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 definitions 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 integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum 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).

nLab > Latest Changes: basic differential form

Bottom of Page

1 to 4 of 4

  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 27th 2019

    just for completeness, am giving this its own little entry, finally

    v1, current

    • CommentRowNumber2.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 27th 2023

    Added:

    For general submersions

    Given a submersion p:→B, one may ask: which differential forms on E are pullbacks of differential forms on B?

    If the fibers of p are connected (otherwise the characterization given below is valid only locally in E), the answer is provided by the notion of a basic form: a form ω is basic if the following two conditions are met:

    • The contraction of ω with any p-vertical vector field is zero.

    • The Lie derivative of ω with respect to any p-vertical vector field is zero.

    Using Cartan’s magic formula, in the presence of the first condition, the second condition can be replaced by the following one:

    • The contraction of dω (where d is the de Rham differential) with any p-vertical vector field is zero.

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeApr 27th 2023

    Thanks.

    We need to do something about the links to contraction… Let me see….

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeApr 27th 2023

    First of all, I have now made the links for “contraction” point to tensor contraction.

    Next to improve the disambiguation at contraction…

    diff, v3, current

1 to 4 of 4

  1.