Induced Vector Fields in a Hypersurface of Riemannian Tangent Bundles

Abstract

In 1963 K. Yano and E.T. Davies [6] considered the notions of horizontal and vertical vectors in the tangent bundle T(M) of an n-dimensional differentiable Riemannian manifold M. That is, they introduced the horizontal and vertical lifts of a vector field attached to M. The notion of a complete lift coincides with the notion of extended vector appearing in Sasaki [4] . The conditions were examined in order that the lifts to be parallel, closed, harmonic and Killing vectors in relation to corresponding properties of the original vector fields defined on M. On the one hand, the author investigated the induced vector fields in a Riemannian hypersurface Mn- 1 of a Riemannian manifold M([1], [2]). That is, the vector fields in Mn- 1 induced from a gradient vector field, a parallel vector field and a Killing vector field have the same properties as the original ones under certain conditions. In this paper we show that the induced vector fields in a hypersurface T(M) 2n-1 from some lifts of the original vector field in M to T(M), with the Sasakian metric, have the same properties as the original ones under certain conditions.