{"paper":{"title":"Results on the homotopy type of the spaces of locally convex curves on $S^3$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Em\\'ilia Alves, Nicolau C. Saldanha","submitted_at":"2017-03-07T20:26:23Z","abstract_excerpt":"A curve $\\gamma: [0,1] \\rightarrow S^n$ of class $C^k$ ($k \\geqslant n$) is locally convex if the vectors $\\gamma(t), \\gamma'(t), \\gamma\"(t), \\cdots, \\gamma^{(n)}(t)$ are a positive orthonormal basis to $R^{n+1}$ for all $t \\in [0,1]$. Given an integer $n \\geq 2$ and $Q \\in SO_{n+1}$, let $LS^n(Q)$ be the set of all locally convex curves $\\gamma: [0,1] \\rightarrow S^n$ with fixed initial and final Frenet frame $F_\\gamma(0)=I$ and $F_\\gamma(1)=Q$. Saldanha and Shapiro proved that there are just finitely many non-homeomorphic spaces among $LS^n(Q)$ when $Q$ varies in $SO_{n+1}$ (in particular, a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.02581","kind":"arxiv","version":2},"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"}