Tangent bundle of the hypersurfaces in a Euclidean space

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

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