Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory infinity integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 19th 2009

    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.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 19th 2009
    This comment is invalid XHTML+MathML+SVG; displaying source. <div> <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> <blockquote> Minkwoski "metric" </blockquote> <p>with a link to <a href="">pseudo-Riemannian metric</a>. We also need an entry on <a href="">Riemannian metric</a> still, it seems...</p> </div>
    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 19th 2009

    I wrote pseudo-Riemannian metric just now, and I have a query there.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeOct 19th 2009

    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.

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 19th 2009

    Ah, I'm sure you're right. I'll fix this in a bit -- thanks.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeOct 19th 2009

    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.

    • CommentRowNumber7.
    • CommentAuthorTobyBartels
    • CommentTimeOct 20th 2009

    We also need an entry on Riemannian metric still, it seems...

    We already have Riemannian manifold, so it could go there too.