摘要：K. T. Chen uses iterated integrals to construct differential forms on loop space C^∞(S¹,N), and Chen’s method becomes an important tool to study homology theory on free loop spaces, however, when the source manifolds M are any differential manifolds, Chen’s mode is not available any more for providing relevant differential forms. In this lecture I will introduce another method coming from variational complex theory to construct differential forms for any spaces C^∞(M,N), and also talk about the corresponding de-Rham cohomology theory and some open problems.