{"paper":{"title":"Metrics with prescribed horizontal bundle on spaces of curve","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Martin Bauer, Philipp Harms","submitted_at":"2015-11-18T17:29:20Z","abstract_excerpt":"We study metrics on the shape space of curves that induce a prescribed splitting of the tangent bundle. More specifically, we consider reparametrization invariant metrics $G$ on the space $\\operatorname{Imm}(S^1,\\mathbb R^2)$ of parametrized regular curves. For many metrics the tangent space $T_c\\operatorname{Imm}(S^1,\\mathbb R^2)$ at each curve $c$ splits into vertical and horizontal components (with respect to the projection onto the shape space $B_i(S^1,\\mathbb R^2)=\\operatorname{Imm}(S^1,\\mathbb R^2)/\\operatorname{Diff}(S^1)$ of unparametrized curves and with respect to the metric $G$). In"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.05889","kind":"arxiv","version":1},"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"}