Not signed in (Sign In)

Start a new discussion

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-categories 2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive constructive-mathematics cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry differential-topology digraphs duality elliptic-cohomology enriched fibration foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity 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 integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monads monoidal monoidal-category-theory morphism motives motivic-cohomology multicategories newpage 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 science set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory string string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory 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.
    • CommentAuthorUrs
    • CommentTimeAug 18th 2010
    • (edited Aug 18th 2010)

    tried to polish a bit the matrial at Chern-Weil theory.

    (not that there is much, yet, but still)

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeSep 19th 2010

    I pasted some historical comments and references (but still incomplete currently) provided by Jim Stasheff into Chern-Weil theory. More to come.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeSep 19th 2010

    Yet more ancient references, this time kindly dug out by Alan Hatcher.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMay 9th 2011

    I have expanded the Idea-section at Chern-Weil theory, giving a more detailed survey of the main constructions.

    (this is a spin-off of me staring to write an “Introduction and Survey”-section on my personal web at infinity-Chern-Simons theory (schreiber))

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeAug 27th 2020

    I have finally dug through the original citations for the “Chern-Weil” constructions. Have added the following to the entry:

    The differential-geometric “Chern-Weil”-construction (evaluating curvature 2-forms of connections in invariant polynomials) is due to

    • Henri Cartan, Section 7 of: Notions d’algébre différentielle; applications aux groupes de Lie et aux variétés où opère un groupe de Lie, Coll. Topologie Algébrique Bruxelles (May 1950) 15-28 (numdam:SHC_1949-1950__2__A18_0)

    and around equation (10) of:

    • {#Chern50} Shiing-shen Chern, Differential geometry of fiber bundles, in: Proceedings of the International Congress of Mathematicians, Cambridge, Mass., (August-September 1950), vol. 2, pages 397-411, Amer. Math. Soc., Providence, R. I. (1952) (pdf)

    It is the independence of this construction under the choice of connection which Chern 50 atributes (below (10)) to

    • André Weil, Géométrie différentielle des espaces fibres, unpublished

    But the main result of Chern 50 is that this differential-geometric “Chern-Weil” construction is equivalent to the topological (homotopy theoretic) construction of pulling back the universal characteristic classes from the classifying space BGB G along the classifying map of the given principal bundle:

    This claim is equation (15) in Chern 50, using (quoting from the same page):

    methods initiated by E. Cartan and recently developed with success by H. Cartan, Chevalley, Koszul, Leray, and Weil [13]

    Here reference 13 is:

    • Jean-Louis Koszul, Homologie et cohomologie des algebres de Lie, Bull. Soc. Math. France vol. 78 (1950) pp. 65-127

    Later, an independent proof of the universal topological “Chern-Weil”-construction inv(𝔤)H (BG)inv(\mathfrak{g}) \to H^\bullet(B G) is given in:

    diff, v23, current

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)