# When a local isometric immersion become global?

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 $$p$$.

• Isometric immersion is a local property. What distinction are you making between a "local" and "global" isometric immersion? Sep 28, 2021 at 6:36
• @Kajelad For example there is a local isometric immersion of $\mathbb H^2$ in $\mathbb R^3$ as a pseudosphere, but there is no global isometric immersion of the complete hyperbolic plane in $\mathbb R^3$, thanks to hilbert's theorem (1901). Sep 28, 2021 at 12:26
• I'm still not sure what you mean by "local isometric immersion". A pseudosphere is not an immersion of $\mathbb{H}^2$ at all. It would be useful, I think, to define the term rather than provide examples. Sep 28, 2021 at 13:52

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.