{"paper":{"title":"The fundamental theorem of curves and classifications in the Heisenberg groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Hung-Lin Chiu, XiuHong Feng, Yen-Chang Huang","submitted_at":"2015-11-17T01:03:08Z","abstract_excerpt":"We study the horizontally regular curves in the Heisenberg groups $H_n$. We show the fundamental theorem of curves in $H_n$ $(n\\geq 2)$ and define the concept of the orders for horizontally regular curves. We also show that the curve $\\gamma$ is of order $k$ if and only if $\\gamma$ lies in $H_k$ but not in $H_{k-1}$ up to a Heisenberg rigid motion; moreover, two curves with the same order differ from a rigid motion if and only if they have the same p-curvatures and contact normality. Thus, combining with our previous work we have completed the classification of horizontally regular curves in $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.05237","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"}