When a local isometric immersion become global?

Question

I have a question that I have not been able to answer clearly. When a local isometric immersion become global? For example, I know the following result, if $\phi:M\to N$ be a differentiable mapping and $g$ a metric in $N$, then $\phi^*g$ is a metric in $M$ if and only if $\phi$ is a immersion (isometric). But that only assures me a local isometric immersion, for example of a pseudosphere and the hyperbolic plane $\mathbb H^2$, as only one horocircle can be immersed but not the entire hyperbolic plane. But in turn, for example Rozendorn uses this same fact to prove that $\mathbb H^2$ is isometrically immersed in $\mathbb R^5$ with a non-injective immersion. In his article he only explicitly gives such a immersion and the idea is to use the result above, after that he concludes that it is a global immersion. I feel a bit tied in my hands with this and I don't know how to get out. I would appreciate some clarification, thank you.

I thought that the definition of immersion was only one, I'm sorry if there are more so: A differentiable map $\phi:M\to N$ is an immersion if the differential $d\phi_p$ is non-singular for all $p$.

Answer 1

You are thinking about the two different concepts of immersion of an entire manifold into another and immersion of a subset of a neighborhood of a point in that manifold into another, and you call those "global" and "local" respectively, though the terms you use are not standard. An immersion is always the former concept. To put these terms in better words, you could call them immersible and "locally immersible".

If $S$ is the pseudosphere as a submanifold of $\mathbb R^3$, and $\tilde S$ is its universal cover, then the projection $\tilde S \rightarrow S \rightarrow \mathbb R^3$ is an immersion of $\tilde S$. The space $\tilde S$ is therefore immersible in $\mathbb R^3$. This space is also isometric to an open subset of $H^2$, a horodisc, so since $H^2$ is homogeneous one could say that $H^2$ is "locally immersible" into $\mathbb R^3$.

The map constructed by Rozendorn is, as you say, a smooth isometric map from the entire space $H^2$ to $\mathbb R^5$. This already proves by example that $H^2$ is immersible in $\mathbb R^5$. No special consideration of locality is required.