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ON DEFORMED-SASAKI METRIC AND HARMONICITY IN TANGENT BUNDLES

  • Received : 2020.01.18
  • Accepted : 2020.05.20
  • Published : 2020.07.31

Abstract

In this paper, we introduce the deformed-Sasaki metric on the tangent bundle TM over an m-dimensional Riemannian manifold (M, g), as a new natural metric on TM. We establish a necessary and sufficient conditions under which a vector field is harmonic with respect to the deformed-Sasaki Metric. We also construct some examples of harmonic vector fields.

Keywords

Acknowledgement

We thank the anonymous reviewers for their insightful comments and suggestions that helped us improve the paper.

References

  1. M. T. K. Abbassi and M. Sarih, On natural metrics on tangent bundles of Riemannian manifolds, Arch. Math. (Brno) 41 (2005), no. 1, 71-92. https://doi.org/10.1007/BF01193825
  2. J. Cheeger and D. Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. of Math. (2) 96 (1972), 413-443. https://doi.org/10.2307/1970819
  3. M. Djaa and J. Gancarzewicz, The geometry of tangent bundles of order r, Boletin Academia, Galega de Ciencias. 4 (1985), 147-165.
  4. P. Dombrowski, On the geometry of the tangent bundle, J. Reine Angew. Math. 210 (1962), 73-88. https://doi.org/10.1515/crll.1962.210.73
  5. J. Eells and L. Lemaire, Another report on harmonic maps, Bull. London Math. Soc. 20 (1988), no. 5, 385-524. https://doi.org/10.1112/blms/20.5.385
  6. J. Eells, Jr., and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109-160. https://doi.org/10.2307/2373037
  7. S. Gudmundsson and E. Kappos, On the geometry of the tangent bundle with the Cheeger-Gromoll metric, Tokyo J. Math. 25 (2002), no. 1, 75-83. https://doi.org/10.3836/tjm/1244208938
  8. F. Latti, M. Djaa, and A. Zagane, Mus-Sasaki metric and harmonicity, Math. Sci. Appl. E-Notes 6 (2018), no. 1, 29-36.
  9. A. Salimov and A. Gezer, On the geometry of the (1, 1)-tensor bundle with Sasaki type metric, Chin. Ann. Math. Ser. B 32 (2011), no. 3, 369-386. https://doi.org/10.1007/s11401-011-0646-3
  10. S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds. II, Tohoku Math. J. (2) 14 (1962), 146-155. https://doi.org/10.2748/tmj/1178244169
  11. K. Yano and S. Ishihara, Tangent and Cotangent Bundles: differential geometry, Marcel Dekker, Inc., New York, 1973.
  12. A. Zagane and M. Djaa, On geodesics of warped Sasaki metric, Math. Sci. Appl. E-Notes 5 (2017), no. 1, 85-92.