Tangent Bundle of Order Two and Biharmonicity
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Abstract
The problem studied in this paper is related to the biharmonicity of a section from a Riemannian manifold $(M,g)$ to its tangent bundle $T^{2}M$ of order two equipped with the diagonal metric $g^{D}$. We show that a section on a compact manifold is biharmonic if and only if it is harmonic. We also investigate the curvature of $(T^{2}M, g^{D})$ and the biharmonicity of section of $M$ as a map from $(M,g)$ to $(T^{2}M, g^{D})$.
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How to Cite
Elhendi, H., Terbeche, M., & Djaa, M.
(2014).
Tangent Bundle of Order Two and Biharmonicity.
Acta Mathematica Universitatis Comenianae, 83(2), 165-179.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/86/46
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References
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2. Djaa M. and Gancarzewicz J., The geometry of tangent bundles of order r, Boletin Academia, Galega de Ciencias, Espagne, 4 (1985), 147-165.
3. Djaa N. E. H., Ouakkas S. and Djaa M., Harmonic sections on the tangent bundle of order two, Ann. Math. Inform. 38 ( 2011), 15-25.
4. Eells J. and Sampson J.H., Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109-60.
5. Ishihara T., Harmonic sections of tangent bundles, J. Math. Univ. Tokushima 13 (1979), 23-27.
6. Jiang G. Y., Harmonic maps and their rst and second variational formulas, Chinese Ann. Math. Ser. A 7 (1986) 389-402.
7. Konderak J. J., On Harmonic Vector Fields, Publications Matmatiques 36 (1992), 217-288.
8. Miron R., The Geometry of Higher Order Hamilton Spaces, arXiv: 1003.2501 [mathDG], 2010.
9. Oproiu V., On Harmonic Maps Between Tangent Bundles, Rend. Sem. Mat, 47 (1989), 47-55.