The usual iterated integral map given by Chen produces an equivalence between the two-sided bar complex on differential forms and the de Rham complex on the path space. This map fails to make sense when considering the curved differential graded algebra of bundle-valued forms with a covariant derivative induced by a connection. In this paper, we define a curved version of Chen’s iterated integral that incorporates parallel transport and maps an analog of the two-sided bar construction on bundle-valued forms to bundle-valued forms on the path space. This iterated integral is proven to be a homotopy equivalence of curved differential graded algebras, and for real-valued forms it factors through the usual Chen iterated integral.

Chen 给出的通常迭代积分图在微分形式上的两侧杆复形与路径空间上的 de Rham 复形之间产生等价。当考虑具有由连接引起的协变导数的丛值形式的弯曲微分分级代数时，该映射没有意义。在本文中，我们定义了 Chen 迭代积分的曲线版本，它结合了并行传输，并将束值形式上的两侧条结构的模拟映射到路径空间上的束值形式。该迭代积分被证明是曲线微分分级代数的同伦等价，并且对于实值形式，它通过通常的 Chen 迭代积分进​​行因式分解。