For a compact Einstein manifold with Einstein constant α>0, sectional curvature at least δ implies a spherical space form when δ > α/n and a homology sphere when δ > α/(n+2).
Math., 142, 435–450 (2000)
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On the Betti and Tachibana numbers of compact Einstein manifolds
For a compact Einstein manifold with Einstein constant α>0, sectional curvature at least δ implies a spherical space form when δ > α/n and a homology sphere when δ > α/(n+2).