New curvature conditions for the Bochner Technique
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New curvature conditions for the Bochner Technique Peter Petersen1 · Matthias Wink1
Received: 30 January 2020 / Accepted: 9 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We show that manifolds with n2 -positive curvature operators are rational homology spheres. This follows from a more general vanishing and estimation theorem for the pth Betti number of closed n-dimensional Riemannian manifolds with a lower bound on the average of the lowest n − p eigenvalues of the curvature operator. This generalizes results due to D. Meyer, Gallot–Meyer, and Gallot. Mathematics Subject Classification 53B20 · 53C20 · 53C21 · 53C23 · 58A14
Introduction A fundamental theme in Riemannian geometry is to understand the relationship between the curvature and the topology of a Riemannian manifold. The Bochner technique addresses this question by studying the existence of harmonic tensors on closed Riemannian manifolds. This is motivated by Hodge’s theorem which asserts that every de Rham cohomology class is represented by a harmonic form.
B Matthias Wink
[email protected] Peter Petersen [email protected]
1
Department of Mathematics, UCLA, 520 Portola Plaza, Los Angeles, CA 90095, USA
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P. Petersen, M. Wink
Bochner [5] opened up a link to geometry and proved that the first Betti number of compact manifolds with positive Ricci curvature vanishes. Berger [1] and Meyer [26] established vanishing results for the Betti numbers of manifolds with positive curvature operators and in particular Meyer showed that they are rational (co)homology spheres. Furthermore, Micallef–Wang [29] proved that the second Betti number of even dimensional manifolds with positive isotropic curvature vanishes. The Ricci flow has been used extensively to obtain classification results, which in particular imply Bochner vanishing-type theorems. For example, Hamilton [20,21], Chen [13] and Böhm–Wilking [11] showed that manifolds with positive, in fact 2-positive, curvature operators are space forms. Brendle– Schoen [10] and Brendle [6] showed that this is more generally the case for manifolds whose product with R2 and R, respectively, have positive isotropic curvature. As a consequence, Brendle–Schoen [10] proved the differentiable sphere theorem. Based on Ricci flow with surgery, compact manifolds with positive isotropic curvature have been classified by Hamilton [22], Chen–Zhu [15] and Chen– Tang–Zhu [14] in dimension n = 4 and by Brendle [8] and Huang [24] in dimensions n ≥ 12. Using different techniques, Micallef–Moore [27] proved that simply connected compact manifolds with positive isotropic curvature are homotopy spheres. Our first main theorem introduces nested curvature conditions that give rise to different vanishing results for the Betti numbers b p (M). Recall that the curvature operator of a Riemannian manifold is called l-positive if the sum of its lowest l eigenvalues is positive. Theorem A Let n ≥ 3 and 1 ≤ p ≤ n2 . If (M, g) is a closed n-dimensional Riemannian manifold with (n − p)-positiv
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