Pullback on differentials of the 2-Sphere

Question

I'm considering the diffeomorphism of the 2-Sphere given by the antipodal map and the pullback given by this map on the differentials $d\theta$ and $d\phi$. Let $\psi$ be such a map. My intuition tells me that $\psi^*d\phi=-d\phi$ and the same for the pullback of $d\theta$. However, I am having trouble coming up with an explicit construction of $\psi$, and showing the results rigorously. First, is my intuition correct? And second how should this be constructed? I was thinking something along the lines of $(\theta,\phi)\to\left(\pi-\theta,\pi+\phi ~~(mod~~2\pi)\right)$. But this isn't going to work...any thoughts?

Answer 1

You need only that $$\psi^* df = d \psi ^* f = d (f\circ \psi)$$ for any function $f$. Now apply $f =\theta$ and $f = \phi$ to get

$$\psi^* d\theta = -d\theta \ \ \text{and } \ \ \psi^* d\phi = d\phi.$$