Geodesic flow on a locally symmetric space

Example

Let $G = \mathrm{PSL}_2(\R)$, acting on the upper half-plane $\mathbb{H} = \set{z \in \C : \operatorname{Im} z > 0}$ by Möbius transformations $z \mapsto (az + b)/(cz + d)$. These maps are the orientation-preserving isometries of the hyperbolic metric $\abs{dz}/\operatorname{Im} z$. The action of $G$ on the unit tangent bundle $T^1\mathbb{H}$ is free and transitive. Let $v_0$ be the upward unit vector at $i$; then $g \mapsto g \cdot v_0$ identifies $G$ with $T^1\mathbb{H}$. For $t \in \R$ let $a_t$ be the class of $\begin{pmatrix} e^{t/2} & 0 \\ 0 & e^{-t/2} \end{pmatrix}$. Then $a_t \cdot i = e^t i$, and the path $t \mapsto e^t i$ is the unit-speed geodesic through $v_0$, since the hyperbolic distance from $i$ to $e^t i$ is $t$ for $t \ge 0$. As $g$ is an isometry, $g a_t \cdot v_0$ is the unit vector reached after time $t$ along the geodesic starting at $g \cdot v_0$. So the geodesic flow on $T^1\mathbb{H}$ is the right multiplication $g \mapsto g a_t$.

Now let $\Gamma$ be a lattice in $G$, for example the modular group or a torsion-free cocompact lattice. Left multiplication by $\Gamma$ commutes with right multiplication by $a_t$. So the flow descends to $\Gamma \backslash G$, which is the unit tangent bundle of the locally symmetric surface $\Gamma \backslash \mathbb{H}$ when $\Gamma$ is torsion-free. It is an action of the Lie group $\R$ on $\Gamma \backslash G$. The same recipe works for a quotient $\Gamma \backslash G / K$ of a Riemannian symmetric space $G/K$. There geodesics are orbits of one-parameter subgroups, and the flow becomes right multiplication on a homogeneous space of $G$.