Hi, I'm a little stuck on Nakahara's treatment about fibre bundles. I hope someone can give me a clear answer on this; they are quite elementary questions, I guess.(adsbygoogle = window.adsbygoogle || []).push({});

We have:

* A principal bundle P(M,G)

* A fibre [itex]G_{p}[/itex] at [itex]p= \pi(u)[/itex]

Then the vertical subspace [itex]V_{u}P[/itex] is defined as a subspace of the tangent space [itex]T_{u}P[/itex] which is tangent to the fibre [itex]G_{p}[/itex] at u. An element [itex]X[/itex] of this vertical subspace is called the fundamental vector field, and is defined as

[tex]

X f(u) = \frac{d}{dt}f(ue^{tA})|_{t=0}

[/tex]

in analogy with tangent vectors on a manifold M. Here f is a curve from P to the real line and A is an element of your left invariant vector field on G. It is then asked to show that (ex. 10.1)

[tex]

\pi_{*}X = 0

[/tex]

for a fundamental vector field X. I don't see this, and it's probably because I don't grasp the meaning of this vertical space. Is it only tangent to the fibre but not to the base manifold M? Is that the reason for the pushforward vector (which is defined on M) is zero (because vectors on M are only defined in the tangent space [itex]T_{p}M[/itex])?

It is further claimed on page 378 that if you act with the Ehresmann connection on a fundamental vector field, you can use

[tex]

d_{P}g_{i} = \frac{dg(ue^{tA})}{dt}|_{t=0}

[/tex]

where d is the exterior derivative on P. Where does this come from? I hope my questions are clear and that someone can help me with this.

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# Elementary questions about fibre bundles

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