An abstract characterization of line integrals

I don’t know if it’s exactly what you’re looking for, but line integration is the unique way to assign a real number $I(omega,c)inmathbb{R}$ to every pair of a smooth $1$-form $omega$ on a smooth manifold $M$ with boundary and smooth path $ccolon[0,1]to M$ such that:

  • (adjunction) if $fcolon Mto N$ is a smooth map of smooth manifolds with boundary, $omega$ is a smooth $1$-form on $N$, and $ccolon[0,1]to M$ is a smooth path in $M$, then$$I(f^*omega,c)=I(omega,fcirc c)$$
  • (normalisation) if $M=[0,1]$, $mathbf 1colon[0,1]rightarrow[0,1]$ is the identity path, and $omega=g(x)mathrm{d}x$ for a smooth function $g$, then $I(omega,mathbf 1)=int_0^1g(x)mathrm{d}x$, where the integral denotes the usual Riemann line integral.

(To show this characterises line integration uniquely, apply the adjunction formula to the smooth path $ccolon[0,1]to M$ to show that $I(omega,c)=I(c^*omega,mathbf 1)=int_0^1c^*omega$.)


Remark:

This is the approach that one takes when defining iterated integration of a sequence $omega_1,dots,omega_n$ of $1$-forms along a path $c$. For instance, we know how to double-integrate over the interval $[0,1]$: the double-integral of $g(x)mathrm dx$ and $h(x)mathrm dx$ is $int_0^1left(int_0^xg(y)mathrm dyright)h(x)mathrm dx$, and by demanding the same adjunction relation you get a way to define a double line integral $I(omega_1omega_2,c)$ for all pairs of smooth $1$-forms $omega_1,omega_2$ on a manifold $M$ with boundary, and all smooth paths $ccolon[0,1]to M$. For more details, see the works of Kuo-Tsai Chen, who was the first to develop this theory systematically

I’ll suggest here another possible characterisation, expanding on a suggestion of the OP in one of the comments. Again, this is an assertion that certain known properties of line integration characterise it uniquely; this doesn’t provide a “new” construction of line integration. Unlike my previous answer, here all the action takes place on the one manifold $M$.

To avoid various technicalities, I’m going to redefine $mathcal C_M$ to be the set of immersed paths, i.e. smooth paths $ccolon[0,1]to M$ such that $dot c(t)neq0$ for all $tin[0,1]$. I think this restriction could probably be removed with enough effort.

Theorem:

For any manifold $M$, line integration is the unique function $IcolonOmega^1(M)timesmathcal C_Mtomathbb R$ satisfying the following properties:

  • (additivity in the path) Suppose that $c_1$ and $c_2$ are two immersed paths that are composable, i.e. all the derivatives $c_1^{(i)}(1)=c_2^{(i)}(0)$. Then $I(omega,c_1c_2)=I(omega,c_1)+I(omega,c_2)$ for all $omegainOmega^1(M)$. Here $c_1c_2$ denotes the composite path, defined by$$c_1c_2(t)=begin{cases}c_1(2t)&0leq tleq1/2\c_2(2t-1)&1/2leq tleq1.end{cases}$$
  • (additivity in the $1$-form) We have $I(omega_1+omega_2,c)=I(omega_1,c)+I(omega_2,c)$ for all $omega_1,omega_2inOmega^1(M)$ and all $cinmathcal C_M$.
  • (locality) If $omegainOmega^1(M)$ satisfies $omega_{c(t)}(dot c(t))=0$ for all $tin[0,1]$, then $I(omega,c)=0$.
  • (exact forms) If $fcolon Mtomathbb R$ is smooth, then $I(mathrm df,c)=f(c(1))-f(c(0))$.

The proof uses two lemmas.

Lemma 1: Let $c$ be an immersed path. Then there is a non-negative integer $N$ such that for all $0leq k<2^N$, the restriction $c|_{[2^{-N}k,2^{-N}(k+1)]}$ of $c$ to the interval $[2^{-N}k,2^{-N}(k+1)]$ is an embedding.

Proof (outline): This follows from the standard fact that an immersion is locally an embedding (see e.g. this MO question), and that $[0,1]$ is compact.

