指标定理（一） (11-5)

As trivializations of E，we take only those maps фα that send orthonormal frames of E (relative to the global metric 〈，〉) to orthonormal frames of ℝⁿ. Then the transition functions gαᵦ will preserve orthonormal frames and hence take values in the orthogonal group O(n).If the determinant of gαᵦ is positive，gαᵦ will actually be in the special orthogonal group SO(n). Similar discussion applies to the complex case.

Functorial operations on vector spaces carry over to vector bundles，For instance，if E and E' are vector bundles over M of rank n and m respectively，their direct sum E ⨁ E' is the vector bundle over M whose fiber at the point x in M is E᙮ ⨁ E'᙮· The local trivializations {фα}，{ф'α} for E and E' induce a local trivialization for E ⨁ E'：

фα ⨁ ф'α：E ⨁ E'|∪α → ∪α × (ℝⁿ ⨁ ℝᵐ).

Similarly we can define the tensor product E ⨁ E'，the dual E*.and Hom (E，E').

Let ξ be a vector bundle with projection π： E → B and let B₁ be an arbitrary topological space. Given any map f：B₁ → B one can construct the induced bundle (pullback) f*ξ over B₁ . The total space E₁ of f*ξ is the subset E₁ ⊂ B₁ × E consisting of all pairs (b，e) with f(b)＝π(e). The projection map π₁：E₁ → B₁ is defined by π₁(b，e)＝b.If (∪，h) is a local coordinate system for ξ. set U₁＝f⁻¹(∪) and define h₁：U₁ × ℝⁿ → π₁⁻¹(∪₁) by h₁(b，x)＝(b，h(f(b)，x)). Then (U₁，h₁) is clearly a local coordinate system for f*ξ . This proves that f*ξ is locally trivial.

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