According to Hilbert, the $\mathbb{H}^2$ cannot be embedded into $\mathbb{R}^3$. But then there's Nash-Kuiper theorem, which states that a $C^1$ embedding exists.
How to reconcile these two results? My understanding so far is that measuring curvature requires at least $C^2$ embedding. Is that the difference here?
Follow-up questions:
- How does the Nash embedding actually look like? Or is it just an existence result?
- Nash embedding is still isometric, meaning I can measure tangent vectors. Intuitively that seems to imply that I can do things such as measuring curve lengths, angles, areas. But wouldn't that mean that I'm effectively able to measure curvature? What am I missing here?