1206.4940 (Tom LaGatta et al.)

Geodesics of Random Riemannian Metrics II: Minimizing Geodesics    [PDF]

Tom LaGatta, Jan Wehr

We continue our analysis of geodesics in quenched, random Riemannian environments. In this article, we prove that a geodesic with randomly chosen initial conditions is almost surely not minimizing. To do this, we show that a minimizing geodesic is guaranteed to eventually pass over a certain "bump surface," which locally has constant positive curvature. By using Jacobi fields, we show that this is sufficient to destabilize the minimizing property.

View original: http://arxiv.org/abs/1206.4940