Consider a surface (with boundary) diffeomorphic to $S^1 \times [0, 1]$ and with constant negative curvature, sitting inside $\mathbb{R}^3$. All the examples I know of such surfaces are "part of" (or "cut out of", if that makes better sense) a surface of revolution (examples are the catenoid, tractricoid, etc.; O'Neill's "Elementary Differential Geometry" has a comprehensive list on page 261, Exercise 7). I was wondering, are all constantly negatively curved $S^1 \times [0, 1]$ obtained this way (that is, cut out from a surface of revolution)?
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3$\begingroup$ No, certainly not. Constant negative curvature embeddings of an annulus can be constructed extremely flexible ways. Consider taking an arbitrary knot as your embedding of $S^1 \times \{1/2\}$ and try to fatten it up to a constant negatively curved annulus. $\endgroup$– Ryan BudneyCommented Jan 4, 2015 at 7:45
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1$\begingroup$ To be a bit more explicit about Ryan's comment: If you take any (smooth) immersed curve $c:S^1\to \mathbb{R}^3$ such that there exists a smooth unit normal $n:S^1\to S^2$ along $c$ (i.e., $c'(\theta)\cdot n(\theta)=0$ for all $\theta\in S^1$) satisfying the property that $c''(\theta)\cdot n(\theta)>0$ for all $\theta\in S^1$, then there exists an immersion $C:S^1\times[-1,1]\to\mathbb{R}^3$ such that $C(\theta,0)=c(\theta)$, $n(\theta)$ is the normal to the immersed surface at $c(\theta)$, and the induced metric has constant Gauss curvature $K\equiv=-1$. $\endgroup$– Robert BryantCommented Jan 5, 2015 at 0:38
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$\begingroup$ By the way, the catenoid has negative curvature, but it is not constant. It has zero constant mean curvature. I suspect that by curvature you meant Gaussian curvature. $\endgroup$– GiuseppeCommented Jan 8, 2015 at 10:47
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$\begingroup$ Yes, he means Gauss curvature. $\endgroup$– Deane YangCommented Jan 8, 2015 at 16:15
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$\begingroup$ This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post - you can always comment on your own posts, and once you have sufficient reputation you will be able to comment on any post. $\endgroup$– user9072Commented Jan 8, 2015 at 16:37
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There is a lot of literature on this: see Zalgaller's Geometry III in the Russian Encyclopedia: https://books.google.com/books?id=2ee_Cw4uWVIC&pg=PA158&lpg=PA158&dq=surface+in+E3+of+constant+negative+curvature&source=bl&ots=5T92bOYiq_&sig=_tG-Ovmk0L7WnOnu7Qw8jc_374Y&hl=en&sa=X&ei=CmCpVPKfIIufgwTAu4SwBg&ved=0CEgQ6AEwCQ#v=onepage&q=surface%20in%20E3%20of%20constant%20negative%20curvature&f=false
There is also the nice paper by Tunitskii (1987):
Tunitskiĭ, D. V.(2-MOSC) Regular isometric immersion in E3 of unbounded domains of negative curvature. (Russian) Mat. Sb. (N.S.) 134(176) (1987), no. 1, 119--134, 143; translation in Math. USSR-Sb. 62 (1989), no. 1, 121–138 53A05 (53C42)
answered Jan 4, 2015 at 19:21