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Year 2019, Volume: 68 Issue: 1, 222 - 235, 01.02.2019
https://doi.org/10.31801/cfsuasmas.443735

Abstract

References

  • Abbassi, M., T., K., Note on the classification theorems of g-natural metrics on the tangent bundle of a Riemannian manifold (M; g), Comment. Math. Univ. Carolin. 45(2004), 591-596.
  • Abbassi, M., T., K., Sarih, M., On some hereditary properties of Riemannian g-natural metrics on tangent bundles of Riemannian manifolds, Differential Geom. Appl. 22(2005), 19-47.
  • Abbassi, M., T., K., Sarih, M., On natural metrics on tangent bundles of Riemannian manifolds, Arch. Math. 41(2005), 71-92.
  • Dida, M., H., Hathout, F., Djaa, M., On the Geometry of the Second Order Tangent Bundle with the Diagonal lift Metric, Int. Journal of Math. Analysis. 3(2009), 443-456.
  • Dombrowski, P., On the Geometry of the Tangent Bundle, J. Reine Angew Math. 210(1962), 73-88.
  • Cheeger, J., Gromoll, D., On the structure of complete manifolds of nonnegative curvature, Ann. of Math, 96(1972), 413-443.
  • García-Río, D., Kupeli, N., Semi-Riemannian Maps and Their Applications, Mathematics and Its Applications, Springer science media, B.V.8 2010.
  • Gezer, A., On the tangent bundle with deformed Sasaki metric, International Electronic Journal of Geometry, 6(2013), 19-31.
  • Gudmundsson, S., Kappos, E., On the Geometry of the Tangent Bundle with the Cheeger-Gromoll metric, Tokyo J. Math. 25(2002), 75-83.
  • Gudmundsson, S., Kappos, E., On the Geometry of the Tangent Bundles, Expo. Math. 20(2002), 1-41.
  • Hathout, F. Dida, H. M., Diagonal lift in the tangent bundle of order two and its applications, Turk. J. Math 30(2006), 373-384.
  • Kowalski, O., Curvature of the induced Riemannian metric of the tangent bundle of a Riemannian manifold, J. Reine Angew.math. 250(1971), 124-129.
  • Musso, E., Tricerri, F., Riemannian metric on tangent bundle, Ann. Math. Pura. Appl. 150(1988), 1-19.
  • O'Neill, B., Semi-Riemannian geometry with applications to relativity, Academic Press, New York, 1983.
  • Oproiu, V., Some new geometric structures on the tangent bundles. Publ Math. Debrecen, 55(1999) 261-281.
  • Oproiu, V., Papaghiuc, N., On the geometry of tangent bundle of a (pseudo-) Riemannian manifold, An Stiint Univ Al I Cuza Iasi Mat 44(1998) 67-83.
  • Sasaki, S., On the differential geometry of tangent bundles of Riemannian manifolds, Tohoku Math. J. 10(1958) 338-358.
  • Sekizawa, M., Curvatures of Tangent Bundles with Cheeger-Gromoll metric, Tokyo J. Math. 14(1991) 407-417.
  • Wang, J., Wang, Y., On the geometry of tangent bundles with the rescaled metric, arXiv:1104.5584v1.
  • Yano, K., Ishihara, S., Tangent and cotangent bundles, Marcel Dekker, Inc., New York 1973.
  • Zayatuev, B. V., On geometry of tangent Hermitian surface, Webs and Quasigroups. T.S.U. (1995) 139-143.
  • Zayatuev, B. V., On some classes of almost-Hermitian structures on the tangent bundle, Webs and Quasigroups. T.S.U. (2002) 103--106.
  • Zhong, H. H., Lei, S., Geometry of tangent bundle with Cheeger--Gromoll type metric, Math. Anal. Appl. 402(2013) 493-504.

On the Geometry of the Tangent Bundle With Vertical Rescaled Metric

Year 2019, Volume: 68 Issue: 1, 222 - 235, 01.02.2019
https://doi.org/10.31801/cfsuasmas.443735

Abstract

Let (M,g) be a n-dimensional smooth Riemannian manifold. In the present paper, we introduce a new class of natural metrics denoted by G^{f} and called the vertical rescaled metric on the tangent bundle TM. We calculate its Levi-Civita connection and Riemannian curvature tensor. We study the geometry of (TM,G^{f}) and several important results are obtained on curvature, Einstein structure, scalar and sectional curvatures

