Abstract. This paper contains a classification of all affine liftings of torsion-free linear connections on n-dimensional manifolds to any linear connections on Weil bundles under the condition that n ≥ 3.A lifting (the so-called complete lifting) of connections to Weil bundles was constructed long ago in [5]. But the problem of finding all such liftings is still unsolved and seems to be hard. The recent result [2] on linear liftings of symmetric tensor fields of type (1, 2) to Weil bundles enables us to cope with a very special case of this problem, namely finding all affine liftings of torsion-free connections to Weil bundles.Let A be a Weil algebra inducing the Weil functor T A (see [4]) and let n be a non-negative integer. We will denote by Co M the set of all linear connections on a manifold M and by ToFrCo M the set of all torsion-free linear connections on M.A lifting of torsion-free linear connections to linear connections on T A is, by definition, a family of maps L M : ToFrCo M → Co T A M indexed by all n-dimensional manifolds and satisfying (1)
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