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Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, Vol. 23, No. 1, pp. 71-87 (2007)
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Tangent bundle of the hypersurfaces in a Euclidean space

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Sharief Deshmukh, Haila Al-Odan and Tahany A. Shaman

King Saud University, Riyadh

**Abstract:** We consider an immersed orientable hypersurface $f\colon M\rightarrow R^{n+1}$ of the Euclidean space ($f$ an immersion), and observe that the tangent bundle $ TM$ of the hypersurface $M$ is an immersed submanifold of the Euclidean space $R^{2n+2}$. Then we show that in general the induced metric on $TM$ is not a natural metric and obtain expressions for the horizontal and vertical lifts of the vector fields on $M$. We also study the special case in which the induced metric on $TM$ becomes a natural metric and show that in this case the tangent bundle $TM$ is trivial.

**Keywords:** Tangent bundle, hypersurfaces, submanifolds, trivial tangent bundle.

**Classification (MSC2000):** 53C25; 53C55

**Full text of the article:**

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