  In this article we extend the computational geometric curve reconstruction
approach to curves in Riemannian manifolds. We prove that the minimal spanning
tree, given a sufficiently dense sample, correctly reconstructs the smooth arcs
and further closed and simple curves in Riemannian manifolds. The proof is
based on the behaviour of the curve segment inside the tubular neighbourhood of
the curve. To take care of the local topological changes of the manifold, the
tubular neighbourhood is constructed in consideration with the injectivity
radius of the underlying Riemannian manifold. We also present examples of
successfully reconstructed curves and show an applications of curve
reconstruction to ordering motion frames.
