Tangent Bundle of the Hypersurfaces in a Euclidean Space

Let M be an orientable hypersurface in the Euclidean space R^{2n} with induced metric g and TM be its tangent bundle. It is known that the tangent bundle TM has induced metric overline{g} as submanifold of the Euclidean space R^{4n} which is not a natural metric in the sense that the submersion pi :(TM,overline{g})rightarrow (M,g) is not the Riemannian submersion. In this paper, we use the fact that R^{4n} is the tangent bundle of the Euclidean space R^{2n} to define a special complex structure overline{J} on the tangent bundle R^{4n} so that % (R^{4n},overline{J},leftlangle ,rightrangle ) is a Kaehler manifold, where leftlangle ,rightrangle is the Euclidean metric which is also the Sasaki metric of the tangent bundle R^{4n}. We study the structure induced on the tangent bundle (TM,overline{g}) of the hypersurface M, which is a submanifold of the Kaehler manifold (R^{4n},overline{J},% leftlangle ,rightrangle ). We show that the tangent bundle TM is a CRsubmanifold of the Kaehler manifold (R^{4n},overline{J},leftlangle ,rightrangle ). We find conditions under which certain special vector fields on the tangent bundle (TM,overline{g}) are Killing vector fields. It is also shown that the tangent bundle TS^{2n1} of the unit sphere % S^{2n1} admits a Riemannian metric overline{g} and that there exists a nontrivial Killing vector field on the tangent bundle (TS^{2n1},% overline{g}).