Examples of ergodic geodesic flow

Question

Are there any good examples of a geodesic flow that is ergodic? I know the result that states that the geodesic flow for manifolds with negative curvature are ergodic, but I'm fishing for some insightful examples.

Answer 1

The geodesic flow I'm most familiar with is billiards on a polygon. This paper gives a description of the geodesic flow corresponding to the motion of a billiard ball, and proves that for almost all right triangles the flow is ergodic. It has also been proven that the flow is ergodic for almost all polygons, but I don't have the reference on hand.

Answer 2

Consider the $n$-torus $\mathbb T^n$, where $\mathbb T = [0,1]$ with $0$ identified with $1$ (i.e. $\mathbb T$ is homeomorphic to the circle). Then I believe the map given by the flow $\varphi_t(x) = x+t u \pmod 1$, where the coordinates of $u$ are independent over the rationals, is ergodic.

For example $\varphi_t(x_1,x_2) = ((x_1+t)\bmod 1,(x_2+t\sqrt 2)\bmod 1)$.

And the flow is clearly geodesic if we use the flat metric.