It’s easy to note that there is a local isometry between the hyperbolic plane $mathbb{H}^2$ and the pseudosphere, since they have constant curvature equal to $-1$. Since Hilbert’s Theorem prevents this local isometry from actually being a global isometry, I don’t know how to reach this conclusion without using that theorem. On the other hand, I have read Hilbert’s theorem in Do Carmo’s book and also in Spikav’s book, I would like to know if it’s possible to prove that the hyperbolic plane $mathbb{H}^2$ can’t be isometrically immersed in $mathbb{R}^3$ in a direct way without using the theorem of Hilbert. With the initials, the following question comes to mind, is it possible that if we remove a point from the hyperbolic plane, it can be isometrically immersed in $mathbb{R}^3$?

I’m a student who is taking the differential geometry course for the first time, it’s for this reason that I ask that his possible answers be detailed in order to understand them to the fullest. Thanks in advance I am attaching an article click here that I found where they present a proof that the hyperbolic plane cannot be isometrically immersed in $mathbb{R}^3$ on page $10$, I find it very interesting I would like to be able to understand it 100%. I want to understand this particular case very well to start investigating hilbert’s theorem thoroughly. Thank you very much in advance.