On the Petrov Type of a 4-manifold
| Autor: | Aazami, Amir Babak |
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| Rok vydání: | 2023 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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 $g$. Comment: 22 pages; v2: significant expansion: main result strengthened, streamlined, and extended to cover all signatures |
| Databáze: | arXiv |
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