Integration of paths—a faithful representation of paths by non-commutative formal power series
Kuo-sai Chen · Transactions of the American Mathematical Society · 1958
In the w-dimensional affine space 3im with the coordinates (xx, • • ■ , xm), the product a-8 of two curves a and 8 is the curve a followed by 8, and the inverse or1 is obtained from a by changing the orientation of a.A curve is irreducible if it cannot be expressed in the form a-y-y^-S, where a, 8, y are curves.As in [3], we associate to each curve a the formal power seriesin the noncommutative indeterminates X\, • • • , Xm.The main result of this paper is briefly as follows: If 6(a) =0(8) tor two irreducible piecewise C1 and continuous curves a and 8, then 8 can be obtained from a by translation.We begin with considering curves in a differentiable manifold 9ft.For the purpose of proving the main result, one may replace the manifold 9ft by the affine space dtm in all arguments.The main result relies heavily upon Fundamental Lemma which asserts that, for each irreducible curve a, there is some iterated integral not vanishing along a.This lemma is indeed the center of this paper, and the majority of the definitions and lemmas in § §1-3 are aimed at its proof.If coi, • • • , wm are differentials in a differentiable manifold 9ft, then we may define for a curve a in 9ft, the formal power series S(a) = 1 + JZ JZ f «4 • * • «V*«.• * * XH-p=*l ** a