{"paper":{"title":"A discrete Gauss-Bonnet type theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GN","authors_text":"Oliver Knill","submitted_at":"2010-09-13T03:44:10Z","abstract_excerpt":"We prove a prototype curvature theorem for subgraphs G of the flat triangular tesselation which play the analogue of \"domains\" in two dimensional Euclidean space: The Pusieux curvature K(p) = 2|S1(p)| - |S2(p)| is equal to 12 times the Euler characteristic when summed over the boundary of G. Here |S1(p)| is the arc length of the unit sphere of p and |S2(p)| is the arc length of the sphere of radius 2. This curvature 12 formula is discrete Gauss-Bonnet formula or Hopf Umlaufsatz. The curvature introduced here is motivated by analogue constructions in the continuum like the Jacobi equations for "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.2292","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"}