Discrete Geodesic Calculus for Complete Riemannian Manifolds

ABSTRACT Complete Riemannian metrics ensure that every geodesic in a Riemannian manifold can be extended indefinitely, making the manifold geodesically complete. Their importance lies in their ability to eliminate the pathological behavior of “falling off the edge” and their fundamental role in the global analysis of manifolds. In this paper, we focus on the complete metrics proposed by Gordon. Despite the simplicity of their tensor representation and inverse, most cases lack explicit expressions for the associated exponential, logarithm, and parallel transport maps. The main goal of this paper is to develop a discrete geodesic calculus based on a computationally inexpensive dissimilarity measure, which will allow efficient computation of discrete exponential, logarithm, and parallel transport maps. As a proof of concept, we present examples for 1‐ and 2‐dimensional manifolds, along with preliminary results on the manifold of planar triangular meshes.