List questions from principal-bundles

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principal bundles references

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Transition fuctions only depend on homotopy

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List Question

Unclear construction in Hitchins paper on Integrable Systems

Invariant differential forms?

Intuition behind the naming of horizontal and vertical tangent spaces?

principal bundles references

Why does the adjoint representation appear in the definition of a connection 1-form?

Intuition behind connection 1-forms and Ehresmann connections

Intuition behind curvature 2-forms

How does $\partial_\mu$ act on $A_\nu$ in the local structure equation for a curvature 2-form written in coordinates?

Local expression of covariant derivative on associated vector bundle

Is the total space of the frame bundle of a smooth manifold $M$ always spin?

Doubts on the proof of the definition of the local connection form in Nakahara's book

Principal bundles: Reductions and lifts of classifying maps

Does parallel transport apply to points or vectors?

A tricky question on $O(2)$ bundles over surfaces

Transition fuctions only depend on homotopy

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