Intermediate Ricci Curvatures and Gromov’s Betti number bound

We consider intermediate Ricci curvatures R i c k on a closed Riemannian manifold M n . 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 R i c k > 0 , we show that Gromov’s upper Bett...

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Bibliographic Details
Published inThe Journal of geometric analysis Vol. 33; no. 12
Main Authors Reiser, Philipp, Wraith, David J.
Format Journal Article
LanguageEnglish
Published New York Springer US 01.12.2023
Springer Nature B.V
Subjects
Abstract Harmonic Analysis
Convex and Discrete Geometry
Curvature
Differential Geometry
Dynamical Systems and Ergodic Theory
Fourier Analysis
Geometry
Global Analysis and Analysis on Manifolds
Mathematics
Mathematics and Statistics
Riemann manifold
Riemannian geometry
53C20
Intermediate Ricci curvatures
Total Betti numbers
Surgery
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Summary:We consider intermediate Ricci curvatures R i c k on a closed Riemannian manifold M n . 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 R i c k > 0 , we show that Gromov’s upper Betti number bound for sectional curvature bounded below fails to hold for R i c k > 0 when ⌊ n / 2 ⌋ + 2 ≤ k ≤ n - 1 . This was previously known only in the case of positive Ricci curvature (Sha and Yang in J Differ Geom 29(1):95–103, 1989, J Differ Geom 33:127–138, 1991).
ISSN:1050-6926
1559-002X
DOI:10.1007/s12220-023-01423-6