1. Show that Ω1 (M ) can be identified with the space of C ∞ (M )-linear maps from the
space of vector fields on M to C ∞ (M ).
2. Let ξ = (E, M, π) be a vector bundle. Let U = {Ui }i be an open cover of M such that
ξ is trivial over Ui for any i; let Φi : π −1 (Ui ) → Ui × Rn be the corresponding isomorphisms.
Then the map
n
n
Φi ◦ Φ−1
j : (Ui ∩ Uj ) × R → (Ui ∩ Uj ) × R
is of the form (x, v) 7→ (x, gij (x)v), where gij : Ui ∩ Ui → GLn (R) is a smooth map. Prove
that these maps have the following properties:
(i) gij (x) = gji (x)−1 ;
(ii) gik (x) = gij (x)gjk (x) for x ∈ Ui ∩ Uj ∩ Uk .
These maps are called coordinate transformations of ξ.
A set {gij }i,j of functions satisfying the above two properties is called a GLn (R)-valued
1-cocycle. Denote by Z 1 (U; GLn (R)) the set of all such cocycles. Prove (see Theorem 3.1.8)
that conversely any cocycle defines a unique (up to isomorphism) vector bundle.
3. Two cocycles g 0 = {gij0 }i,j and g 00 = {gij00 }i,j in Z 1 (U; GLn (R)) are called cohomologous
if there exist smooth functions gi : Ui → GLn (R) such that
gij0 (x) = gi (x)gij00 (x)gj (x)−1
for x ∈ Ui ∩ Uj . Prove that cohomologous cocycles define isomorphic vector bundles. Conversely, show that if ξ and ζ are isomorphic vector bundles such that they are trivial on Ui
for any i, then the corresponding cocycles are cohomologous.
4. Let ξ be the tautological line bundle over RPn , so that the fiber of ξ over a line L in
R
is the line L itself. Denote by ξ ⊥ the bundle such that the fiber over L is the hyperplane
L⊥ ⊂ Rn+1 . Prove that
τ (RPn ) ∼
= Hom(ξ, ξ ⊥ )
n+1
as follows. For L ∈ RPn consider {v, −v} = L ∩ S n . Then a curve in RPn passing through
L can locally be lifted to a curve γ in S n such that γ(0) = v. Then γ 0 (0) ⊥ L. Define a
linear map L → L⊥ by λv → λγ 0 (0). This way we get a well-defined map from τL (RPn ) to
Hom(L, L⊥ ).
Denoting the trivial n-dimensional bundle by εn conclude that
τ (RPn ) ⊕ ε1 ∼
= Hom(ξ, ξ ⊥ ) ⊕ Hom(ξ, ξ) ∼
= Hom(ξ, ξ ⊥ ⊕ ξ)
∼
= Hom(ξ, εn+1 ) ∼
= ξ∗ ⊕ . . . ⊕ ξ∗ .
{z
}
|
n+1
For the first isomorphism prove that since ξ is one-dimensional, the bundle Hom(ξ, ξ) is
trivial.