[proofplan] We pass from the tangent bundle to the cotangent bundle using the Riemannian metric. The geodesic flow is the Hamiltonian flow of the kinetic-energy Hamiltonian. Hamiltonian flows preserve the canonical symplectic form, hence preserve the Liouville volume form. Restricting this invariant volume to the unit-energy hypersurface gives invariance of Liouville measure on the unit tangent bundle. [/proofplan] [step:Realize geodesic flow as a Hamiltonian flow] Use the metric $g$ to identify $TM$ and $T^*M$. On $T^*M$, let $\omega$ be the canonical symplectic form and define the Hamiltonian \begin{align*} H(q,p):=\frac12|p|_g^2. \end{align*} The Hamiltonian vector field $X_H$ is determined by \begin{align*} \iota_{X_H}\omega=dH. \end{align*} Under the metric identification, the Hamiltonian flow of $H$ is exactly the geodesic flow on $TM$. [/step] [step:Show that the Hamiltonian flow preserves symplectic volume] Since $\omega$ is closed, Cartan's formula gives \begin{align*} \mathcal{L}_{X_H}\omega =d(\iota_{X_H}\omega)+\iota_{X_H}d\omega =d(dH)+0 =0. \end{align*} Thus the Hamiltonian flow preserves $\omega$. If $\dim M=m$, it also preserves the symplectic volume form \begin{align*} \frac{\omega^m}{m!}. \end{align*} [/step] [step:Restrict to the unit tangent bundle] The unit cotangent bundle is the regular energy hypersurface \begin{align*} S^*M=\{(q,p):H(q,p)=1/2\}. \end{align*} Because $H$ is constant along its Hamiltonian flow, this hypersurface is invariant. The Liouville measure on $S^*M$, and hence on $T^1M$ under the metric identification, is the hypersurface measure induced by the invariant symplectic volume and the Hamiltonian flow direction. Since the ambient symplectic volume and the hypersurface are both preserved, the induced Liouville measure is preserved. Therefore \begin{align*} \nu(\phi_t(B))=\nu(B) \end{align*} for every Borel $B\subseteq T^1M$ and every $t\in\mathbb{R}$. [/step]