Period of a smooth one-form along a closed curve

Let $M$ be a smooth $n$-manifold, meaning a topological $n$-manifold equipped with a maximal smooth atlas, and let $\omega \in {\mathrm{\Omega }}^1(M)$ be a smooth real one-form, meaning a smooth section of the cotangent bundle $T^{\ast }M\to M$. Let $\gamma :[0,1]\to M$ be a continuous piecewise-$C^1$ curve with $\gamma (0)=\gamma (1)$: there is a finite subdivision $0=t_0<\cdots <t_m=1$ such that on each closed piece a chart contains the curve image, its coordinate representative is $C^1$ on the piece interior, and its derivative extends continuously to the endpoints. At a piece endpoint, use the corresponding one-sided derivative as the tangent. On each piece, pair this tangent $\dot{\gamma }(t)\in T_{\gamma (t)}M$ with ${\omega }_{\gamma (t)}\in T_{\gamma (t)}^{\ast }M$. Define the oriented period by

${\mathrm{Per}}_{\gamma }(\omega ):={\sum }_j{\int }_{t_j}^{t_{j+1}}{\omega }_{\gamma (t)}(\dot{\gamma }(t))dt,$

where $0=t_0<\cdots <t_m=1$ is any admissible chartwise subdivision with the regularity specified above. The covector-vector pairing is intrinsic, and additivity makes this value independent of further subdivision. No invariance under homotopy is asserted by this definition.

Smooth manifolds and their smooth charts

A smooth differential $k$-form

Piecewise c one curve on a manifold

A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion

For $a<c<b$: $f$ is integrable on $[a,b]$ if and only if it is integrable on $[a,c]$ and on $[c,b]$, and then ${\int }_a^{b}f={\int }_a^{c}f+{\int }_c^{b}f$; with the oriented form for arbitrary $a,b,c$