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You should be looking at the theory of Bäcklund transformations for surfaces of Gaussian curvature $K=-1$. There is a large literature on this, and there are many examples of pseudospherical immersio …

I was working on answering another question involving integrating the geodesic equations on a surface, and the links there lead me back to this question, which I hadn't noticed before. In case anyone …

The simplest way I know is to use the second hyperbolic law of cosines to find the lengths of the sides of the triangle, which then reduces the problem to taking two square roots plus some rational ma …

Probably, you can find a discussion of this in Thurston's notes on hyperbolic 3-manifolds, or maybe some of the expositions by his students. However, what you are asking for is actually pretty simple: …

Things will simplify if you just consider the circles on the Riemann sphere $S^2 = \mathbb{C}\cup\{\infty\}$, for your space is simply the space of circles on the sphere (with the lines in $\mathbb{C} …

This notion of 'the sphere at infinity' is commonly encountered in hyperbolic geometries. Gromov, in particular, has used it in studying the behavior of discrete transformation groups on hyperbolic m …

If you just want examples for which it's not hard to figure out how the geodesics behave, here's a class of examples with negative and non-constant curvauture in the plane where the geodesics are rela …

I believe that the answer is that the curvature of $S$ has to vanish, i.e., the surface is locally isometric to the plane. I haven't checked all of the details, which are somewhat messy in my analysi …

I don't have a complete answer, but here are a few remarks about this that you may find interesting or useful: The OP didn't specify exactly what was meant by 'complex hyperbolic space' $\mathbb{CH}^ …

To answer Joseph's questions: First, it's not impossible to integrate the geodesic flow of the hyperbolic plane in these coordinates, but the formulae I got aren't very nice, so I'm not going to type …

I'm not sure exactly what you mean by 'absolute geometry' that unifies spherical, Euclidean, and hyperbolic geometry, but in the sense of Klein's identification of geometries with group actions, so th …

This 'isomorphism' does not hold for $n>2$, in the sense that the 'natural' $\mathrm{SL}(n,\mathbb{R})$-invariant metric on what you are calling $\mathcal{SP}(n)$ and the constant sectional curvature …

This is not really an answer, but rather a longish comment and a suggestion about how one might focus the question a bit better. First, when one asks for a 'canonical' isometric embedding into some E …

I'll just point out an answer based on somewhat different criteria than explicitly knowing features of the hyperbolic metric: As is well-known, every oriented, compact hyperbolic surface $C$ is canon …

The answer is 'no'. Suppose that $M\subset\mathbb{E}^3$ is a smooth connected surface. If the ratio of the Gauss curvature $K$ and $p^4$ is constant (where $p(x)$ is the distance from $T_xM$ to the o …