# Download Connections, Curvature, and Cohomology: Cohomology of by Werner Hildbert Greub PDF By Werner Hildbert Greub

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Extra resources for Connections, Curvature, and Cohomology: Cohomology of principal bundles and homogeneous spaces

Sample text

We can even limit our search to tangent bundles. Thus we ask the following question. Is it true that for every smooth manifold M the tangent bundle T M is trivial (as a vector bundle)? Let us look at some positive examples. 45. T S 1 ∼ = RS 1 Let θ denote the angular coordinate on the circle. Then ∂ is a globally defined, nowhere vanishing vector field on S 1 . We thus get a map ∂θ RS 1 → T S 1 , (s, θ) → (s ∂ , θ) ∈ Tθ S 1 ∂θ which is easily seen to be a bundle isomorphism. Let us carefully analyze this example.

A bundle isomorr phism E → KM is called a trivialization of E, while an isomorphism Kr → E is called a framing of E. A pair (trivial vector bundle, trivialization) is called a trivialized, or framed bundle. 44. Let us explain why we refer to a bundle isomorphism ϕ : KrM → E as a framing. Denote by (e1 , . . , er ) the canonical basis of Kr . , as (special) sections of KrM . The isomorphism ϕ determines sections fi = ϕ(ei ) of E with the property that for every x ∈ M the collection (f1 (x), . .

Often, when the field of scalars is clear from the context, we will use the simpler notation E ⊗ F . The tensor product has the following universality property. 1. For any bilinear map φ : E × F → G there exists a unique linear map Φ : E ⊗ F → G such that the diagram below is commutative. ι '' w E ⊗ F '' Φ ') u φ E×F . G The proof of this result is left to the reader as an exercise. Note that if (ei ) is a basis of E, and (fj ) is a basis of F , then (ei ⊗ fj ) is a basis of E ⊗ F , and therefore dimK E ⊗K F = (dimK E) · (dimK F ).