Harmonic maps on the tangent bundle according to the ciconia metric

Abstract

The focus of this paper revolves around investigating the harmonicity aspects of various mappings. Firstly, we explore the harmonicity of the canonical projection $\pi :\left( TM,\tilde{g}\right) \rightarrow \left( M_{2n},J,g\right) $, where $\left( M_{2n},J,g\right) $ represents an anti-paraK\"{a}hler manifold and $\left( TM,\tilde{g}\right) $ its tangent bundle with the ciconia metric. Additionally, we study the harmonicity of a vector field $\xi$, treated as mappings from $M$ to $TM$ . In this context, we consider the harmonicity relations between the ciconia metric $\tilde{g}$ and the Sasaki metric $^{S}g$, examining their mutual interactions. Furthermore, we investigate the Schoutan-Van Kampen connection and the Vr\~{a}nceanu connection, both associated with the Levi-Civita connection of the ciconia metric. Our analysis also includes the computation of the mean connections for the Schoutan-Van Kampen and Vr\~{a}nceanu connections, thereby providing insights into their properties. Finally, our exploration extends to the second fundamental form of the identity mapping from $\left( TM,\tilde{g}\right) $ to $\left(TM,\overline{\nabla }^{m}\right) ~$ and $\left( TM,\widetilde{\nabla }^{\ast m}\right) $. Here $\overline{\nabla }^{m}$ and $\widetilde{\nabla }^{\ast m}$ denote the mean connections associated with the Schoutan-Van Kampen and Vr\~{a}nceanu connections, respectively.

Keywords

• tangent bundle
• ciconia metric
• harmonic maps

References

1. [1] M. T. K. Abbassi and M. Sarih, On some hereditary properties of Riemannian gnatural metrics on tangent bundles of Riemannian manifolds, Difer. Geom. Appl. 22, 19–47, 2005.
2. [2] R. Albuquerque, The ciconia metric on the tangent bundle of an almost Hermitian manifold, Ann. Mat. Pura Appl. 199 (3), 969–984, 2020.
3. [3] M. Altunbas, R. Simsek and A. Gezer, Some harmonic problems on the tangent bundle with a Berger-type deformed Sasaki metric, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 82 (2), 37–42, 2020.
4. [4] P. Baird and C. L. Bejan, Quasi-harmonic maps between almost symplectic manifolds, Chapman & amp; Hall/CRC Res. Notes Math., 413 Chapman & Hall/CRC, Boca Raton, FL, 2000, 75–95.
5. [5] P. Baird and J. C. Wood, Harmonic morphisms between Riemannian manifolds, London Mathematical Society Monograph (New Series), Oxford University Press, New York (2003).
6. [6] A. Balmus, C. Oniciuc and N. Papaghiuc, Harmonic properties on the tangent bundle, An. Univ. Vest Timisoara, Ser. Mat.-Inform 42 , 17–28, 2004.
7. [7] C. L. Bejan, Quasi-harmonic morphisms, Ital. J. Pure Appl. Math. 18 , 51– 68, 2005.
8. [8] C. L. Bejan and S. L. Druta-Romaniuc, Connections which are harmonic with respect to general natural metrics, Diff. Geom. Appl. 30 (4), 306–317, 2012.

9. [9] C. L. Bejan and S. L. Druta-Romaniuc, Harmonic almost complex structures with respect to general natural metrics, Mediterr. J. Math. 11 (1), 123–136, 2014.
10. [10] A. Bejancu and H. Faran, Foliations and Geometric Structures, Math. Appl. 580, Springer, Dordrecht, 2006.
11. [11] A. Bejancu and H. Faran, Vrãnceanu connections and foliations with bundle-like metrics, Proc. Indian Acad. Sci. Math. Sci. 118 (1), 99113, 2008.
12. [12] M. Benyounes, E. Loubeau and C. M.Wood, Harmonic sections of Riemannian vector bundles, and metrics of Cheeger–Gromoll type, Differ. Geom. Appl. 25 (3), 322–334, 2007.
13. [13] B. Y. Chen and T. Nagano, Harmonic metrics, harmonic tensors and Gauss maps, J. Math. Soc. Jpn. 36, 295–313, 1984.
14. [14] C. T. J. Dodson, P. M. Trinidad and M. E. Vazquez-Abal, Harmonic-Killing vector fields, Bull. Belg. Math. Soc. Simon Stevin 9 (4), 481–490, 2002.
15. [15] J. Eells and L. Lemaire, Selected topics in harmonic maps, Conf. Board of the Math.Sci. A.M.S. 50, 85 pp., 1983.
16. [16] J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86, 109–160, 1964.
17. [17] E. Garcia-Rio, L. Vanhecke and M. E. Vazquez-Abal, Harmonic connections, Acta Sci. Math. (Szeged) 62 (3–4), 583–607, 1996.
18. [18] E. Garcia-Rio, L. Vanhecke and M. E. Vazquez-Abal, Harmonic endomorphism fields, J. Math. 41 (1), 23–30, 1997.
19. [19] A. Gezer, L. Bilen and U. C. De, Conformal vector fields and geometric solitons on the tangent bundle with the ciconia metric, Filomat 37 (24), 8193–8204, 2023.
20. [20] S. Ishihara, Harmonic sections of tangent bundles, J. Math. Tokushima Univ 13, 23–27, 1979.
21. [21] O. Nouhaud, Applications harmoniques d’une variete Riemanniene dans son fibre tangent, C. R. Acad. Sci. Paris Sér. 284 (14), 815–818, 1997.
22. [22] Z. Olszak, The Schouten Van-Kampen affine connection adapted to an almost(para) contact metric structure, Publications De l’institut Mathématique, 94 (114), 31–42, 2013.
23. [23] C. Oniciuc, Tangency and Harmonicity Properties, Ph.D. Thesis, DGDS Monographs 3, Geometry Balkan Press, Bucuresti, 2003.
24. [24] V. Oproiu, On the harmonic sections of cotangent bundles, Rend. Sem. Fac. Sci. Univ. Cagliari 59 (2), 177–184, 1989.
25. [25] M. P. Piu, Campi di vettori ed applicazioni armoniche, Rend. Sem. Fac. Sci. Univ. Cagliari 52, 85–94, 1982.
26. [26] A. A. Salimov, M. Iscan and F. Etayo, Paraholomorphic B-manifold and its properties, Topology Appl. 154 (4), 925-933, 2007.
27. [27] S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds, Tohoku Math. J. 10 (3), 338-354, 1958.
28. [28] J. A. Schouten and E. R. Van Kampen, Zur Einbettungs-und Krümmungstheorie nichtholonomer Gebilde, Math. Ann. 103, 752-783, 1930.
29. [29] M. Spivak, A Comprehensive Introduction to differential geometry vol 2, Publish or perish, INC Houston, Texas 1999.
30. [30] G. Vrãnceanu, Sur quelques points de la théories des espaces non holonomomes, Bull. Fac. St. Cernãuti 5, 177–205, 1931.
31. [31] K. Yano and M. Ako, On certain operators associated with tensor field, Kodai Math. Sem. Rep. 20, 414-436, 1968.
32. [32] K. Yano and S. Ishihara, Tangent and Cotangent Bundles, Marcel Dekker, Inc. New York, USA, 1973.