References

  • Abbassi, M., T., K., Note on the classification theorems of g-natural metrics on the tangent bundle of a Riemannian manifold (M; g), Comment. Math. Univ. Carolin. 45(2004), 591-596.
  • Abbassi, M., T., K., Sarih, M., On some hereditary properties of Riemannian g-natural metrics on tangent bundles of Riemannian manifolds, Differential Geom. Appl. 22(2005), 19-47.
  • Abbassi, M., T., K., Sarih, M., On natural metrics on tangent bundles of Riemannian manifolds, Arch. Math. 41(2005), 71-92.
  • Dida, M., H., Hathout, F., Djaa, M., On the Geometry of the Second Order Tangent Bundle with the Diagonal lift Metric, Int. Journal of Math. Analysis. 3(2009), 443-456.
  • Dombrowski, P., On the Geometry of the Tangent Bundle, J. Reine Angew Math. 210(1962), 73-88.
  • Cheeger, J., Gromoll, D., On the structure of complete manifolds of nonnegative curvature, Ann. of Math, 96(1972), 413-443.
  • García-Río, D., Kupeli, N., Semi-Riemannian Maps and Their Applications, Mathematics and Its Applications, Springer science media, B.V.8 2010.
  • Gezer, A., On the tangent bundle with deformed Sasaki metric, International Electronic Journal of Geometry, 6(2013), 19-31.
  • Gudmundsson, S., Kappos, E., On the Geometry of the Tangent Bundle with the Cheeger-Gromoll metric, Tokyo J. Math. 25(2002), 75-83.
  • Gudmundsson, S., Kappos, E., On the Geometry of the Tangent Bundles, Expo. Math. 20(2002), 1-41.
  • Hathout, F. Dida, H. M., Diagonal lift in the tangent bundle of order two and its applications, Turk. J. Math 30(2006), 373-384.
  • Kowalski, O., Curvature of the induced Riemannian metric of the tangent bundle of a Riemannian manifold, J. Reine Angew.math. 250(1971), 124-129.
  • Musso, E., Tricerri, F., Riemannian metric on tangent bundle, Ann. Math. Pura. Appl. 150(1988), 1-19.
  • O'Neill, B., Semi-Riemannian geometry with applications to relativity, Academic Press, New York, 1983.
  • Oproiu, V., Some new geometric structures on the tangent bundles. Publ Math. Debrecen, 55(1999) 261-281.
  • Oproiu, V., Papaghiuc, N., On the geometry of tangent bundle of a (pseudo-) Riemannian manifold, An Stiint Univ Al I Cuza Iasi Mat 44(1998) 67-83.
  • Sasaki, S., On the differential geometry of tangent bundles of Riemannian manifolds, Tohoku Math. J. 10(1958) 338-358.
  • Sekizawa, M., Curvatures of Tangent Bundles with Cheeger-Gromoll metric, Tokyo J. Math. 14(1991) 407-417.
  • Wang, J., Wang, Y., On the geometry of tangent bundles with the rescaled metric, arXiv:1104.5584v1.
  • Yano, K., Ishihara, S., Tangent and cotangent bundles, Marcel Dekker, Inc., New York 1973.
  • Zayatuev, B. V., On geometry of tangent Hermitian surface, Webs and Quasigroups. T.S.U. (1995) 139-143.
  • Zayatuev, B. V., On some classes of almost-Hermitian structures on the tangent bundle, Webs and Quasigroups. T.S.U. (2002) 103--106.
  • Zhong, H. H., Lei, S., Geometry of tangent bundle with Cheeger--Gromoll type metric, Math. Anal. Appl. 402(2013) 493-504.
There are 23 citations in total.

Details

Primary Language English
Journal Section Review Articles
Authors

Hamou Mohammed Dida This is me

Fouzi Hathout

Abdelhalim Azzouz This is me

Publication Date February 1, 2019
Submission Date February 15, 2017
Acceptance Date December 6, 2017
Published in Issue Year 2019 Volume: 68 Issue: 1

Cite

APA Dida, H. M., Hathout, F., & Azzouz, A. (2019). On the Geometry of the Tangent Bundle With Vertical Rescaled Metric. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 68(1), 222-235. https://doi.org/10.31801/cfsuasmas.443735
AMA Dida HM, Hathout F, Azzouz A. On the Geometry of the Tangent Bundle With Vertical Rescaled Metric. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. February 2019;68(1):222-235. doi:10.31801/cfsuasmas.443735
Chicago Dida, Hamou Mohammed, Fouzi Hathout, and Abdelhalim Azzouz. “On the Geometry of the Tangent Bundle With Vertical Rescaled Metric”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68, no. 1 (February 2019): 222-35. https://doi.org/10.31801/cfsuasmas.443735.
EndNote Dida HM, Hathout F, Azzouz A (February 1, 2019) On the Geometry of the Tangent Bundle With Vertical Rescaled Metric. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68 1 222–235.
IEEE H. M. Dida, F. Hathout, and A. Azzouz, “On the Geometry of the Tangent Bundle With Vertical Rescaled Metric”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 68, no. 1, pp. 222–235, 2019, doi: 10.31801/cfsuasmas.443735.
ISNAD Dida, Hamou Mohammed et al. “On the Geometry of the Tangent Bundle With Vertical Rescaled Metric”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68/1 (February 2019), 222-235. https://doi.org/10.31801/cfsuasmas.443735.
JAMA Dida HM, Hathout F, Azzouz A. On the Geometry of the Tangent Bundle With Vertical Rescaled Metric. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68:222–235.
MLA Dida, Hamou Mohammed et al. “On the Geometry of the Tangent Bundle With Vertical Rescaled Metric”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 68, no. 1, 2019, pp. 222-35, doi:10.31801/cfsuasmas.443735.
Vancouver Dida HM, Hathout F, Azzouz A. On the Geometry of the Tangent Bundle With Vertical Rescaled Metric. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68(1):222-35.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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