Smooth Embedding

Question

I need help showing that the following smooth map is not a smooth embedding: $f:\mathbb{S}^1 \to \mathbb{R}$ defined by $f(z)= \operatorname{Re}(z)$.

I know that this map is not a submersion because the $dF$ vanishes at $0$, and I know it is not a covering map because it doesn't cover all of $\mathbb{R}$, just an interval of it.

Thanks for any help!

Answer 1

The same reason that you gave for $f$ not being a submersion also tells you that $f$ is not an immersion (indeed, for maps between manifolds of the same dimension the concepts of immersion and submersion coincide). Any smooth embedding is in particular an immersion; and your map isn't.