Tutorial 6—Global Analysis
2.11.2021
1. Suppose M = R2
with coordinates (x, y). Consider the vector fields ξ(x, y) = y ∂
∂x
and η(x, y) = x2
2
∂
∂y
on M. We computed in class their flows and saw that they are
complete. Compute [ξ, η] and its flow? Is [ξ, η] complete?
2. Let M be a (smooth) manifold and ξ, η ∈ X(M) two vector fields on M. Show that
(a) [ξ, η] = 0 ⇐⇒ (Flξ
t )∗
η = η, whenever defined ⇐⇒ Flξ
t ◦ Flη
s = Flη
s ◦ Flξ
t ,
whenever defined.
(b) If N is another manifold, f : M → N a smooth map, and ξ and η are f-related
to vector fields ˜ξ resp. ˜η on N, then [ξ, η] is f-related to [˜ξ, ˜η].
3. Suppose αi
j for i = 1, ..., k and j = 1, ..., n are smooth real-valued functions defined
on some open set U ⊂ Rn+k
satisfying
∂αi
j
∂xk
+ αk
∂αi
j
∂z
=
∂αi
k
∂xj
+ αj
∂αi
k
∂z
,
where we write (x, z) = (x1
, ..., xn
, z1
, ..., zk
) for a point in Rn+k
. Show that for
any point (x0, z0) ∈ U there exists an open neighbourhood V of x0 in Rn
and a
unique C∞
-map f : V → Rk
such that
∂fi
∂xj
(x1
, ..., xn
) = αi
j(x1
, ..., xn
, f1
(x), ..., fk
(x)) and f(x0) = z0.
In the class/tutorial we proved this for k = 1 and j = 2.
4. Which of the following systems of PDEs have solutions f(x, y) (resp. f(x, y) and
g(x, y)) in an open neighbourhood of the origin for positive values of f(0, 0) (resp.
f(0, 0) and g(0, 0))?
(a) ∂f
∂x
= f cos y and ∂f
∂y
= −f log f tan y.
(b) ∂f
∂x
= exf
and ∂f
∂y
= xeyf
.
(c) ∂f
∂x
= f and ∂f
∂y
= g; ∂g
∂x
= g and ∂g
∂y
= f.
5. Suppose E → M is a (smooth) vector bundle of rank k over a manifold M. Then
E is called trivializable, if it isomorphic to the trivial vector bundle M × Rk
→ M.
1
2
(a) Show that E → M is trivializable ⇐⇒ E → M admits a global frame, i.e.
there exist (smooth) sections s1, ..., sk of E such that s1(x), ..., sk(x) span Ex
for any x ∈ M.
(b) Show that the tangent bundle of any Lie group G is trivializable.
(c) Recall that Rn
has the structure of a (not necessarily associative) division algebra
over R for n = 1, 2, 4, 8. Use this to show that the tangent bundle of the
spheres S1
⊂ R2
, S3
⊂ R4
and S7
⊂ R8
is trivializable.