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There are well known classicalexamples of vector fields on the tangent bundle TM which can be constructed from a vector field on the base manifold M, namely the verticalliftand the complete lift. Furthermore, if we consider the tangent bundle over an affine manifold (M, 7), we can define the horizontal lift of a vector field on (M, 7) to TM. As we shall see in Section 1, the classicalconstructions are examples of " natural transformations of the second order ". We have two goals in this paper. The first is to describe explicitly all second order natural transformations of vector fieldson manifolds into vector fieldson their respective tangent bundles. The second is to describe explicitly all pointwise second order natural transformations of vector fieldson manifolds with symmetric affine connections to vector fields on their respective tangent bundles. As we have done in previous papers [1], [2], [7], we shall use for our purposes the concepts and methods developed by D. Krupka [3]-[5]. This leads to a system of partial differential equations to solve, and to the problem of geometric interpretation of all solutions. Our main results are formulated in Theorems 2.4 and 3.3. I would like to thank Professor 0. Kowalski and Professor D. Krupka for some helpful discussions.

Natural transformations of vector fields on manifolds to vector fields on tangent bundles