Local diffeomorphism between a disk and a sphere

Question

This may be a silly question, but I’ll make it anyway. Let $f: D^2 \to S^2$ be a local diffeomorphism between the closed unit disk and the unit sphere. Is it necessarily injective?

Answer 1

No, I don't think so. For instance, think about stretching out $D^2$ into a very long and thin oval and wrapping it twice around the equator of $S^2$. This wrapping is locally a diffeomorphism, but it is not injective. If this is unclear, I can try to attach a picture.

You could imagine wrapping $[0,4\pi]\times [\pi/4,3\pi/4]$ around the equator of $S^2$ using the usual spherical coordinate parametrization where we view $[0,4\pi]$ as the $\theta$ coordinate and $[\pi/4,3\pi/4]$ as the azimuthal coordinate $\phi$.