MATH 623: GRASSMANNIAN 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. Given an n-dimensional vector space Vconsider the Grassmannian Gr(k, V )
of k-planes in V. The tautological vector bundle π:T → Gr(k, V ) over Gr(k, V ) is a rank k
vector bundle. The fibre over ξ∈Gr(k, V ) is the k-dimensional vector space ξ⊂V. That
is,
T={(ξ, v)∈Gr(k , V )×V|v∈ξ},
and the projection π:T → Gr(k, V ) maps (ξ , v)7→ ξ.
Example. Recall that Gr(1, V ) = PV.
HW. Prove Tis a vector bundle, and identify the transition functions.
Date: November 18, 2008.
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