Fibre Bundles and Chern-Weil Theory

1-1 correspondence adjoint representation algebra g Altk bijection bundle map bundle with fibre called characteristic classes commutative diagram compact connection forms curvature form defined Definition denote diffeomorphism differentiable manifold differentiable mapping differential forms equation equivalent equivariant Example Exercise F(Vp fibre bundle Fo(V follows frame bundle frame bundle F(V G-invariant gauge transformation given GL(n GL(V H C G HAR(M hence inclusion induced inner product Iso(R k-forms Lemma Let G Lie algebra Lie subgroup linear map local trivializations matrix Maurer-Cartan connection n-dimensional neighborhood Notation orthogonal p-equivariant P(Fk principal G-bundle principal GL(n product bundle proj projection Proof Proposition 6.8 R)-bundle reduction Remark Riemannian metric Rn+1 satisfying SO(n submanifold tangent Theorem 7.2 total space transition functions trivializations vector bundle vector field vector space w₁ Wn,k Αα θυ σα ση υκ φα Ω¹ მ მ