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By Nicolas Bourbaki

Bourbaki Library of Congress Catalog #66-25377 published in France 1966

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However, to get some understanding via analysis of vector bundles, it is necessary to introduce a generalized notion of function (reflecting the geometry of the vector bundle) to which we can apply the tools of analysis. , s maps a point in the base space into the fibre over that point. S(X, E) will denote the S-sections of E over X. , S(U, E) = S(U, E|U ) [we shall also occasionally use the common notation (X, E) for sections, provided that there is no confusion as to which category we are dealing with].

We note that Ur,n (R) → Gr,n (R) is a real-analytic (and hence also differentiable) R-vector bundle and that Ur,n (C) → Gr,n (C) is a holomorphic vector bundle. The reason for the name “universal bundle” will be made more apparent later in this section. 7: Let π: E → X be an S-bundle and U an open subset of X. Then the restriction of E to U , denoted by E|U is the S-bundle π|π −1 (U ) : π −1 (U ) −→ U. , πE : E → X and πF : F → X. , f commutes with the projections and is a K-linear mapping when restricted to fibres.

9: Let E −→X be an S-bundle. An S-submanifold F ⊂ E is said to be an S-subbundle of E if (a) F ∩ Ex is a vector subspace of Ex . , there exist local trivializations for E and F which are compatible as in the following diagram: E|U O i F |U ∼ / U × Kr O id × j ∼ / U × Ks, s ≤ r, where the map j is the natural inclusion mapping of K s as a subspace of K r and i is the inclusion of F in E. We shall frequently use the language of linear algebra in discussing homof morphisms of vector bundles. As an example, suppose that E −→F is a vector bundle homomorphism of K-vector bundles over a space X.

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