Equivalently, it is a manifold equipped with an atlas—called the affine structure—such that all transition functions between charts are affine transformations (that is, have constant Jacobian matrix); two atlases are equivalent if the manifold admits an atlas subjugated to both, with transitions from both atlases to a smaller atlas being affine.
Equivalently, it is a manifold equipped with an atlas—referred to as the affine structure—where all transition functions between charts are affine transformations (with constant Jacobian matrix), and two atlases are considered equivalent if there exists a common atlas that can be used for both, with transitions from each original atlas to this common one being linear.