Tangent bundle of the hypersurfaces in a Euclidean space

Deshmukh, Sharief and Al-Odan, Haila and Shaman, Tahany A. (2007) Tangent bundle of the hypersurfaces in a Euclidean space. ACTA MATHEMATICA ACADEMIAE PAEDAGOGICAE NYÍREGYHÁZIENSIS, 23 (1). pp. 71-87. ISSN 0866-0174

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Abstract

We consider an immersed orientable hypersurface f : M → Rn+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 R2n+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.

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