Every Lipschitz metric has ${\mathcal {C}}^{1}$-geodesics

Closing geodesics in C1 topology

Given a closed Riemannian manifold, we show how to close an orbit of the geodesic flow by a small perturbation of the metric in the C topology.

Isometry - invariant geodesics with Lipschitz obstacle 1

Given a linear isometry A0 : Rn → Rn of finite order on Rn , a general 〈A0〉-invariant closed subset M of Rn is considered with Lipschitz boundary. Under suitable topological restrictions the existence of A0-invariant geodesics of M is proven.

Geodesics in Asymmetric Metric Spaces

In a recent paper [16] we studied asymmetric metric spaces; in this context we studied the length of paths, introduced the class of runcontinuous paths; we noted that there are different definitions of “length space” (also known as “path-metric space” or “intrinsic space”). In this paper we continue the analysis of asymmetric metric spaces. We propose possible definitions of completeness and (l...

Metric Embeddings and Lipschitz Extensions

Bi-Lipschitz Decomposition of Lipschitz functions into a Metric space

We prove a quantitative version of the following statement. Given a Lipschitz function f from the k-dimensional unit cube into a general metric space, one can be decomposed f into a finite number of BiLipschitz functions f |Fi so that the k-Hausdorff content of f([0, 1] r ∪Fi) is small. We thus generalize a theorem of P. Jones [Jon88] from the setting of R to the setting of a general metric spa...