  There has been increasing interest in statistical analysis of data lying in
manifolds. This paper generalizes a smoothing spline fitting method to
Riemannian manifold data based on the technique of unrolling and unwrapping
originally proposed in Jupp and Kent (1987) for spherical data. In particular
we develop such a fitting procedure for shapes of configurations in general
$m$-dimensional Euclidean space, extending our previous work for two
dimensional shapes. We show that parallel transport along a geodesic on Kendall
shape space is linked to the solution of a homogeneous first-order differential
equation, some of whose coefficients are implicitly defined functions. This
finding enables us to approximate the procedure of unrolling and unwrapping by
simultaneously solving such equations numerically, and so to find numerical
solutions for smoothing splines fitted to higher dimensional shape data. This
fitting method is applied to the analysis of some dynamic 3D peptide data.
