I'm researching the Hilbert's Theorem.- There is no isometric immersion of the hyperbolic plane $\mathbb{H}^2$ in $\mathbb{R}^3$. But on the way I have come across several results, such as:
Efimov's theorem.- There is no isometric immersion of a complete surface with Gaussian curvature bounded superiorly by a negative constant, at $\mathbb{R}^3$.
Then the question arose, where is it possible to have an isometric immersion of $\mathbb{H}^2$? I read that there is an isometric immersion in $\mathbb{R}^5$, but I only got this result,
Blanusa's theorem.- There is an isometric embedding proper to the hyperbolic plane $\mathbb{H}^2$ in $\mathbb{R}^6$.
From Efimov's theorem I have managed to find references to read, but from Blanusa's Theorem or other similar ones I have not obtained much information. If someone knows the topics and could give me some references or perhaps other theorems that can enrich my research, I would appreciate it very much.
Some documents i have found:
- D. Brander - Isometric Embeddings between Space Forms
- T. Milnor - Efimov's theorem about complete immersed surface of negative curvature
- Do Carmo - Differential geometry
- And others Here