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Compact manifolds with a little negative curvature. (English) Zbl 0677.53050

The main result in this interesting research announcement generalizes the classical vanishing theorem of Bochner for harmonic 1-forms on a compact Riemannian manifold of positive Ricci curvature to the case where the Ricci curvature is allowed to be negative on a subset of sufficiently small diameter, where this diameter depends on bounds for the Ricci curvature. Related results about the finiteness of the fundamental group and about the vanishing of harmonic p-forms for \(p>1\) are also stated. Proofs are sketched.
Reviewer: M.Min-Oo

MSC:

53C20 Global Riemannian geometry, including pinching
58J35 Heat and other parabolic equation methods for PDEs on manifolds
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