Curvature operator and Euler number
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908471263232000 |
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| author | Teng, Huang Qiang, Tan |
| author_facet | Teng, Huang Qiang, Tan |
| contents | Let $(X,g)$ be a compact $n$-dimensional smooth Riemannian manifold with a lower bound on the average of the lowest $n-p$ eigenvalues of the curvature operator and the diameter of $X$ is bounded above by $D>0$. In this article, we investigate the relationship between the curvature operator and the Euler number of $X$. Our analysis is based on more general vanishing theorems for a Dirac operator associated with a smooth $1$-form on $X$. As a consequence, we obtain partial affirmative answers to Question 4.6 posed by Herrmann, Sebastian, and Tuschmann in \cite{HST}. Specifically, we prove that if a compact $2m$-dimensional manifold admits an almost nonnegative curvature operator (ANCO) and has a nontrivial first de Rham cohomology group, then its Euler number vanishes. Furthermore, in the case where $m=2$, we show that the Euler number is nonnegative. This result provides a complete resolution to their question in the four-dimensional setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_22091 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Curvature operator and Euler number Teng, Huang Qiang, Tan Differential Geometry Let $(X,g)$ be a compact $n$-dimensional smooth Riemannian manifold with a lower bound on the average of the lowest $n-p$ eigenvalues of the curvature operator and the diameter of $X$ is bounded above by $D>0$. In this article, we investigate the relationship between the curvature operator and the Euler number of $X$. Our analysis is based on more general vanishing theorems for a Dirac operator associated with a smooth $1$-form on $X$. As a consequence, we obtain partial affirmative answers to Question 4.6 posed by Herrmann, Sebastian, and Tuschmann in \cite{HST}. Specifically, we prove that if a compact $2m$-dimensional manifold admits an almost nonnegative curvature operator (ANCO) and has a nontrivial first de Rham cohomology group, then its Euler number vanishes. Furthermore, in the case where $m=2$, we show that the Euler number is nonnegative. This result provides a complete resolution to their question in the four-dimensional setting. |
| title | Curvature operator and Euler number |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2507.22091 |