# “Hyperboloid like surface” as hyperbolic plane / pseudosphere

A pseudosphere is an surface wth a constant negative curvature.

In most publications, it is almost given that the tracioid (rotated tractrix) is the surface that has a constant negative curvature, and "tracioid" "pseudosphere" are used interchangable.

This made me wonder are there other pseudospheres?

In Klein's "Vorlesungen uber Nicht-Euclidische Geometrie" (1928) \$4, page 286, figure 220, KLein mentions an "single sheet hyperboloid-like plane" as hyperbolic plane, and even gives its represetation in the Klein disk model.

I tried to find more recent information on this surface but could not find much more.

(everything I found refers to the hyperboloid model, but this is different, this is not a model of the hyperbolic plane but a surface with a contant negative curvature.)

Still can this made me wonder, what is the equation of this surface. (is it an hyperboloid or doe it only look like one) how does it look in the Poincare disk model.

Or is there a proof that Klein was wrong, such a surface cannot have a constant negative curvature?

I am at a dead end, is there more?

## 1 Answer

The pseudo-sphere is not isometric to any portion of the hyperbolic plane. Instead, as explained here, the pseudo-sphere is isometric to a quotient of a portion of the hyperbolic plane, namely the quotient of a horoball under the action of an infinite cyclic group of parabolic isometries of the hyperbolic plane which leave the horoball invariant.

• Thanks , not sure what you mean by "namely the quotient of a horoball under the action of an infinite cyclic group of parabolic isometries of the hyperbolic plane which leave the horoball invariant. ", I thought the tracicioid (pseudosphere) is an area on the hyperbolic plane limited by two concentric horospheres and two horoparallel lines going to the same (ideal) centre. my questionis "is the tracioid the only (euclidean) surface with an constant negative curvature?" i think the other surface i mentioned could be an en.wikipedia.org/wiki/Catenoid but am not sure – Willemien Sep 12 '14 at 14:17
• @Willemien: the "area on the hyperobolic plane" in your comment is a shape limited by four segments, and is homeomorphic to a rectangle which is homeomorphic to a closed unit disc. In particular it is simply connected. So that area cannot possibly by isometric much less homeomorphic to the pseudosphere which is not simply connected, in fact its fundamental group is infinite cyclic. – Lee Mosher Sep 13 '14 at 13:20
• To address your question about my quotient construction, you might like to read a book about hyperbolic geometry and how it is used to construct hyperbolic structures on surfaces. The book "Geometry of Surfaces" by John Stillwell is a good start. – Lee Mosher Sep 13 '14 at 13:22
• THanks for the reference to Stillwells book added it to my reading list, but i fear we misundersand eachother did make new question that is maybe clearer see math.stackexchange.com/q/930847/88985 – Willemien Sep 14 '14 at 12:03