{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.affine-connection","code":"thing-q1371213","url":"https://ver.cy/models/thing/q1371213/","name":"affine connection","alternateNames":["Levi-Civita connection"],"kind":"thing","status":"published","version":"0.2.0-wave.1","language":"en","classifiers":{"family":"Thing Registry","category":"Cross-cutting context","entryKind":"thing","plane":"XCT","domain":["XCT.REL"],"industry":[],"navPath":"","tags":[],"facets":{}},"whatItIs":"In differential geometry, a structure on a smooth manifold that specifies how to differentiate vector fields along curves and to parallel transport vectors, defining geodesics and curvature; the Levi-Civita connection is the unique torsion-free connection compatible with a Riemannian metric and underlies general relativity.","purpose":"Let an agent explain affine connections by definition, Christoffel symbols, parallel transport, curvature and applications, for students and researchers.","scope":{"in":[],"out":[],"boundaries":[]},"distinguishingFeatures":["Defines covariant differentiation","Gives parallel transport","May have torsion and curvature","Levi-Civita is metric-compatible and torsion-free"],"structure":{"bundles":[{"id":"definition","name":"Definition","description":"What it is.","layers":[{"id":"axioms","name":"Axioms","description":"Defining properties.","findings":[{"id":"axioms-finding","name":"Axioms","description":"Axioms.","questions":[{"text":"What properties define an affine connection?","kind":"definition"},{"text":"How does it differ from a metric?","kind":"boundary"}]}]},{"id":"christoffel","name":"Christoffel","description":"Coordinates.","findings":[{"id":"christoffel-finding","name":"Christoffel","description":"Christoffel symbols.","questions":[{"text":"What are the Christoffel symbols for this metric?","kind":"measurement"},{"text":"Which sign and index conventions are used?","kind":"boundary"}]}]}]},{"id":"geometry","name":"Geometry","description":"Transport and curvature.","layers":[{"id":"transport","name":"Transport","description":"Parallel transport.","findings":[{"id":"transport-finding","name":"Transport","description":"Parallel transport.","questions":[{"text":"How does parallel transport work along this curve?","kind":"definition"},{"text":"What are the geodesics?","kind":"definition"}]}]},{"id":"curvature","name":"Curvature","description":"Riemann tensor.","findings":[{"id":"curvature-finding","name":"Curvature","description":"Curvature.","questions":[{"text":"How is curvature computed from the connection?","kind":"definition"},{"text":"What does torsion measure?","kind":"definition"}]}]}]},{"id":"physics","name":"Physics","description":"Relativity.","layers":[{"id":"relativity","name":"Relativity","description":"General relativity.","findings":[{"id":"relativity-finding","name":"Relativity","description":"General relativity.","questions":[{"text":"How does the Levi-Civita connection describe free fall in general relativity?","kind":"definition"},{"text":"Which equations use it?","kind":"definition"}]}]},{"id":"gauge","name":"Gauge","description":"Gauge theories.","findings":[{"id":"gauge-finding","name":"Gauge","description":"Gauge theories.","questions":[{"text":"How do connections on bundles appear in gauge theories?","kind":"definition"},{"text":"Which references explain it?","kind":"provenance"}]}]}]},{"id":"learning","name":"Learning","description":"Study.","layers":[{"id":"references","name":"References","description":"Textbooks.","findings":[{"id":"references-finding","name":"References","description":"Textbooks.","questions":[{"text":"Which textbooks introduce affine connections?","kind":"provenance"},{"text":"What prerequisites are needed?","kind":"definition"}]}]},{"id":"tools","name":"Tools","description":"Computation.","findings":[{"id":"tools-finding","name":"Tools","description":"Computation.","questions":[{"text":"Which software computes Christoffel symbols and curvature?","kind":"provenance"},{"text":"How can results be checked?","kind":"action"}]}]}]}]},"agentConduct":{"may":["State definitions and standard results about affine connections, citing the literature.","Compute connection coefficients for given metrics."],"mustNot":["Present unproven claims as theorems.","Misattribute results to mathematicians or physicists without a source."],"requiresHuman":["Publishing a claimed new result in a reference work."]},"ethics":{"considerations":["Correct attribution matters to scholarship.","Errors in teaching material mislead learners."],"affectedParties":["Students and researchers"]},"owners":{"steward":"Nobody: a mathematical concept held in common.","roles":[],"masterSystems":[]},"relations":[{"target":"Q31924","type":"parent","note":"registry parent class"},{"target":"connection (mathematics)","type":"related","note":"category"},{"target":"geodesic","type":"related","note":"curve"},{"target":"continuous function","type":"related","note":"smooth maps"},{"target":"general relativity","type":"related","note":"application"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.affine-connection","Wikidata: Q1371213 (https://www.wikidata.org/wiki/Q1371213)"]},"properties":{"applicability":"not-applicable","items":[]},"recognition":{"applicability":"optional","items":["Covariant derivative notation","Christoffel symbols","A metric alone is a different structure","Not physical; formulas with Christoffel symbols and diagrams of parallel transport."]},"capabilities":{"applicability":"required","items":["explain the definition and axioms","compute Christoffel symbols for a metric","describe parallel transport and geodesics","relate connections to curvature and relativity"]},"hazards":{"applicability":"required","items":["Index and sign convention errors","Confusing connection and metric"]},"interfaces":{"applicability":"required","items":["No specific regulation"]},"context":{"applicability":"required","items":["Differential geometry, physics and geometric modelling.","Levi-Civita connection","connections with torsion","flat connections","projective and conformal connections","connections on vector bundles (generalisation)"]}},"sources":[{"title":"Wikidata item Q1371213: affine connection","url":"https://www.wikidata.org/wiki/Q1371213","note":"identity and sense of the item"},{"title":"Wikipedia: Affine connection","url":"https://en.wikipedia.org/wiki/Affine_connection","note":"general description of the item"}],"openQuestions":["Should the Levi-Civita connection be a separate entry?","How should textbooks be linked?","How should computational tools be linked?"],"resources":{"spec":"/models/things/publications/thing-q1371213/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q1371213/spec.json"],"providers":["Claude"],"researchStatus":"unreviewed","generatedAt":"2026-09-11T15:11:43Z","builder":"tools/build_cards.py@1.0.0"},"completeness":{"sections":{"classifiers":"filled","whatItIs":"filled","purpose":"filled","distinguishingFeatures":"filled","structure":"filled","agentConduct":"filled","ethics":"filled","owners":"filled","relations":"filled","interaction.identity":"filled","interaction.properties":"not-applicable","interaction.recognition":"filled","interaction.capabilities":"filled","interaction.hazards":"filled","interaction.interfaces":"filled","interaction.context":"filled","sources":"filled"},"notes":{"_gate":"Published through the thing publication gate 1.0 on 2026-10-06T19:51:54+00:00: completeness 1.00; sense: Wikidata description; 2 live sources; review by codex: pass","interaction.properties":"Plane XCT: no invented physical properties."},"score":1.0}}