Let $f : T^2 \to S^3$ be the smooth map of a 2-torus into $S^3$. Let $\omega$ be a closed 2-form on $S^3$. Show that $$\int_{T^2}f^*\omega = 0.$$
So apparently, if I can use the theorem on Guillemin and Pollack's Differential Topology, so I change $\int_{T^2}f^*\omega$ to $\int_{S^3}\omega$ then I can prove it by Stokes Theorem.
Theorem [Page 168] If $f: Y \to X$ is an orientation-preserving diffeomorphism, then $$\int_X \omega = \int_Y f^* \omega$$ for every compactly supported, smooth $k$-form on $X$ ($k = \dim X = \dim Y$).
However, the problem is, how can I justify $f$ is an orientation-preserving diffeomorphism?
Thank you. :-)
The first fact you need to know is that every closed $2$-form on $S^3$ is exact. If you're happy with algebraic topology, then this is precisely the fact that $H^2_{\text{dR}}(S^3) = 0$. Hence, $\omega = d\eta$ for some $1$-form $\eta$ on $S^3$.
The second fact you need to know is that taking pullbacks commutes with taking exterior derivatives. Hence, $$ f^\ast \omega = f^\ast d \eta = d f^\ast \eta. $$
Now, applying this last observation, observe that $$ \int_{T^2} f^\ast \omega = \int_{T^2} d f^\ast \eta. $$ So, what is $\partial T^2$, and what does this imply about $\int_{T^2} d f^\ast \eta$?