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Some sort of vector bundle appears in most every aspect differential geometry. This chapter introduces these objects and gives the first few examples. By way of a warning, I give my favorite definition first; the standard definition is given subsequently in Chapter 3b. (These definitions describe the same objects). 3a) The definition Let M denote a smooth manifold of dimension m. A real vector bundle over M of fiber dimension n is a smooth manifold, E, of dimension n + m with the following additional structure: • • • • There is a smooth map π: E → M.

Define a rank n vector bundle E → Sm by using the coordinate patches U+ = Sm−(0, …, -1) and U- = Sm−(0, …, +1) with the transition function that g U- U+ send a point (x, xn+1) ∈ Sn ⊂ Rn × R to g(x/|x|). The resulting bundle ψ*E → M is isomorphic to the bundle constructed in Chapter 1c using the data consisting of Λ and the set {(Up, ϕp, gp = g)}p∈Λ. 5b) Pull-backs and Grassmannians As it turns out, any given vector bundle is isomorphic to the pull-back of a tautological bundle over a Grassmannian, these as described in Chapter 1f.

46 4b) Quotient bundles Suppose that V is a vector space and V´ ⊂ V a vector subspace. One can then define the quotient space V/V´, this the set of equivalence classes that are defined by the rule that has v ~ u when v - u ∈ V´. The quotient V/V´ is a vector space in its own right, its dimension is dim(V) - dim(V´). Note that in the case when V is identified with Rn, the orthogonal projection from Rn to the orthogonal complement of V´ identifies the quotient bundle V/V´ with this same orthogonal complement.

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