{"paper":{"title":"Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Martin Bauer, Martins Bruveris, Peter W. Michor, Philipp Harms","submitted_at":"2011-05-02T13:42:31Z","abstract_excerpt":"We study Sobolev-type metrics of fractional order $s\\geq0$ on the group $\\Diff_c(M)$ of compactly supported diffeomorphisms of a manifold $M$. We show that for the important special case $M=S^1$ the geodesic distance on $\\Diff_c(S^1)$ vanishes if and only if $s\\leq\\frac12$. For other manifolds we obtain a partial characterization: the geodesic distance on $\\Diff_c(M)$ vanishes for $M=\\R\\times N, s<\\frac12$ and for $M=S^1\\times N, s\\leq\\frac12$, with $N$ being a compact Riemannian manifold. On the other hand the geodesic distance on $\\Diff_c(M)$ is positive for $\\dim(M)=1, s>\\frac12$ and $\\dim("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.0327","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}