In this paper, we introduce new results in the analysis of convergence of nonlinear systems. The point of view we take is the one of contraction theory and we focus in particular on convergence to smooth manifolds. A main characteristic of contraction theory is that it does not require nor use any knowledge about the asymptotic trajectory of the system. Our contribution is to extend the core body of contraction results to include such knowledge in the analysis. As a result, this approach naturally leads to the definition of a new type of commutator for vector fields. We will show that the vanishing of this commutator, together with a contraction assumption, yields a sufficient condition for convergence and we will illustrate the results on the Andronov-Hopf oscillator.