1999), 33–37. https://doi.org/10.5642/hmnj.199901.19.14 doi: 10.5642/hmnj.199901.19.14
    [21] K. Yano, Tensor fields and connections on cross-sections in the tangent bundle of a differentiable manifold, Proc. Roy. Soc. Edinburgh Sect. A, 67 (1967), 277–288. https://doi.org/10.1017/S0080454100008141 doi: 10.1017/S0080454100008141
    [22] K. Yano, E. T. Davis, Metrics and connections in the tangent bundle, Kodai Math. Sem. Rep., 23 (1971), 493–504. https://doi.org/10.2996/kmj/1138846418 doi: 10.2996/kmj/1138846418
    [23] K. Yano, S. Ishihara, Tangent and cotangent bundles: Differential geometry, New York: Marcel Dekker, Inc., 1973.
    [24] K. Yano, S. Ishihara, Differentiable geomtry of tangent bundles of order 2, Kodai Math. Sem. Rep., 20 (1968), 318–354. https://doi.org/10.2996/kmj/1138845701 doi: 10.2996/kmj/1138845701
  • Reader Comments
  • © 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1582) PDF downloads(97) Cited by(11)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog