Intermediate Ricci curvatures and Gromov's Betti number bound

2022/08/29 by Philipp Reiser, Reiser, Philipp, David J. Wraith +1 · 5 citations
Mathematics · Medicine · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Ophthalmology and Eye Disorders

paper · pdf · doi:10.48550/arxiv.2208.13438

Abstract

We consider intermediate Ricci curvatures Ric_k on a closed Riemannian manifold Mⁿ. These interpolate between the Ricci curvature when k=n-1 and the sectional curvature when k=1. By establishing a surgery result for Riemannian metrics with Ric_k>0, we show that Gromov's upper Betti number bound for sectional curvature bounded below fails to hold for Ric_k>0 when \lfloor n/2 \rfloor+2 ≤ k ≤ n-1. This was previously known only in the case of positive Ricci curvature.

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