Differentiability of Minimal Geodesics in Metrics of Low Regularity

In Riemannian metrics that are only Hölder continuous of order α, 0 α 1, let minimal geodesics be the continuous curves realizing the shortest distance between two points. It is shown that for 0 < α 1, minimal geodesics are differentiable with a derivative which is at least Hölder continuous of order α=2 for 0 < α < 1 and which is Hölder continuous of order 1 for α= 1.