An elementary approach to constructing pseudo-cubic C^1-splines on the Stiefel manifold
A standard approach to interpolating a manifold-valued function is to map the given data set to the tangent space at a suitable base point, perform the actual interpolation operations in the tangent space, which is a flat vector space, and to map the interpolation results back to the manifold. The practical execution of this approach requires methods for computing the Riemannian normal coordinates or invertible retractions on the manifold in question, the latter of which can be thought thought of as an approximation of the former. However, the approach is local in nature and ceases to work, when the data set exceeds the domain on which the coordinate maps are invertible.
In this talk, we present an elementary approach to construct non-local manifold C^1-interpolants. More precisely, we will develop a method for constructing a course of pseudo-cubic manifold splines that is differentiable also at the connecting points.
The method will be demonstrated by means of numerical experiments, where we consider interpolation problems on the Stiefel manifold of orthogonal frames.