New curvature conditions for the Bochner Technique

Abstract

We show that manifolds with \( \lceil \frac{n}{2} \rceil \)-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.

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Acknowledgements

We would like to thank Christoph Böhm and the referee for constructive comments, for suggestions that improved the exposition and for pointing out Example 4.3(b).

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Correspondence to Matthias Wink.

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Petersen, P., Wink, M. New curvature conditions for the Bochner Technique. Invent. math. (2020). https://doi.org/10.1007/s00222-020-01003-3

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Mathematics Subject Classification

  • 53B20
  • 53C20
  • 53C21
  • 53C23
  • 58A14