Reference request: integration of *one*-forms along curves on a differentiable manifold.

Question

Could somebody please direct me to a book/lecture notes with an introduction to integration of one-forms along curves in a differentiable/Riemannian manifold -- preferably leaning more towards analytical side rather than algebraic? I'm aware that there are many books dealing with calculus on manifolds but the ones I have looked at seem to deal with integration of general $k$-forms -- which is a bit too complicated for me (I'm a young probabilist who deals exclusively with paths). Likewise, I've found several resources which deal with integration of curves in $\mathbb R^n$ as opposed to general differentiable manifolds, but again this is not quite what I want. Many thanks!

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EDIT: To explain myself better, in probability the type of paths that arise are typically continuous paths which are nowhere differentiable (usually only Holder continuous or of finite $p$-variation), so in particular do not have classical tangent vectors along the path. So of course our integration theory requires more structure on the manifold. For example, let $G$ be a Lie group which I am thinking of as being a topological subgroup of some Banach algebra $A$. Then if I have a (nondifferentiable) path in the group $X:[0,1]\to G$, I can describe its increments using the group operation in the following way $X_{s,t}:=X_s^{-1}\circ X_t$. Then I can define a 'one-form' as a (smooth) map $\alpha:G\to \mathbb L(A,\mathbb R)$. It turns out that if $X$ is not too rough (of finite $p$-variation for some $p<2$ for example), the following is actually well-defined $$\int_0^1\alpha(X_u)\mathrm{d}X_u:=\lim_{|D|\to 0}\sum_{[t_i,t_{i+1}]\in D}\alpha_{t_i}(X_{t_i})X_{t_i,t_{i+1}},$$ where $D$ is a finite partition of $[0,1]$.

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What I'm seeking is something that considers such constructions and answers standard analytical questions (what norm can we put on the $\alpha$'s to have continuity, can we weaken the condition of $\alpha$ being smooth to say $C^k$ and what's the optimal $k$ given the roughness of $X$ etc.?).

Answer 1

I can't think of a reference for this in particular, but (assuming basic familiarity with manifolds) I feel like it's a simple enough topic to just explain.

Given a differentiable manifold $M$, a curve in $M$ is a differentiable function from some interval $[a,b]$ to $M$, and comes equipped with a tangent vector $\dot\gamma(t) = d\gamma(t)/dt$.

A one-form on $M$ is a function that takes a vector tangent to $M$ and produces a real number. (A one-form $\theta$ must also be linear at each point - in the Riemannian case, this is equivalent to being of the form $\theta(v) = X \cdot v$ for some vector field $X$).

These two objects pair together quite well: we can compose our one-form (a map $TM \to \mathbb R$) with our tangent vector field (a map $[a,b] \to TM$) to get a familiar function $[a,b] \to \mathbb R$. The integral of the one-form along the curve is then simply the integral of this real-valued function; i.e. $$\int_\gamma \theta = \int_a^b \theta(\dot \gamma(t)) dt.$$

One intuition here is that $\theta$ is some way of "measuring" tiny vectors, and the integral of $\theta$ along $\gamma$ is what you get when you split $\gamma$ up in to tiny vectors, measure them all with $\theta$ and then add up the results.

Answer 2

You might look at Chapter 11 in my Introduction to Smooth Manifolds (2nd ed.), which has an extensive discussion of integration of $1$-forms along curves.

Answer 3

The only case I am aware of when you can integrate a smooth $1$-form $\omega$ along a continuous path with no differentiability property is when $\omega$ is closed: $d\omega=0$.
You then have the
Recipe
Consider the universal covering $p:\tilde X\to X$ of your manifold, lift your path $\gamma:[0,1]\to X$ to a path $\tilde {\gamma}:[0,1]\to \tilde X$, find a primitive $F\in \mathcal C^\infty (\tilde X)$ for $\tilde \omega$ (i.e. $dF=\tilde \omega$), and define $$\int_\gamma \omega=F(\tilde {\gamma}(1))-F(\tilde {\gamma}(0))$$

Explanations for the cook
The existence of a primitive $F$ for the closed form $\tilde \omega$ follows from the triviality of the De Rham cohomology space $H^1_{DR}(\tilde X)=0$, due to the simple-connectedness of $\tilde X$.
The integral is obviously independant of the choice of $F$ since two choices differ by a constant.
Finally the definition is independant of the lifting $\tilde {\gamma}$ of $\gamma$ since the univeral covering $p:\tilde X\to X$ is Galois (=normal), so that one lifting can be transformed to any other one by an automorphism of the covering .