DISTRIBUTIONS ON THE COTANGENT BUNDLE FROM TORSION-FREE CONNECTIONS

Summary: Let m and q be arbitrary integers such that m ≥ 1 and 0 ≤ q ≤ 2m. We study the problem how a torsion-free classical linear connection ∇ on an m-dimensional manifold M induces canonically a smooth (C∞) distribution A(∇) ⊂ TT ∗M on the cotangent bundle T ∗M of M such that dim(A(∇)ω) = q for any ω ∈ T ∗M. We have the following simple examples: (a) The zero distribution A[1](∇) such that A[1](∇)ω = {0} for any ω ∈ T ∗M; (b) The vertical distribution A[2](∇) = V T ∗M