{"id": "card_joint_bochner_vanishing", "kind": "note", "title": "Positive Ricci curvature kills harmonic one-forms", "body": "Bochner (1946): a closed manifold with positive Ricci curvature carries no non-zero harmonic one-form, so its first Betti number is zero - a Hodge-theoretic conclusion from the very hypothesis Hamilton's 1982 Ricci-flow theorem starts from. Myers (1941): under Ric >= (n - 1)k the diameter is at most pi/sqrt(k), with equality on the round sphere - sealed on the unit three-sphere.", "source": {"label": "Narrow Highway - the Millennium floor, a joint found in the literature (operator seed)", "url": "", "domain": "mathematics", "authority_tier": "engine_derived"}, "shelf": "codex", "box": "joint", "bands": ["joint", "two trees", "millennium", "floor", "bochner", "myers", "ricci curvature", "harmonic forms"], "subject": "Positive Ricci curvature kills harmonic one-forms", "connections": [{"to_card_id": "card_floor_millennium", "relationship": "part_of", "evidence": "a joint that is proven or observed - a part of the floor"}, {"to_card_id": "card_question_hodge", "relationship": "connects_at", "evidence": "Positive Ricci curvature kills harmonic one-forms. bochner 1946; myers 1941; hamilton 1982. Sealed: myers_diameter_bound_on_the_unit_s3 (Poincare stick)"}, {"to_card_id": "card_question_poincare", "relationship": "connects_at", "evidence": "Positive Ricci curvature kills harmonic one-forms. bochner 1946; myers 1941; hamilton 1982. Sealed: myers_diameter_bound_on_the_unit_s3 (Poincare stick)"}, {"to_card_id": "card_src_mill_bochner_1946", "relationship": "cites", "evidence": "the record the joint stands on"}, {"to_card_id": "card_src_mill_myers_1941", "relationship": "cites", "evidence": "the record the joint stands on"}, {"to_card_id": "card_src_mill_hamilton_1982", "relationship": "cites", "evidence": "the record the joint stands on"}], "author": "engine", "created_at": 0.0, "updated_at": 0.0, "visibility": "public", "lifecycle_stage": "public", "volatility": "permanent", "surface": "secular", "generated": false, "call": "codex.joint", "facets": {"subject": ["bochner", "floor", "harmonic forms", "millennium", "myers", "ricci curvature", "two trees"]}, "presentation": {"glyph": "•", "kind_label": "joint", "by": "Narrow Highway - the Millennium floor, a joint found in the literature (operator seed)", "authority": "engine_derived", "posted": "", "standing": ""}, "neighbors": [{"id": "card_floor_millennium", "title": "The Millennium floor - seven open questions, and where they connect", "relationship": "part of", "why": "a joint that is proven or observed - a part of the floor", "href": "/card/card_floor_millennium", "resolved": true}, {"id": "card_question_hodge", "title": "The Hodge conjecture", "relationship": "connects at", "why": "Positive Ricci curvature kills harmonic one-forms. bochner 1946; myers 1941; hamilton 1982. Sealed: myers_diameter_bound_on_the_unit_s3 (Poincare stick)", "href": "/card/card_question_hodge", "resolved": true}, {"id": "card_question_poincare", "title": "The Poincare conjecture", "relationship": "connects at", "why": "Positive Ricci curvature kills harmonic one-forms. bochner 1946; myers 1941; hamilton 1982. Sealed: myers_diameter_bound_on_the_unit_s3 (Poincare stick)", "href": "/card/card_question_poincare", "resolved": true}, {"id": "card_src_mill_bochner_1946", "title": "S. Bochner 1946 — Vector fields and Ricci curvature", "relationship": "cites", "why": "the record the joint stands on", "href": "/card/card_src_mill_bochner_1946", "resolved": true}, {"id": "card_src_mill_myers_1941", "title": "S. B. Myers 1941 — Riemannian manifolds with positive mean curvature", "relationship": "cites", "why": "the record the joint stands on", "href": "/card/card_src_mill_myers_1941", "resolved": true}, {"id": "card_src_mill_hamilton_1982", "title": "R. S. Hamilton 1982 — Three-manifolds with positive Ricci curvature", "relationship": "cites", "why": "the record the joint stands on", "href": "/card/card_src_mill_hamilton_1982", "resolved": true}], "house": {"door": "FIND", "kind": "card", "trail": "connections", "seal": "/card/card_joint_bochner_vanishing", "next_step": {"do": "follow a connection: what this card rests on, and what rests on it", "door": "FIND", "tool": "card_connections", "params": {"id": "card_joint_bochner_vanishing"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}