References for integrating Lie-group valued paths I have a suspicion that integration of curves on a Lie group should be a fairly well treated subject. More specifically, what I need is a generalisation of the integral $\int y_t dx_t$ or $\int f(x_t) dx_t$ for Lie-group valued paths. 
In the flat case, given two Banach spaces $E$ and $F$ and paths $x:[0,T] \to E$ and $y:[0,T]\to \mathcal{L}(E,F)$ (both satisfying some regularity conditions), one can make sense of the above integral by using a linear Riemann sum $\sum y_{t_i}(x_{t_{i+1}}-x_{t_{i}})$.
Now take $G_1$ and $G_2$ two Lie groups (with "norms" if absolutely necessary), and paths $x:[0,T] \to G_1$ and $y:[0,T]\to \mathcal{F}(G_1,G_2)$ (the latter being some space of maps between $G_1$ and $G_2$), does there exist already a theory making sense of the integral using for example a generalised Riemann sum $\prod y_{t_i}(x_{t_{i}}^{-1} x_{t_{i+1}})$? I would be very surprised if none existed. I came upon references that seem to insinuate somehow that such a thing could exist, but never in a direct manner. If not, is this a consequence of a theory of integration on manifolds?
 A: I think this is pretty much relegated to the discussion of integrating forms on manifolds.  For a general manifold you are given a $k$-form $\omega$ which in a coordinate frame $(x_1, \ldots, x_n)$ may be expressed as 
$$
\omega \;\; =\;\; \omega_{i_1, \ldots, i_k} dx^{i_1}\wedge\ldots \wedge dx^{i_k}
$$
where summation is implied over the indices $(i_1, \ldots, i_k)$, and the $\omega_{i_1,\ldots,i_k}$ forms a  parameter-dependent set of functions representing the $k$-form.  Integration here is then interepreted as integrating these functions over the desired indices:
$$
\int_U \omega \;\; =\;\; \int_{\varphi^{-1}(U)} \omega_{i_1,\ldots,i_k}dx^{i_1}\ldots dx^{i_k}
$$
where $\varphi:\mathbb{R}^n \to U\subseteq M$ represents the specific parameterization in which $\omega$ is written in.
Now, when we adapt this to Lie groups this formalism still holds.  The only additional thing to add is when dealing with a Riemannian volume form we can integrate functions $f \in C^\infty(M)$ as:
$$
\int_Uf \text{dVol}_M \;\; =\;\; \int_{\varphi^{-1}(U)} (f\circ \varphi^{-1})\sqrt{\det g}\text{dVol}_{\mathbb{R}^n}
$$
where $g$ is the Riemannian metric and $\sqrt{\det g}$ is in viewing $g$ through its matrix representation as a bilinear, symmetric positive-definite form.  When we discuss Lie groups, $g$ is often taken to be the left-invariant metric with respect to the group structure, and we can also interpret this through the lens of left-invariant volume form.   
