TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 358, Number 7, Pages 2917–2926 S 0002-9947(06)04195-X Article electronically published on February 14, 2006 ON H ¨ OLDER CONTINUOUS RIEMANNIAN AND FINSLER METRICS ALEXANDER LYTCHAK AND ASLI YAMAN Abstract. We discuss smoothness of geodesics in Riemannian and Finsler metrics. 1. Introduction In this paper we improve and generalize a result of Calabi and Hartmann in [CH70]. They study smoothness of isometries between equal-dimensional manifolds with α-H¨older continuous Riemannian metrics and prove that each isometry is of class C 1,α . The main tool in the proof is Theorem 3.1 of [CH70], stating that geodesics in α-H¨older Riemannian metrics are uniformly C 1,α , but the proof of this theorem is not correct. Although the result on the smoothness of isometries is true, as Taylor has shown by a different method in [Tay], Theorem 3.1 is wrong for α< 1. Indeed one of our results is the following: Theorem 1.1. For each 0 <α ≤ 1 set β = α 2−α . There is an α-H¨olderRiemannian metric on R 2 , such that geodesics near the origin are not uniformly C 1,l for any l>β. Actually it seems to be possible but technically not trivial to construct an α- H¨older continuous Riemannian metric for which some geodesic γ is not C 1,l for all l>β. This would show that there are manifolds (of different dimensions) with α-H¨older Riemannian metrics and a distance-preserving embedding of one of them into another that is at best C 1,β , in contrast to the C 1,α smoothness of isometries shown by Taylor. Remark 1.1. If α goes to 1, then the regularity of geodesics is almost C 1,α , whereas if α goes to 0, one only gets a bit more than half of the expected regularity! In contrast to this result the first author proved in [Lyt] that geodesics in C 1,α submanifolds of smooth Riemannian manifolds are uniformly C 1,α . This gives us the following non-embeddability result: Corollary 1.2. For 0 <α< 1 and β = α 2−α there are α-H¨older continuous Rie- mannian metrics that (even locally) do not admit C 1,l arcwise isometric embeddings into any smooth Riemannian manifold, for any l>β. Received by the editors March 25, 2004. 2000 Mathematics Subject Classification. Primary 53B40, 53B20. Key words and phrases. Geodesics, isometric embeddings. c 2006 American Mathematical Society Reverts to public domain 28 years from publication 2917 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use