The geometry corresponding to a constant negative Gaussian curvature Manifold.

It can be derived by embedding one sheet of a Hyperboloid in $\mathbb{R}^3$, with the Minkowski metric. A manifold with hyperbolic geometry is called Hyperbolic space

Mark Norfleet - Hyperbolic Geometry

Connections between hyperbolic models and complex analysis (Moebius transformations and $z \rightarrow -\bar{z}$ form a basis for the group of all isometries in the upper half-plane model)

Pseudosphere model

As seen here (and as it is evident after some thought), the pseudosphere is only a model of a section of the full hyperbolic space (as defined by the hyperboloid model for instance) (note that because it has the same curvature, it should be isometric, that's why we know it should be a model of it even if only of a region). Furthermore the pseudosphere is not homeomorphic to hyperbolic space, as it involves identifying to sides of this region.

As mentioned here:

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:

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.

However, patches of the full Hyperbolic space can be embedded as patches of the Pseudosphere (note that, as the theorem above says, the full pseudosphere has a singularity).

See here in the lecture to see the hyperbolic plane introduced as an example of a surface which can't be embbeded in R^3

As explained in here, any surface with constant negative Gaussian curvature is locally (that is inside Local patches) isometric to a piece of a Pseudosphere

Hyperboloid model

The Elliptic hyperboloid embedded in $\mathbb{R}^3$, with the Minkowski metric.

Geodesics (lines) in the hyperboloid model

Lines can be constructed by taking a plane that passes through the origin and finding its intersection with the hyperboloid. Note that this is analogous to how great circles are constructed in Spherical geometry

Projective models of hyperbolic geometry

Different ways of projecting the points in the hyperboloid to planes or subsets of planes give other models.

Projective disk model

projection

definition

Poincare model (aka conformal disk model)

projection

definition

Upper half-plane model

definition

Visualizing Hyperbolic Honeycombs

Fun With Hyperbolic Space

Non-euclidean virtual reality