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.
Keywords
Curvature tensor field lifts econd-order tangent bundle statistical structure weakly symmetry
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.
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.
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Research Article |
| Authors | |
| Project Number | AR-GE 3001 Project No. 118F190 |
| Publication Date | October 15, 2020 |
| Acceptance Date | June 22, 2020 |
| Published in Issue | Year 2020 |