## Tags » Hyperbolic Geometry

#### Ergodicity of the Geodesic Flow

### The geodesic flow

Given any Riemannian manifold *M*, we may define a **geodesic flo****w** on the unit tangent bundle which sends a point 1,058 more words

#### Hyperbolic Geometry Note #2: On Möbius transformations as isometries of the upper half plane

Möbius transformations are conformal (i.e., angle-preserving), orientation-preserving, area-preserving *isometries* in the upper half-plane model of hyperbolic space. It piqued my interest to discover that the form of the metric in this model (which I discussed in… 732 more words

#### More Maths Art

Wednesday is here, the middle of the week, just a couple of days until the weekend. In the past weeks I have done more artistic and relaxing Wednesdays and this one will be the same. 273 more words

#### Hyperbolic Geometry Note #1: Strange behaviour of length calculations in the Poincaré half-plane model of hyperbolic space

In the Poincaré half-plane model of hyperbolic space, the upper half-plane is the set of complex numbers with positive imaginary part:

The boundary of , , is the real axis together with the point . 762 more words

#### The many faces of the hyperbolic plane

is the unique (up to isometry) complete simply-connected 2-dimensional Riemann manifold of constant sectional curvature -1.

- It is diffeomorphic to as a topological space (or, indeed, isometric as a Riemannian manifold, if we give a left-invariant metric): to show this, we note that has isometry group , and the subgroup of isometries which stabilize any given is isomorphic to . 211 more words

#### Fuchsian vs. Kleinian

Or, all the fun you could have in one additional dimension.

The former, of course, may be viewed as a special case of the latter; the more general Kleinian case shares many, but not all, of the characteristics of the simpler Fuchsian case. 776 more words