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| Format: | Preprint |
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2025
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| Accès en ligne: | https://arxiv.org/abs/2512.19050 |
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| _version_ | 1866918258933760000 |
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| author | Aazami, Amir Babak |
| author_facet | Aazami, Amir Babak |
| contents | We study oriented Riemannian $4n$-manifolds whose Thorpe $2n^{\text{th}}$ curvature operator $\hat{R}_{2n}\colonΛ^{2n} \longrightarrow Λ^{2n}$, or its Weyl analogue $\hat{W}_{2n}$, commutes with the Hodge star. For pure curvature operators this commuting condition becomes a finite system of hafnian identities in the eigenvalues of the curvature operator, which we analyze in two subclasses, including the locally conformally flat case. We further observe that $*\hat{W}_{2n} = \hat{W}_{2n}*$ is a new conformal invariant in dimensions $4n$, providing higher-dimensional analogues of self-duality. Finally, we give sufficient conditions ensuring nonnegativity of the Euler characteristic and relate these conditions to normal forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_19050 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the curvature operator in dimensions $4n$ Aazami, Amir Babak Differential Geometry We study oriented Riemannian $4n$-manifolds whose Thorpe $2n^{\text{th}}$ curvature operator $\hat{R}_{2n}\colonΛ^{2n} \longrightarrow Λ^{2n}$, or its Weyl analogue $\hat{W}_{2n}$, commutes with the Hodge star. For pure curvature operators this commuting condition becomes a finite system of hafnian identities in the eigenvalues of the curvature operator, which we analyze in two subclasses, including the locally conformally flat case. We further observe that $*\hat{W}_{2n} = \hat{W}_{2n}*$ is a new conformal invariant in dimensions $4n$, providing higher-dimensional analogues of self-duality. Finally, we give sufficient conditions ensuring nonnegativity of the Euler characteristic and relate these conditions to normal forms. |
| title | On the curvature operator in dimensions $4n$ |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2512.19050 |