Convergence and smoothness analysis of subdivision rules in Riemannian and symmetric spaces

After a discussion on definability of invariant subdivision rules we discuss rules for sequential data living in Riemannianmanifolds and in symmetric spaces, having in mind the space of positive definite matrices as a major example. We show that subdivision rules defined with intrinsic means in Cartan-Hadamard manifolds converge for all input data, which is a much stronger result than those usually available for manifold subdivision rules. We also show weaker convergence results which are true in general but apply only to dense enough input data. Finally we discuss C1 and C2 smoothness of limit curves.