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I try to understand the notion of connections on fibre bundles from the lecture notes http://arxiv.org/abs/math-ph/9902027" by George Svetlichny.
On page 27 stands what the attached picture shows.
(I don't know whether the attached picture will be visible, so I copy the text here:
I don't see, why must have this projection function must have this form. Why must [tex]\Gamma[/tex] not depend on y, and why must it be linear function of ξ ?
On page 27 stands what the attached picture shows.
(I don't know whether the attached picture will be visible, so I copy the text here:
)Let us examine this in a trivialization. One has T(x,f)(U×F) ≃ TxU×TfF.
Let (ξ, y) ∈ TxU × TfF. Such a vector is vertical if and only if ξ = 0. Thus
πv
(x,f) must have the form
πv
(x,f)(ξ, y) = (0, y + (x, f)ξ)
where (x, f) is a linear map TxU → TfF.
I don't see, why must have this projection function must have this form. Why must [tex]\Gamma[/tex] not depend on y, and why must it be linear function of ξ ?
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