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 $p$.