Lemma 2: Let $c$ be an embedded path in $M$ and $omegainOmega^1(M)$. Then there exists a smooth function $fcolon Mtomathbb R$ such that $omega_{c(t)}(dot c(t))=mathrm df_{c(t)}(dot c(t))$ for all $0<t<1$.

Proof: The pullback $c^*omega$ is a smooth $1$-form on $[0,1]$, hence is $mathrm df_0$ for some smooth $f_0colon[0,1]tomathbb R$. We want to show that $f_0$ extends to a smooth map $fcolon Mtomathbb R$ (i.e. $f_0=fcirc c$).

To do this, we first extend $c$ to a smooth map $ccolon(-epsilon,1+epsilon)to M$ for some $epsilon>0$. This is possible by Borel’s Lemma, which says that we can choose smooth maps $(-epsilon,0]to M$ and $[1,1+epsilon)to M$ having the same higher-order derivatives at $0$ and $1$ as $c$, respectively.

Decreasing $epsilon$ if necessary, we may even assume that $ccolon(-epsilon,1+epsilon)hookrightarrow M$ is an embedding. The tubular neighbourhood theorem then implies that the embedding $c$ extends to an embedding $tilde ccolon(-epsilon,1+epsilon)times (-1,1)^{d-1}hookrightarrow M$, where $d=dim(M)$. In other words, we have $c(t)=tilde c(t,0,dots,0)$ for all $t$.

We now extend $f_0$ as follows. By Borel’s Lemma again, we may extend $f_0$ to a smooth function $f_0colon(-epsilon,1+epsilon)tomathbb R$, and then extend this again to a smooth function $f_0colon(-epsilon,1+epsilon)times(-1,1)^{d-1}tomathbb R$. Multiplying by an appropriate bump function if necessary, we may assume that $f_0$ vanishes outside $(-frac12epsilon,1+frac12epsilon)times(-frac12,frac12)^{d-1}$.

We’ve now constructed an extension $f=f_0circtilde c^{-1}$ of $f$ on the open neighbourhood $mathrm{im}(tilde c)$ of the image of $c$. Moreover, we’ve ensured that this extension has compact support (it vanishes outside a compact subspace), so we can extend $f$ to all of $M$ by specifying that it is $0$ outside $mathrm{im}(tilde c)$. This yields the desired $f$. This proves Lemma 2.


Proof of Theorem: We show unicity. Let $I$ and $I’$ be two functions $Omega^1(M)timesmathcal C_Mtomathbb R$ which satisfy the given conditions. We need to show that $I(omega,c)=I'(omega,c)$ for all $omegainOmega^1(M)$ and all immersed paths $c$.

To do this, suppose first that $c$ is embedded. By Lemma 2 we can choose a smooth map $fcolon Mtomathbb R$ such that $omega_{c(t)}(dot c(t))=mathrm df_{c(t)}(dot c(t))$ for all $cin[0,1]$. Using additivity in the $1$-form, locality and the condition about exact forms, we find that $I(omega,c)=I(mathrm df,c)=f(1)-f(0)$. Since the exact same argument applies to $I’$, we have $I(omega,c)=I'(omega,c)$.

Now let us deal with the general case. By Lemma 1 we can choose a non-negative integer $N$ such that $c|_{[2^{-N}k,2^{-N}(k+1)]}$ is an embedded path for all $0leq k<2^N$. A repeated application of the additivity property implies that $I(omega,c)=sum_{k=0}^{2^N-1}I(omega,c|_{[2^{-N}k,2^{-N}(k+1)]})$ and similarly for $I’$. Since we already know that $I$ and $I’$ agree on embedded paths, we obtain that $I(omega,c)=I'(omega,c)$, as desired. This concludes the proof.


Remark:

If one only cares about the integrals of closed 1-forms, then this whole setup can be significantly simplified: one doesn’t need to restrict to immersed paths, and one can replace the locality condition above with the more natural condition:

  • (locality’) If $omega$ vanishes on an open neighbourhood of the image of $c$, then $I(omega,c)=0$.

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