Hilbert's Theorem tells us that we cannot immerse a surface of constant negative curvature into $\mathbb{R}^3$.
My questions:
1, What is the difference between an "immersion" and an "embedding"?
2, When we talk about $\mathbb{R}^3$ above, we are not specifying a metric so surely we can think of Minkowski space as $M^{2,1}=(\mathbb{R}^3,g_M)$ where $g_M$ is the Minkowski metric. Why can we not construct an immersion of $\mathbb{H}^2$ into $\mathbb{R}^3$ using this approach - what fails?
3, Why is the hyperboloid $\mathbb{H}^2$ referred to as a "model" of hyperbolic space?
4, Hilbert's Theorem is only stated for $\mathbb{R}^3$. What happens when we go to more than 3 dimensions?