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Some Problems Concerning with Sasaki Metric on the Second-Order Tangent Bundles

Year 2020, , 75 - 86, 15.10.2020
https://doi.org/10.36890/iejg.750905

Abstract

In this paper, we consider a second-order tangent bundle equipped with
Sasaki metric over a Riemannian manifold. All forms of curvature tensor
fields are computed. We obtained the relation between the scalar curvature
of the base manifold and the scalar curvature of the second-order tangent
bundle and presented some geometric results concerning with kinds of
curvature tensor fields. Also, we search the weakly symmetry property of the
second-order tangent bundle. Finally, we end our paper with statistical
structures on the second-order tangent bundle.

Supporting Institution

TUBİTAK

Project Number

AR-GE 3001 Project No. 118F190

Thanks

The paper is supported by the Scientific and Technological Research Council of Turkey, AR-GE 3001 Project No. 118F190

References

  • [1] Bejan, C. L., Crasmareanu, M.: Weakly-symmetry of the Sasakian lifts on tangent bundles, Publ. Math. Debrecen, 83 (1-2), 63–69 (2013).
  • [2] Binh, T. Q., Tamassy, L.: On recurrence or pseudo-symmetry of the Sasakian metric on the tangent bundle of a Riemannian manifold, Indian J. Pure Appl. Math., 35 ( 4), 555–560 (2004).
  • [3] de Leon, M., Vazquez, E.: On the geometry of the tangent bundle of order 2, An. Univ. Bucureşti Mat., 34, 40–48 (1985).
  • [4] De, U. C., Bandyopadhyay, S.: On weakly symmetric Riemannian spaces, Publ. Math. Debrecen, 54 (3-4), 377–381 (1999).
  • [5] Dida, M. H., Hathout, F., Djaa, M.: On the geometry of the second order tangent bundle with the diagonal lift metric, Int. J. Math. Anal., 3 (9-12), 443–456 (2009).
  • [6] Djaa, M., Gancarzewicz, J.: The geometry of tangent bundles of order r, Boletin Academia, Galega de Ciencias, 4, 147-165 (1985).
  • [7] Dodson, C. T. J., Radivoiovici, M. S.: Tangent and frame bundles of order two, Analele stiintifice ale Universitatii "Al. I. Cuza", 28, 63-71 (1982).
  • [8] Gezer, A., Magden, A.: Geometry of the second-order tangent bundles of Riemannian manifolds, Chin. Ann. Math. Ser. B, 38 (4), 985–998 (2017).
  • [9] Hathout, F, Dida, H. M.: Diagonal lift in the tangent bundle of order two and its applications, Turkish J. Math., 30 (4), 373–384 (2004).
  • [10] Ishikawa, S.: On Riemannian metrics of tangent bundles of order 2 of Riemannian manifolds, Tensor (N.S.), 34 (2), 173–178 (1980).
  • [11] Lauritzen, S. L.: Statistical manifolds( In: Differential Geometry in Statistical Inferences, IMS Lecture Notes Monogr. Ser., 10, Inst. Math. Statist., Hayward California, 1987, 96- 163).
  • [12] Nomizu, K., Sasaki, T.: Affine Differential Geometry Geometry of Affine Immersions, vol. 111 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge 1994.
  • [13] Oniciuc, C.: The tangent bundles and harmonicity, An. ¸Stiin¸t. Univ. Al. I. Cuza Ia¸si. Mat. (N.S.), 43 (1), 151–172 (1987).
  • [14] Schwenk-Schellschmidt, A., Simon, U.: Codazzi-equivalent affine connections, Result Math., 56 (1–4), 211–229 (2009).
  • [15] Yano, K., Ishihara, S.: Tangent and cotangent bundles, Marcel Dekker, Inc., New York, 1973.
Year 2020, , 75 - 86, 15.10.2020
https://doi.org/10.36890/iejg.750905

