Abstract
We use the fact that the tangent bundle TM of an orientable hypersurface M in the Euclidean space R n+1 is a submanifold of the Euclidean space R 2n+2, and use the induced metric on TM as submanifold to study its geometry. This induced metric is not a natural metric in the sense that the projection π : TM → M is not a Riemannian submersion (which holds for Sasaki and other metrics, used to study geometry of the tangent bundle). First we prove that there is a reduction in the codimension of the submanifold TM and thus the tangent bundle TM is a hypersurface of the Euclidean space R 2n+1. As a consequence of our study, we infer that the induced metric on TS n the tangent bundle of the unit sphere S n makes TS n a Riemannian manifold of nonnegative sectional curvature. We also obtain a condition under which the tangent bundle TM of a hypersurface M in a Euclidean space is trivial.
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References
Cheeger J., Gromoll D.: On the structure of complete manifolds of nonnegative curvature. Ann. Math. 96, 413–443 (1972)
Deshmukh S., Al-Odan H., Shaman T.A.: Tangent bundle of the hypersurfaces in a Euclidean space. Acta Math. Acad. Paedagog. Nyházi, (N.S.) 23(1), 71–87 (2007)
Dombrowski P.: On the geometry of the tangent bundle. J. Reine Angew. Math. 210, 73–88 (1962)
Erbacher J.: Reduction of the codimensions of an isometric immersion. J. Differ. Geom. 5, 333–340 (1971)
Gudmundsson S., Kappos E.: On the geometry of tangent bundles. Expo. Math. 20(1), 1–41 (2002)
Kowalski O.: Curvature of the induced Riemannian metric on the tangent bundle of a Riemannian manifold. J. Reine Angew. Math. 250, 124–129 (1970)
O’Neill B.: The fundamental equations of a submersion. Mich. Math. J. 13, 459–469 (1966)
Oproiu, V., Papaghiuc, N.: Locally Symmetric Space Structure on the Tangent Bundle. Differential Geometry and Application, Masaryk Univ. Brno, pp 99–109 (1999)
Sasaki S.: On the geometry of the tangent bundle of Riemannian manifolds. Tohoku Math. J., II.Ser. 10, 338–354 (1958)
Sekizawa M.: Curvatures of tangent bundles with Cheeger–Gromoll metric. Tokyo J. Math 14, 407–417 (1991)
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Deshmukh, S., Al-Shaikh, S.B. Tangent bundle of a hypersurface of a Euclidean space. Beitr Algebra Geom 52, 29–44 (2011). https://doi.org/10.1007/s13366-011-0003-4
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DOI: https://doi.org/10.1007/s13366-011-0003-4