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I wrote Poincare group as an entree to the project of carrying on in nLab the blog discussion on unitary representations of the Poincare group. I'm not a specialist of course, so I ask the experts to please examine for accuracy.
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<p>thanks, Todd! That's very useful. I quickly added a toc and a list of some related entries (some of which need to be written).</p>
<p>When I have more time, I would like to replace</p>
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Minkwoski "metric"
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<p>with a link to <a href="https://ncatlab.org/nlab/show/pseudo-Riemannian+metric">pseudo-Riemannian metric</a>. We also need an entry on <a href="https://ncatlab.org/nlab/show/Riemannian+metric">Riemannian metric</a> still, it seems...</p>
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I wrote pseudo-Riemannian metric just now, and I have a query there.
Thanks again, Todd!
but wait, I am not sure I entirely agree with your current definition there:
it looks to me like what you are defining is what i would call a flat Lorentzian manifold: a pseudo-Riemannian manifold whose metric is locally the standard one on R^n. We happen to have an entry on the supergeometry analog of that at Euclidean supermanifold.
I would have said a Riemannian manifold is a smooth manifold equipped with a smooth section of the symmetrized second power of its cotangent bungle which is pointwise a positive definite bilinear form on tangent vectors.
(Then a pseudo-Riemannian manifold is the same with pos def replaced by nondegenerate.)
This or something equivalent.
Ah, I'm sure you're right. I'll fix this in a bit -- thanks.
I would join you, but I badly need to take care of something else.
So I just drop remarks here, since that is quicker:
eventually I am thinking we should try to find the the-nice-nLab-way of saying it. I am actually thinking that:
the traditional def of Riemannian structure with a symmetric rank two tensor (the one I just mentioned) is actualy BAD, at least as soon as metric compatible connections come into the game.
Most every textbook on this manages to make the notion of connection on a Riemannian manifold appear as something logically disconnected from connections on fiber bundles. Which is misleading.
I think the right answer is the "first order formalism" that every text that aims towards Spin geometry chooses: represent the metric and the Levi-Civita connection in terms of "vielbein" and "spin connection".
But that in turn just says: a Riemannian metric is a connection for the Euclidean (or, in the pseudo-Riemannian case: Poincare) group, subject to a non-degeneracy condition.
This is the definition that makes the closest contact to the most abstract nonsense. It also serves to make immediate the description of gravity and supergravity theories as gauge theories for connections. it generalizes correctly and seamlessly to spin- and supergeometry and then to higher connextions and makes everything clear.
I think the notion "connection on a bundle" is very deeply rooted in abstract nonsense, and whenever there is any ordinary concept that has an equivalent definition in terms of connections, then that definition tends to be the "right" definition.
We also need an entry on Riemannian metric still, it seems...
We already have Riemannian manifold, so it could go there too.
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