On Topology of Compact Hessian Manifolds
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866908661175025664 |
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| author | Liu, Hanwen |
| author_facet | Liu, Hanwen |
| contents | For a differentiable manifold $M$, a pair $(M, \nabla)$ is called an affine manifold if $\nabla$ is a flat and torsion-free connection on the tangent bundle $TM\rightarrow M$. A Riemannian metric $g$ on $M$ is said to be a Hessian metric on $(M, \nabla)$ if the $(0,3)$-tensor $\nabla g$ is totally symmetric.
We investigate general topological properties of compact Hessian manifolds, in particular confirm a long-standing conjecture of S.S.Chern in this special case, and then apply these constraints to low dimensions, providing a thorough topological classification of complete Hessian surfaces and closed orientable Hessian manifolds of dimension less than $5$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_01176 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Topology of Compact Hessian Manifolds Liu, Hanwen Differential Geometry For a differentiable manifold $M$, a pair $(M, \nabla)$ is called an affine manifold if $\nabla$ is a flat and torsion-free connection on the tangent bundle $TM\rightarrow M$. A Riemannian metric $g$ on $M$ is said to be a Hessian metric on $(M, \nabla)$ if the $(0,3)$-tensor $\nabla g$ is totally symmetric. We investigate general topological properties of compact Hessian manifolds, in particular confirm a long-standing conjecture of S.S.Chern in this special case, and then apply these constraints to low dimensions, providing a thorough topological classification of complete Hessian surfaces and closed orientable Hessian manifolds of dimension less than $5$. |
| title | On Topology of Compact Hessian Manifolds |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2509.01176 |