MATH 623: VECTOR BUNDLE HW
C. ROBLES
Definition. Avector bundle Eof rank rover a manifold Mis a smooth manifold with
submersion π:E→Msuch that
(i) Ex=π−1(x) is a vector space of dimension rfor all x∈M,
(ii) there exists a covering {Uα|α∈A}of Mby open sets and diffeomorphisms ϕαsuch
that the diagram below commutes,
π−1(Uα)Uα×Rr
Uα
JJJJ
J^
πproj 1
-
ϕα
(iii) the restriction of ϕαto fibres is a linear isomorphism Ex→Rrfor all x∈Uα.
Defintion. The ϕαare local trivializations of E. It follows from the definition that the
map ϕβ◦ϕ−1
α,
(Uα∩Uβ)×Rrϕ−1
α
−→ π−1(Uα∩Uβ)ϕβ
−→ (Uα∩Uβ)×Rr,
is of the form
(x, v)7→ (x, gβα (x)v)
for some smooth map gβα :Uα∩Uβ→GLrR. The gβα are the transition functions associated
to the local trivializations.
Definition. Let Ebe a vector bundle over M. Define the dual bundle of Eto be
E∗={λ∈Ex
∗|x∈M}
with projection map π∗:λ∈Ex
∗7→ x∈M.
1.) Prove that E∗is a vector bundle.
Definition. Let Eand Fbe vector bundles over M. Define the product bundle of Eand
Fto be
E⊗F={ξ∈Ex⊗Fx|x∈M}
with projection map π:ξ∈Ex⊗Fx7→ x∈M.
2.) Prove that E⊗Fis a vector bundle.
Date: November 13, 2008.
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