Abstract

Project Number

AR-GE 3001 Project No. 118F190

References

  • [1] Bejan, C. L., Crasmareanu, M.: Weakly-symmetry of the Sasakian lifts on tangent bundles, Publ. Math. Debrecen, 83 (1-2), 63–69 (2013).
  • [2] Binh, T. Q., Tamassy, L.: On recurrence or pseudo-symmetry of the Sasakian metric on the tangent bundle of a Riemannian manifold, Indian J. Pure Appl. Math., 35 ( 4), 555–560 (2004).
  • [3] de Leon, M., Vazquez, E.: On the geometry of the tangent bundle of order 2, An. Univ. Bucureşti Mat., 34, 40–48 (1985).
  • [4] De, U. C., Bandyopadhyay, S.: On weakly symmetric Riemannian spaces, Publ. Math. Debrecen, 54 (3-4), 377–381 (1999).
  • [5] Dida, M. H., Hathout, F., Djaa, M.: On the geometry of the second order tangent bundle with the diagonal lift metric, Int. J. Math. Anal., 3 (9-12), 443–456 (2009).
  • [6] Djaa, M., Gancarzewicz, J.: The geometry of tangent bundles of order r, Boletin Academia, Galega de Ciencias, 4, 147-165 (1985).
  • [7] Dodson, C. T. J., Radivoiovici, M. S.: Tangent and frame bundles of order two, Analele stiintifice ale Universitatii "Al. I. Cuza", 28, 63-71 (1982).
  • [8] Gezer, A., Magden, A.: Geometry of the second-order tangent bundles of Riemannian manifolds, Chin. Ann. Math. Ser. B, 38 (4), 985–998 (2017).
  • [9] Hathout, F, Dida, H. M.: Diagonal lift in the tangent bundle of order two and its applications, Turkish J. Math., 30 (4), 373–384 (2004).
  • [10] Ishikawa, S.: On Riemannian metrics of tangent bundles of order 2 of Riemannian manifolds, Tensor (N.S.), 34 (2), 173–178 (1980).
  • [11] Lauritzen, S. L.: Statistical manifolds( In: Differential Geometry in Statistical Inferences, IMS Lecture Notes Monogr. Ser., 10, Inst. Math. Statist., Hayward California, 1987, 96- 163).
  • [12] Nomizu, K., Sasaki, T.: Affine Differential Geometry Geometry of Affine Immersions, vol. 111 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge 1994.
  • [13] Oniciuc, C.: The tangent bundles and harmonicity, An. ¸Stiin¸t. Univ. Al. I. Cuza Ia¸si. Mat. (N.S.), 43 (1), 151–172 (1987).
  • [14] Schwenk-Schellschmidt, A., Simon, U.: Codazzi-equivalent affine connections, Result Math., 56 (1–4), 211–229 (2009).
  • [15] Yano, K., Ishihara, S.: Tangent and cotangent bundles, Marcel Dekker, Inc., New York, 1973.
There are 15 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Abdullah Mağden 0000-0002-9513-7012

Aydın Gezer 0000-0001-7505-0385

Kübra Karaca 0000-0002-6329-4435

Project Number AR-GE 3001 Project No. 118F190
Publication Date October 15, 2020
Acceptance Date June 22, 2020
Published in Issue Year 2020

Cite

APA Mağden, A., Gezer, A., & Karaca, K. (2020). Some Problems Concerning with Sasaki Metric on the Second-Order Tangent Bundles. International Electronic Journal of Geometry, 13(2), 75-86. https://doi.org/10.36890/iejg.750905
AMA Mağden A, Gezer A, Karaca K. Some Problems Concerning with Sasaki Metric on the Second-Order Tangent Bundles. Int. Electron. J. Geom. October 2020;13(2):75-86. doi:10.36890/iejg.750905
Chicago Mağden, Abdullah, Aydın Gezer, and Kübra Karaca. “Some Problems Concerning With Sasaki Metric on the Second-Order Tangent Bundles”. International Electronic Journal of Geometry 13, no. 2 (October 2020): 75-86. https://doi.org/10.36890/iejg.750905.
EndNote Mağden A, Gezer A, Karaca K (October 1, 2020) Some Problems Concerning with Sasaki Metric on the Second-Order Tangent Bundles. International Electronic Journal of Geometry 13 2 75–86.
IEEE A. Mağden, A. Gezer, and K. Karaca, “Some Problems Concerning with Sasaki Metric on the Second-Order Tangent Bundles”, Int. Electron. J. Geom., vol. 13, no. 2, pp. 75–86, 2020, doi: 10.36890/iejg.750905.
ISNAD Mağden, Abdullah et al. “Some Problems Concerning With Sasaki Metric on the Second-Order Tangent Bundles”. International Electronic Journal of Geometry 13/2 (October 2020), 75-86. https://doi.org/10.36890/iejg.750905.
JAMA Mağden A, Gezer A, Karaca K. Some Problems Concerning with Sasaki Metric on the Second-Order Tangent Bundles. Int. Electron. J. Geom. 2020;13:75–86.
MLA Mağden, Abdullah et al. “Some Problems Concerning With Sasaki Metric on the Second-Order Tangent Bundles”. International Electronic Journal of Geometry, vol. 13, no. 2, 2020, pp. 75-86, doi:10.36890/iejg.750905.
Vancouver Mağden A, Gezer A, Karaca K. Some Problems Concerning with Sasaki Metric on the Second-Order Tangent Bundles. Int. Electron. J. Geom. 2020;13(2):75-86.