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Positive Ricci curvature kills harmonic one-forms
joint
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
Narrow Highway - the Millennium floor, a joint found in the literature (operator seed)
card id
card_joint_bochner_vanishing
address
WIT.codex.EXP/positive-ricci-curvature-kills-harmonic-one-form/REF.WITNESSED@narrow-highway-the-mille
adjoining cards
- part of → The Millennium floor - seven open questions, and where they connect — a joint that is proven or observed - a part of the floor
- connects at → The Hodge conjecture — Positive Ricci curvature kills harmonic one-forms. bochner 1946; myers 1941; hamilton 1982
- connects at → The Poincare conjecture — Positive Ricci curvature kills harmonic one-forms. bochner 1946; myers 1941; hamilton 1982
- cites → S. Bochner 1946 — Vector fields and Ricci curvature — the record the joint stands on
- cites → S. B. Myers 1941 — Riemannian manifolds with positive mean curvature — the record the joint stands on
- cites → R. S. Hamilton 1982 — Three-manifolds with positive Ricci curvature — the record the joint stands on
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