On the Petrov Type of a 4-manifold
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866910425687261184 |
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| author | Aazami, Amir Babak |
| author_facet | Aazami, Amir Babak |
| contents | On an oriented 4-manifold, we examine the geometry that arises when the curvature operator of a Riemannian or Lorentzian metric $g$ commutes, not with its own Hodge star operator, but rather with that of another semi-Riemannian metric $h$ that is a suitable deformation of $g$. We classify the case when one of these metrics is Riemannian and the other Lorentzian by generalizing the concept of Petrov Type from general relativity; the case when $h$ is split-signature is also examined. The "generalized Petrov Types" so obtained are shown to relate to the critical points of $g$'s sectional curvature, and sometimes yield unique normal forms. They also carry topological information independent of the Hitchin-Thorpe inequality, and yield a direct geometric formulation of "almost-Einsten" metric via the Ricci or sectional curvature of $g$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_13717 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the Petrov Type of a 4-manifold Aazami, Amir Babak Differential Geometry On an oriented 4-manifold, we examine the geometry that arises when the curvature operator of a Riemannian or Lorentzian metric $g$ commutes, not with its own Hodge star operator, but rather with that of another semi-Riemannian metric $h$ that is a suitable deformation of $g$. We classify the case when one of these metrics is Riemannian and the other Lorentzian by generalizing the concept of Petrov Type from general relativity; the case when $h$ is split-signature is also examined. The "generalized Petrov Types" so obtained are shown to relate to the critical points of $g$'s sectional curvature, and sometimes yield unique normal forms. They also carry topological information independent of the Hitchin-Thorpe inequality, and yield a direct geometric formulation of "almost-Einsten" metric via the Ricci or sectional curvature of $g$. |
| title | On the Petrov Type of a 4-manifold |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2309.13717 |