Integral of Hodge Star of Differential Form over Submanifold of Complementary Dimension which Intersects Transversely Let $([M]^n, g)$ be a closed, smooth, oriented, Riemannian manifold, and let $[S]^k$ and $[T]^{n-k}$ be closed, smooth, oriented submanifolds of $M$ which intersect transversely. Let $\omega$ be a differential $k$-form on $M$ with $\displaystyle \oint\limits_{[S]} \omega = r \ne 0$. Let $\eta$ be the Hodge star of $\omega$.
What can be said about $\displaystyle \oint\limits_{[T]} \eta$? Would it help if the $\mathbb{Z}\pi$-intersection number of $S$ and $T$ were $\pm 1$ (where $\pi = \pi_1(M)$)?
 A: As a partial answer, in the case $M = S \times T$, let $[S]$ and $[T]$ be $S$ and $T$ with orientations, let $[M]$ be $M$ with its induced orientation from $[S]$ and $[T]$, and let $\zeta_0$ be the volume form on $[M]$. Then every $n$-form on $M$ may be written $f\xi_0$ for some function $f: M \to \mathbb{R} \in \Lambda^0(M)$. Similarly, let $\omega_0$ and $\eta_0$ be the volume forms on $[S]$ and $[T]$ respectively. Then 1) $\omega_0 \wedge \eta_0 = \zeta_0$, 2) $\displaystyle\text{Vol}(M) = \oint\limits_{[M]} \zeta_0 = \oint\limits_{[S] \times [T]} \omega_0 \wedge \eta_0 = \left(\oint\limits_{[[S]]} \omega_0\right)\left(\oint\limits_{[T]} \eta_0\right) = \text{Vol}(S)\text{Vol}(T)$, and 3) for any forms $\omega \in \Lambda^k(S)$ and $\eta \in \Lambda^{n-k}(T)$, we have functions $g_1$ and $g_2$ with $\omega = g_1\omega_0$ and $\eta = g_2\eta_0$. Hence, $\displaystyle \oint\limits_{[S]} \omega = \oint\limits_{[S]} g_1\omega_0 = \pm\left(\int\limits_S g_1\right)\text{Vol}(S)$.
Suppose $\displaystyle \oint\limits_{[S]} \omega = r \ne 0$
Now, ${}^*\omega$ has $\omega \wedge {}^*\omega = \langle\omega,\omega\rangle\zeta_0 = \langle\omega,\omega\rangle(\omega_0 \wedge \eta_0) = g_1^2(\omega_0 \wedge \eta_0)$, so ${}^*\omega = g_1\eta_0$. Hence, $\displaystyle \oint\limits_{[T]} {}^*\omega = \oint\limits_{[T]} g_1\eta_0 = \pm\left(\int\limits_T g_1\right)\text{Vol}(T) = \pm\frac{\left(\int\limits_T g_1\right)\left(\int\limits_S g_1\right)\text{Vol}(S)\text{Vol}(T)}{\left(\int\limits_S g_1\right)\text{Vol}(S)} = \pm\frac{\left(\int\limits_T g_1\right)\left(\int\limits_S g_1\right)\left(\oint\limits_{[S]} \omega_0\right)\left(\oint\limits_{[T]} \eta_0\right)}{\left(\int\limits_S g_1\right)\text{Vol}(S)} = \pm\frac{\left(\int\limits_{S \times T} g_1^2\right)\left(\oint\limits_{[S] \times [T]} \omega_0 \wedge \eta_0\right)}{\left(\int\limits_S g_1\right)\text{Vol}(S)} = \pm\frac{\left(\int\limits_{M} g_1^2\right)\left(\oint\limits_{[M]} \zeta_0\right)}{\left(\int\limits_S g_1\right)\text{Vol}(S)} = \pm\frac{\langle\omega,\omega\rangle\text{Vol}(M)}{r}\ \blacksquare$
