Let $\alpha : I \to \mathbb R^2$, where $ I \subset \mathbb R$, be a simple closed curve with positive speed and curvature, does it follow that $\alpha’$ is also injective, as a vector field on $\alpha$, as well, except at the end points?
This came out while I was trying to prove that $\int_{a}^{b} \kappa(t) dt = 2\pi$, where $\alpha$ is such a curve with unit speed, $\kappa$ is the curvature of $\alpha$ and $I = [a,b]$. I was trying to prove that the derivative map $\alpha$ is injective to show that the integral cannot be greater than $2\pi$.
Intuitively, this should be obvious but I couldn’t find a way to show it, thank you.