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It is known that for any smooth, orientable, compact manifold $X$ without boundary and $\alpha \in \Omega^{r}(X), \beta \in \Omega^{r-1}(X)$ it holds

\begin{equation} (d\beta,\alpha)= (\beta, d^{\dagger}\alpha), \end{equation}

where

\begin{equation} (d\beta,\alpha) = \int_X d\beta \wedge \star \alpha \end{equation}

and $d^{\dagger}$ (acting on a $r-$form) such that

\begin{equation} d^{\dagger} = (-1)^{mr+m+1}\star d \star \end{equation}

with $m=\dim X$ for $X$ Riemannian and with an additional $(-1)$ for Lorentzian manifolds.

This is actually not so hard to prove and I don't see any further assumptions required.

Now, suppose I have a manifold $X$ with the assumptions above (if possible!) but which admits torsional cycles. For instance, to the corresponding torsional cycle one can then consider a globally well-defined $r-$form $\alpha_r$ s.t. $d\alpha_r = k \beta_{r+1}$ with some integer $k$. Is it then still possible to write for example

\begin{equation} (d\alpha_r, d\alpha_r) = (\alpha_r,d^{\dagger}d\alpha_r)? \end{equation}

Can the existence of torsional cycles spoil this relation? Is the integration well-defined at all? I mean, I could imagine that one constructs a torus by twisting one end of a cylinder when identifying both ends. Then, intuitively, I would doubt that this surface is orientable and allows integration of forms over this manifold.

Unfortunately, I almost don't know anything about torsional cycles and forms. Hence, I'm sorry if my question is sloppy and not precise. This is also, why I'm asking it here in the physics section: 1. At this level, I probably can't follow a mathematician. 2. This question is related to field theory.

I'd be grateful for any help and hints. Thanks!

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