指标定理（一） (11-6)

Consider two vector bundles ξ：E(ξ) → B and η：E(η) → B over the same base space B with E(ξ) ⊂ E(η)； then ξ is a subbundle of η if each fiber Fb(ξ) is a sub-vector-space of the corresponding fiber Fb(η). Given a subbundle ξ ⊂ η， if η is provided with a Euclidean metric then a complementary summand can be constructed as follows.

Let Fb(ξ⊥) denote the subspace of Fb(η) consisting of all vectors e such that υ · ω＝0 for all ω ∈ Fb(ξ). Let E(ξ⊥)⊂ E(η) denote the union of the Fb(ξ⊥).

Lemma 2 E(ξ⊥) is the total space of α sub-bundle ξ⊥ ⊂ η. Furthermore η is isomorphic to the Whitney sum ξ ⨁ ξ⊥ .

One may construct the normal bundle of an immersion in the obvious way.

1.3 Grassmann manifolds

The Grassmann manifold Gₙ(ℝⁿ⁺ᵏ) is the set of all n-dimensional planes through the origin of the coordinate space ℝⁿ⁺ᵏ . This is to be topologized as a quotient space，as follows.

4

An n-frame in ℝⁿ⁺ᵏ is an n-tuple of linearly independent vectors of ℝⁿ⁺ᵏ . The collection of all n-frames in ℝⁿ⁺ᵏ forms an open subset of the n-fold Cartesian product ℝⁿ⁺ᵏ × · · · × ℝⁿ⁺ᵏ，called the Stiefel manifold Vₙ(ℝⁿ⁺ᵏ). There is a canonical map q：Vₙ(ℝⁿ⁺ᵏ) → Gₙ (ℝⁿ⁺ᵏ) which maps each n-frame to the n-plane which it spans. Now give Gₙ(ℝⁿ⁺ᵏ) the quotient topology.

Grassmann manifolds can also be viewed as the collection of all orthogonal projections of rank n.

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