Decomposition of the Curvature Operator and Applications to the Hopf Conjecture
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918104129339392 |
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| author | Huang, Teng Wang, Weiwei |
| author_facet | Huang, Teng Wang, Weiwei |
| contents | In this article, we investigate the interplay between the curvature operator, Weyl curvature, and the Hopf conjecture on compact Riemannian manifolds of even dimension. By decomposing the curvature operator into Hermitian components, we develop eigenvalue criteria for sectional curvature and prove vanishing theorems for Betti numbers under integral bounds on the Weyl tensor. Our results confirm the Hopf conjecture for manifolds with sufficiently small Weyl curvature, including locally conformally flat cases, and provide new rigidity theorems under harmonic Weyl curvature conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_15237 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Decomposition of the Curvature Operator and Applications to the Hopf Conjecture Huang, Teng Wang, Weiwei Differential Geometry In this article, we investigate the interplay between the curvature operator, Weyl curvature, and the Hopf conjecture on compact Riemannian manifolds of even dimension. By decomposing the curvature operator into Hermitian components, we develop eigenvalue criteria for sectional curvature and prove vanishing theorems for Betti numbers under integral bounds on the Weyl tensor. Our results confirm the Hopf conjecture for manifolds with sufficiently small Weyl curvature, including locally conformally flat cases, and provide new rigidity theorems under harmonic Weyl curvature conditions. |
| title | Decomposition of the Curvature Operator and Applications to the Hopf Conjecture |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2507.15237 |