Abstract
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 (g) over bar as submanifold of the Euclidean space R-4n which is not a natural metric in the sense that the submersion pi : (TM, (g) over bar) -> (M,g) is not the Riemannian submersion. In this paper, we use the fact that R4n is the tangent bundle of the Euclidean space R-2n to define a special complex structure (J) over bar on the tangent bundle R-4n so that (R-4n, (J) over bar,<,>) is a Kaehler manifold, where <,> 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, (g) over bar) of the hypersurface M, which is a submanifold of the Kaehler manifold (R-4n, (J) over bar,<,>). We show that the tangent bundle TM is a CR-submanifold of the Kaehler manifold (R4", (J) over bar,<,>). We find conditions under which certain special vector fields on the tangent bundle (TM,) are Killing vector fields. It is also shown that the tangent bundle TS2n-1 of the unit sphere S2n-1 admits a Riemannian metric and and that there exists a nontrivial Killing vector field on the tangent bundle (TS2n-1, (g) over bar).