Exterior derivatives on fibre bundles

In summary, the conversation discusses the use of the Ehresmann connection on a principal bundle P, which is represented by a Lie-algebra valued one-form omega. This one-form can be pulled back to the basemanifold using a section sigma, where it is interpreted as a gauge potential. To ensure that omega fulfills the two axioms of a connection one-form, one must check its identification using a canonical local trivialization g_i and a fundamental vector field A^*. This identification is given by d_P g_i(A^*) = dg(ue^tA)/dt|_{t=0}. The conversation ends with the problem being solved by applying X(f) = df(X), specifically A(g) = dg(A).
  • #1
haushofer
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Hi, I have a small question about exterior derivatives d on defined on principal bundles P.

We have the Ehresmann connection on a principal bundle P, represented by a Lie-algebra valued one-form omega. We can use the section sigma to pull this one-form back to our basemanifold, where the interpretation of a gaugepotential comes in.

Now one postulates this omega on P and checks if it fullfills the 2 axioms of this connection one-form. Now they (Nakahara chapter 10 ) use the following identification, which I don't see:

[tex]
d_{P} g_{i} (A^{*} ) = \frac{dg(ue^{tA})}{dt} \ |_{t=0}
[/tex]

Here [itex]g_{i}[/itex] is the canonical local trivialization,

[tex]
\phi_{i}^{-1}(p,g_{i}) = (p,g) \, \ \ \ u = \sigma_{i}(p) g_{i}
[/tex]

, A is an element of the Lie-algebra and [itex]A^{*}[/itex] is the fundamental vector field lying in the vertical subspace:

[tex]
A^{*}f(u) = \frac{d}{dt}f(ue^{tA}) \ |_{t=0}
[/tex]

Anyone an idea?
 
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  • #2
Okiedokie, solved the problem. :')

Just an application of X(f) = df(X), so A(g) = dg(A).
 

Related to Exterior derivatives on fibre bundles

1. What is an exterior derivative on a fibre bundle?

An exterior derivative on a fibre bundle is a mathematical operation that maps differential forms from one fibre to another. It is used to study the topology and geometry of a fibre bundle, and is an essential tool in differential geometry and differential topology.

2. How is an exterior derivative defined on a fibre bundle?

The exterior derivative on a fibre bundle is defined as a linear map from the space of differential forms on one fibre to the space of differential forms on another fibre. It is defined using the concept of a connection on a fibre bundle, which allows for the comparison of differential forms on different fibres.

3. What are the applications of exterior derivatives on fibre bundles?

Exterior derivatives on fibre bundles have various applications in mathematics and physics. They are used in the study of differential equations, gauge theories, and general relativity. They also have applications in geometry, topology, and algebraic geometry.

4. How are exterior derivatives related to other mathematical concepts?

An exterior derivative on a fibre bundle is closely related to the concept of a differential form, which is a mathematical object used to study calculus on manifolds. It is also related to the notion of a vector field, which is a mathematical object used to describe the behavior of vector quantities in a space.

5. Are there any limitations to using exterior derivatives on fibre bundles?

One limitation of using exterior derivatives on fibre bundles is that it requires a smooth manifold structure on the fibres. This restricts its application to only smooth fibre bundles. Additionally, the computation of exterior derivatives can be challenging, especially for higher-dimensional fibres.

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