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associated bundle base space bijection bundle map closed sets cohomology groups commutator compact components convergence coordinate chart coordinate system cotangent cross-section defined denoted diffeomorphisms differentiable manifold differential forms differential geometry equation equivalence class Euclidean space example exists exterior derivative fibre bundle Figure filter base function G-action G-bundle gauge transformation generalised GL(m GL(n GL+(n global hence homomorphism horizontal lift idea implies important induced infinite-dimensional integral curve isomorphism lattice left-invariant vector field Lie algebra Lie group Lie group G linear map map h map TT matrix metric space n-form neighbourhood Note one-form open sets open subset pair particular precisely principal bundle product bundle Proof pull-back push-forward QED Comments real numbers real vector space relation respect satisfies sequence smooth spacetime subspace surjective tangent space tangent vector tensor theorem theoretical physics topological space topology trivialisation unique vector bundle Yang-Mills field Yang-Mills theory
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