Equation of the saddle-like surface with constant negative curvature? What is the equation for the saddle-like 2d surface (embeded in 3d Euclidean space with cartesian coordinates x, y and z) with constant negative curvature frequently used to illustrate open universe (for example in the following image is taken from Wikipedia)?

Flat universe (zero curvature) is illustrated by a plane, with equation say
$z = 0$
Closed universe (constant positive curvature) is illustrated by a surface of sphere, say
$x^2 + y^2 + z^2 = r$
 A: One shouldn't expect to have a "good" formula for the local isometric embeddings of a constant negative curvature surface in Euclidean $\mathbb{R}^3$. This is due to a little theorem proved by David Hilbert around 1901:
Theorem There does not exist a smooth immersion of the hyperbolic plane into Euclidean 3 space. 
The theorem has been further studied in the years following. In 1961 Efimov showed that any complete surface with curvature strictly bounded above (that is to say, if there exists a negative number $K_0 < 0$ such that the Gaussian curvature is always strictly less than $K_0$) cannot admit a smooth (twice continuously differentiable) isometric immersion into Euclidean three space. 

That is to say, if you try to "extend" any surface in Euclidean 3 space that satisfies constant negative curvature, you are guaranteed to hit a singularity. In particular, you cannnot expect the surface to be described by $F(x,y,z) = 0$ where $F(x,y,z)$ has a nice algebraic expression (say, polynomial) and has smooth level sets. 

Typically the image one usually use to illustrate the notion of negative (but not constant) curvature is the graph 
$$ z = x^2 - y^2 $$
which produces a classical saddle, or the catenoid whose Gaussian curvature, while everywhere negative, is not constant. (Though it has constant [in fact everywhere vanishing] mean curvature.)

Lastly, however, despite the above, it is possible to embed "patches" of hyperbolic plane into Euclidean 3 space. There are many ways of doing so (one can search for the term pseudosphere; though some people use the same term for the hyperboloid/de Sitter spaces embedded in higher dimensional Minkowski space), but one of the more well-known is the tractricoid. (See Wiki entry here.) Parametrically in cylindrical coordinates $(z,r,\theta)$ the surface can be described by:
$$ \mathbb{R}_+\times\mathbb{S}^1 \ni (t,\omega) \mapsto \left(z=\frac{1}{\cosh t},r=t-\tanh t , \theta = \omega\right) $$
and has constant negative curvature. 
A: I don't think you can embed a surface of constant negative curvature in Euclidean space. However you can embed it in Minkowski space. See http://en.wikipedia.org/wiki/De_Sitter_space for details. If you have Minkowski space defined by:
$$ ds^2 = -dx_0^2 + dx_1^2 + dx_2^2 + dx_3^2 $$
then the corresponding surface is:
$$ -x_0^2 + x_1^2 + x_2^2 + x_3^2 = a^2$$
