Discrete geodesic calculus for complete Riemannian manifolds

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Loayza Romero, Estefania and Wirth, Benedikt (2026) Discrete geodesic calculus for complete Riemannian manifolds. Proceedings in Applied Mathematics and Mechanics, PAMM, 26 (2). e70118. ISSN 1617-7061 (https://doi.org/10.1002/pamm.70118)

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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.

ORCID iDs

Loayza Romero, Estefania ORCID logoORCID: https://orcid.org/0000-0001-7919-9259 and Wirth, Benedikt;
  • Item type:Article
    ID code:96104
    Dates:
    Date
    Event
    1 June 2026
    Published
    15 April 2026
    Published Online
    26 February 2026
    Accepted
    2025
    Submitted
    Subjects:Science > Mathematics
    Department:Faculty of Science > Mathematics and Statistics
    Depositing user: Pure Administrator
    Date deposited:27 Apr 2026 11:25
    Last modified:01 May 2026 00:23
    URI:https://strathprints.strath.ac.uk/id/eprint/96104
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