  Interpolation of data on non-Euclidean spaces is an active research area
fostered by its numerous applications. This work considers the Hermite
interpolation problem: finding a sufficiently smooth manifold curve that
interpolates a collection of data points on a Riemannian manifold while
matching a prescribed derivative at each point. We propose a novel procedure
relying on the general concept of retractions to solve this problem on a large
class of manifolds, including those for which computing the Riemannian
exponential or logarithmic maps is not straightforward, such as the manifold of
fixed-rank matrices. We analyze the well-posedness of the method by introducing
and showing the existence of retraction-convex sets, a generalization of
geodesically convex sets. We extend to the manifold setting a classical result
on the asymptotic interpolation error of Hermite interpolation. We finally
illustrate these results and the effectiveness of the method with numerical
experiments on the manifold of fixed-rank matrices and the Stiefel manifold of
matrices with orthonormal columns.
