Tangent bundle of a hypersurface of a Euclidean space

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.