Iterated path integrals

The classical calculus of variation is a critical point theory of certain differentiable functions (or functional) on a smooth or piecewise smooth path space, whose differentiable structure is defined implicitly. Because of the importance of path spaces to analysis, geometry and other fields, it is desirable to develop a geometric integration theory or a de Rham theory for path spaces. Having in mind this general goal, we are going to consider a large class of path space differential forms, which can be constructed from usual differential forms by a method of iterated integration.