Conformally Kähler structures
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915415201939456 |
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| author | Dunajski, Maciej Gover, A. Rod |
| author_facet | Dunajski, Maciej Gover, A. Rod |
| contents | We establish a one-to-one correspondence between Kähler metrics in a given conformal class and parallel sections of a certain vector bundle with conformally invariant connection, where the parallel sections satisfy a set of non--linear algebraic constraints that we describe. The vector bundle captures 2-form prolongations and is isomorphic to $Λ^3(\cT)$, where ${\cT}$ is the tractor bundle of conformal geometry, but the resulting connection differs from the normal tractor connection by curvature terms.
Our analysis leads to a set of obstructions for a Riemannian metric to be conformal to a Kähler metric. In particular we find an explicit algebraic condition for a Weyl tensor which must hold if there exists a conformal Killing-Yano tensor, which is a necessary condition for a metric to be conformal to Kähler. This gives an invariant characterisation of algebraically special Riemannian metrics of type $D$ in dimensions higher than four. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_17347 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Conformally Kähler structures Dunajski, Maciej Gover, A. Rod Differential Geometry Mathematical Physics We establish a one-to-one correspondence between Kähler metrics in a given conformal class and parallel sections of a certain vector bundle with conformally invariant connection, where the parallel sections satisfy a set of non--linear algebraic constraints that we describe. The vector bundle captures 2-form prolongations and is isomorphic to $Λ^3(\cT)$, where ${\cT}$ is the tractor bundle of conformal geometry, but the resulting connection differs from the normal tractor connection by curvature terms. Our analysis leads to a set of obstructions for a Riemannian metric to be conformal to a Kähler metric. In particular we find an explicit algebraic condition for a Weyl tensor which must hold if there exists a conformal Killing-Yano tensor, which is a necessary condition for a metric to be conformal to Kähler. This gives an invariant characterisation of algebraically special Riemannian metrics of type $D$ in dimensions higher than four. |
| title | Conformally Kähler structures |
| topic | Differential Geometry Mathematical Physics |
| url | https://arxiv.org/abs/2409.17347 |