Affine Connection

An affine connection on M is a principal Aff(n)-bundle Q over M, together with a principal GL(n)-subbundle P of Q and a principal Aff(n)-connection α (a 1-form on Q with values in aff(n)) which satisfies the following (generic) Cartan condition. The Rn component of pullback of α to P is a horizontal equivariant 1-form and so defines a bundle homomorphism from TM to P ×_GL(n) Rn: this is required to be an isomorphism.