指标定理（一） (11-4)

A cross-section (sometimes just called a section) of a vector bundle ξ with base space B is a continuous function s : B → E(ξ) which takes each b ∈ B into the corresponding fiber Fb(ξ). Such a cross-section is nowhere zero if s(b) is a non-zero vector of Fb(ξ) for each b.

Lemma 1 An Rⁿ-bundle ξ is trivial if and only if ξ αdmits n cross-sections s₁，. . .，sₙ ωhich αre noωhere dependent.

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¹Index theory. Encyclopedia of Mathematics.

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Definition 1 A Euclidean υector bundle is α real vector bundle ξ together ωith α contin- uous function

μ：E(ξ) → ℝ

such thαt the restriction of μ to each fiber of ξ is positiυe definite αnd quadratic. The function μ itself ωill be called α Buclideαn metric on the υector bndle ξ .

A Euclidean structure on a tangent bundle is indeed a Riemannian metric.

For a vector bundle E over a manifold (which admits partition of unity)，we now show that the structure group of E may be reduced to the orthogonal group. First，we can endow E with a Riemannian structure as follows.Let {∪α} be an open cover of M which trivializes E. On each ∪α， choose a frame for E|∪α . and declare it to be orthonormal. This defines a Riemannian structure on E|∪α . Now use a partition of unity {ρα} to form a global Riemannian metric.

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