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De Rham Cohomology of Manifolds and Vector Bundles
Summary of volume I
62 other sections not shown
a e G a e Sec action of G Assume bundle map called closed subgroup cohomology commutative diagram complex vector bundle connected Lie group coordinate representation corresponding covariant exterior derivative cross-section curvature defined denotes determines diffeomorphism differential forms equivariant Example fibre follows function G is connected G-invariant given GL(F Hence horizontal induced inner product isometries left action left invariant Lemma Let G Lie algebra Lie group G linear connection linear isomorphism linear map manifold maximal torus Moreover morphism obtain orbit orientation Pontrjagin principal bundle principal connection problem projection Proof Proposition VII Q.E.D. Corollary quaternionic real vector respect restricts Riemannian connection Riemannian metric right action satisfies Show smooth map subalgebra subbundle subgroup of G subset subspace surjective tangent bundle Te(G tensor Theorem trivial unique vector field vector space volume Z'-bundle zero