Fractional Order Riemann Curvature Tensor in Differential Geometry

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DOI: https://doi.org/10.28924/2291-8639-20-2022-26
Published: May 27, 2022
Cite as:
Wedad Saleh, Fractional Order Riemann Curvature Tensor in Differential Geometry, Int. J. Anal. Appl., 20 (2022), 26.

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Wedad Saleh

Abstract

This study discussed some interesting aspects and features of fractional curvature in the differential manifold. In particular, Riemannian fractional curvature tensor, Livi-Civita fractional connection and Bianchi fractional identity are presented.

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References

  1. I.D. Albu, M. Neamtu, D. Opris, The Geometry of Fractional Osculator Bundle of Higher Order and Applications, in: Processing of International Conference on Differential Geometry Lagrange and Hamilton Spaces, Dedicated to Acad. Radu Miron at eighty, September 3-8, 2007, Iasi, Romania https://doi.org/10.48550/arXiv.0709.2000.
  2. M. Axtell, M.E. Bise, Fractional Calculus Application in Control Systems, in: IEEE Conference on Aerospace and Electronics, IEEE, Dayton, OH, USA, 1990: pp. 563–566. https://doi.org/10.1109/NAECON.1990.112826.
  3. Y. Ding, X. Wu, Fractional Integrals on Product Manifolds, Potential Anal. 30 (2009), 371–383. https://doi.org/10.1007/s11118-009-9120-1.
  4. G. Jumarie, Riemann-Christoffel Tensor in Differential Geometry of Fractional Order Application to Fractal SpaceTime, Fractals. 21 (2013), 1350004. https://doi.org/10.1142/S0218348X13500047.
  5. I. D. Albu and D. Opris, The Geometry of Fractional Tangent Bundle and Applications, in: BSG Proceedings 16. The Int. Conf. of Diff. Geom. and Dynamical Systems (DGDS-2008) and The V-th Int. Colloq. of Mathematics in Engineering and Numerical Physics (MENP-5) - math. sections, August 29 - September 2, 2008, Mangalia, Romania, pp. 1-11. http://repository.utm.md/handle/5014/17678.
  6. A. Kilicman, W. Saleh, Note on the Fractional Mittag-Leffler Functions by Applying the Modified Riemann-Liouville Derivatives, Bol. Soc. Paranaense Mat. 40 (2022), 1–16. https://doi.org/10.5269/bspm.44103.
  7. K.M. Kolwankar, Studies of Fractal Structures and Processes Using Methods of Fractional Calculus, ArXiv:ChaoDyn/9811008. (1998). http://arxiv.org/abs/chao-dyn/9811008.
  8. M. Delkhosh, Introduction of Derivatives and Integrals of Fractional Order and Its Applications, Appl. Math. Phys. 1 (2013), 103–119 .
  9. K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, Inc. New York, (1993).
  10. K.B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, (1974).
  11. I. Podlubny, Fractional-Order Systems and Fractional-Order Controllers, Technical Report UEF-03-94, Slovak Acad. Sci. (1994).
  12. B. Ross, ed., Fractional Calculus and Its Applications: Proceedings of the International Conference Held at the University of New Haven, June 1974, Springer Berlin Heidelberg, Berlin, Heidelberg, 1975. https://doi.org/10.1007/BFb0067095.
  13. W. Saleh, A. Kılıçman, Some Remarks on Geometry of Fractional Calculus, Appl. Math. Inform. Sci. 9 (2015), 1265–1275.
  14. W. Saleh and A. Kılıçman, Some Inequalities for Generalized s-Convex Functions, JP J. Geom. Topol. 17 (2015), 63–82.
  15. I.N. Sneddon, Special Functions of Mathematical Physics and Chemistry, Oliver & Boyd, Edinburgh, 1956.
  16. M.H. Tavassoli, A. Tavassoli, M.R. Ostad Rahimi, The Geometric and Physical Interpretation of Fractional Order Derivatives of Polynomial Functions, Differ. Geom. Dyn. Syst. 15 (2013), 93–104.
  17. A. Kılıçman and B. Fisher, Some Results on the Composition of Singular Distribution, Appl. Math. Inform. Sci. 5 (2011), 597–608.
  18. A. Kılıçman, W. Saleh, NOtions of Generalized s-Convex Functions on Fractal Sets, J. Inequal. Appl. 2015 (2015), 312. https://doi.org/10.1186/s13660-015-0826-x.
  19. A. Kılıçman, W. Saleh, Some Generalized Hermite-Hadamard Type Integral Inequalities for Generalized S-Convex Functions on Fractal Sets, Adv. Differ. Equ. 2015 (2015), 301. https://doi.org/10.1186/s13662-015-0639-8.
  20. B. Fisher, B. Jolevsaka-Tuneska, A. KiliÇman, On Defining the Incomplete Gamma Function, Integral Transforms Spec. Funct. 14 (2003), 293–299. https://doi.org/10.1080/1065246031000081667.
  21. B. Fisher, A. Kilicman, D. Nicholas, On the Beta Function and the Neutrix Product of Distributions, Integral Transforms Spec. Funct. 7 (1998), 35–42. https://doi.org/10.1080/10652469808819184.
  22. B. Fisher, A. Kılıçman, Results on the Gamma Function, Appl. Math. Inform. Sci. 6 (2012), 173–176.