  We consider the evolution of a compact segment of an analytic curve on the
unit tangent bundle of a finite volume hyperbolic $n$-manifold under the
geodesic flow. Suppose that the curve is not contained in a stable leaf of the
flow. It is shown that under the geodesic flow, the normalized parameter
measure on the curve gets asymptotically equidistributed with respect to the
normalized natural Riemannian measure on the unit tangent bundle of a closed
totally geodesically immersed submanifold.
  Moreover, if this immersed submanifold is a proper subset, then a lift of the
curve to the universal covering space $T^1(H^n)$ is mapped into a proper
subsphere of the ideal boundary sphere $\partial H^n$ under the visual map.
This proper subsphere can be realized as the ideal boundary of an isometrically
embedded hyperbolic subspace in $H^n$ covering the closed immersed submanifold.
  In particular, if the visual map does not send a lift of the curve into a
proper subsphere of $\partial H^n$, then under the geodesic flow the curve gets
asymptotically equidistributed on the unit tangent bundle of the manifold with
respect to the normalized natural Riemannian measure.
  The proof uses dynamical properties of unipotent flows on finite volume
homogeneous spaces of SO(n,1).
