Talk:Einstein manifold
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Dimension discussion
[edit]I would like to propose a section discussing how Einstein metrics behave by dimension, maybe something like:
- 2-dimensions -- Einstein metrics are constant curvature metrics -- so every closed surface has one.
- 3-dimensions -- Einstein metrics are constant curvature metrics -- so it is not difficult to tell if a 3-manifold admits an Einstein metric
- 4-dimensions -- Einstein metrics need not have constant sectional curvature (e.g. CP^2 with Fubini-Study metric). Some discussion on obstructions, including Hitchin-Thorpe inequality.
- 5-dimension and higher -- what little is known —Preceding unsigned comment added by Jjauregui (talk • contribs) 17:53, 6 March 2009 (UTC)
CP(n)
[edit]Please add the Einstein constant for the complex protective space. — Preceding unsigned comment added by 80.187.102.66 (talk) 13:34, 28 March 2015 (UTC)
Einstein and Lorentz Manifold
[edit]What is the difference? As stated, the metric of an Einstein manifold can have an arbitrary signature. If that were all, then the Lorentz manifold would be a special case of the Einstein manifold? I'm going to guess that it isn't this way: I'm guessing that the manifolds of Einstein–Cartan theory are Lorentz manifolds, but not Einstein manifolds, since there is not only curvature, but also torsion. Could an expert please clarify this (with reference in the article)? Thanks a lot!--Ernsts (talk) 22:19, 6 June 2022 (UTC)
- Lorentzian refers to the signature of the metric (-+++). Einstein requires the curvature to be proportional to the metric. These are not related to each other. In particular there are Lorentzian metrics which are not Einstein and Einstein metrics which are not Lorentzian. In general relativity one searches for Lorentzian Einstein metrics, and in Riemannian geometry one searches for Riemannian Einstein metrics, and so on. Tazerenix (talk) 08:14, 7 June 2022 (UTC)
- One considers the Ricci curvature of the Levi-Civita connection-the unique torsion free connection preserving the pseudo-Riemannian metric. So there is no torsion. ~2026-12880-40 (talk) 16:51, 27 February 2026 (UTC)