The linear frame bundle over a smooth manifold is considered. The mapping dе defined by the differentials of the first-order frame e is a lift to the normal N, i. e., a space complementing the first-order tangent space to the second-order tangent space to this bundle. In particular, the mapping defined by the differentials of the vertical vector of this frame is a vertical lift into normal N. The lift dе allows us to construct a prolongation both of the tangent space and its vertical subspace into the second-order tangent space, more precisely into the normal N. The normal lift dе defines the normal prolongation of the tangent space (i. e. the normal N) and its vertical subspace. The vertical lift defines the vertical prolongation of the tangent space and its vertical subspace. The differential of an arbitrary vector field on the linear frame bundle is a complete lift from the first-order tangent space to the second-order tangent space to this bundle. It is known that the action of vector fields as differential operators on functions coincides with action of the differentials of these functions as 1-forms on these vector fields. Horizontal vectors played a dual role in the fibre bundle. On the one hand, the basic horizontal vectors serve as operators for the covariant differentiation of geometric objects in the bundle. On the other hand, the differentials of these geometric objects can be considered as forms (including tangential-valued ones) and their values on basic horizontal vectors give covariant derivatives of these geometric objects. For objects which covariant derivatives require the second-order connection, the covariant derivatives are equal to the values of the differentials of these objects on horizontal vectors in prolonged affine connectivity. Prolongations of the basic horizontal vectors, i. e., the second-order horizontal vectors for prolonged connection, were constructed. The second-order tangent space is represented as a straight sum of the first-order tangent space, vertical prolongations of the vertical and horizontal subspaces, and horizontal prolongation of the horizontal subspace.
Natural operators transforming vector fields to the second order tangent bundle
