Geodesics of Random Riemannian Metrics: Supplementary Material

TL;DR

This supplementary material provides detailed mathematical background, proofs, and constructions related to the study of geodesics in random Riemannian metrics, supporting the main article's theoretical developments.

Contribution

It offers rigorous mathematical results, constructions, and explanations that underpin the analysis of geodesics in random Riemannian metrics.

Findings

General results on Gaussian random fields

Restatement of the Shape Theorem in this context

Geometric consequences for geodesics in random metrics

Abstract

This is supplementary material for the main Geodesics article by the authors. In Appendix A, we present some general results on the construction of Gaussian random fields. In Appendix B, we restate our Shape Theorem, specialized to the setting of this article. In Appendix C, we state some straightforward consequences on the geometry of geodesics for a random metric. In Appendix D, we provide a rapid introduction to Riemannian geometry for the unfamiliar reader. In Appendix E, we present some analytic estimates which we use in the article. In Appendix F, we present the construction of the conditional mean operator for Gaussian measures. In Appendix G, we describe Fermi normal coordinates, which we use in our construction of the bump metric.