# Is there integration of $df : TM \rightarrow TN$?

If $$f:M \rightarrow N$$ is a smooth function, there is the differentiation of it, $$df:TM \rightarrow TN$$.

Is there something like antiderivative or integration of this?

If $$N= \mathbb{R}$$, I know the integration of the differential form $$df$$.

• If v is a tangent vector at x, then df(v) is a tangent vector at f(x)... So you can just remember only the basepoints and recover f from df. For R, all of the tangent spaces are identified, so you are forgetting the basepoints and have to do more to recover the map (and then only up to a constant translation / chosen starting point). If there was a natural way to identify the tangent spaces of N, then maybe you could ask for something analogous. (For example, if N and M are Lie groups, I think there is something you can do.) – Lorenzo Feb 7 at 15:24
• See the correspondence here (homomorphism theorem) : en.wikipedia.org/wiki/… ... This is much less general than what you are asking for, however. – Lorenzo Feb 7 at 19:11

Integration is, in some sense, summation pushed to infinity. When you integrate functions $$X \rightarrow \mathbb R$$, you secrely use the structure of vector space on $$\mathbb R$$. In general, to integrate a function $$f : X \rightarrow Y$$, you need $$Y$$ to be a vector space (or something similar - exact definitions may differ), so that you can form linear combinations of values of $$f$$.
While $$TN_p$$ is actually a vector space (where $$p \in N$$), it is a separate vector space for each point $$p$$, and there is, in general, no well-defined operation that adds vectors from two unrelated vector spaces. Since your function $$f : M \rightarrow N$$ can map different points of $$M$$ to different points of $$N$$, $$df$$ will map tangent spaces of different points of $$M$$ to tangent spaces of different points of $$N$$, so adding the values of $$df$$, in general, is undefined, and no integration can be defined.
• $TM,TN$ are vector bundles, so if this structure is not enough, you should explain why. – Wojowu Feb 7 at 15:23