{"paper":{"title":"The homogeneous Sobolev metric of order one on diffeomorphism groups on the real line","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Martin Bauer, Martins Bruveris, Peter W. Michor","submitted_at":"2012-09-13T09:58:14Z","abstract_excerpt":"In this article we study Sobolev metrics of order one on diffeomorphism groups on the real line. We prove that the space $\\operatorname{Diff}_{1}(\\mathbb R)$ equipped with the homogenous Sobolev metric of order one is a flat space in the sense of Riemannian geometry, as it is isometric to an open subset of a mapping space equipped with the flat $L^2$-metric. Here $\\operatorname{Diff}_{1}(\\mathbb R)$ denotes the extension of the group of all either compactly supported, rapidly decreasing or $H^\\infty$-diffeomorphisms, that allows for a shift towards infinity. In particular this result provides "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.2836","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"}