Abstract. In this paper rst we propose a formula to lift a connection on M to its higher order tangent bundles TrM, r 2 N. More precisely, for a given connection r on TrM, r 2 N [ f0g, we construct the connection rc on Tr+1M. Setting rci = rci1 c, we show that rc1 = lim rci exists and it is a connection on the Frechet manifold T1M = lim TiM and the geodesics with respect to rc1 exist. In the next step, we will consider a Riemannian manifold (M; g) with its Levi-Civita connection r. Under suitable conditions this procedure gives a sequence of Riemannian manifolds f(TiM, gi)gi2N equipped with a sequence of Riemannian connections frcigi2N. Then we show that
