Homework 4—Global Analysis
Due date:1.12.2020
1. 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].
2. Consider the general linear group GL(n, R). For A ∈ GL(n, R) denote by
λA : GL(n, R) → GL(n, R) λA(B) = AB
ρA : GL(n, R) → GL(n, R) ρA(B) = BA
left respectively right multiplication by A, and by µ : GL(n, R) × GL(n, R) →
GL(n, R) the multiplication map.
(a) Show that λA and ρA are diffeomorphisms for any A ∈ GL(n, R) and that
TBλA(B, X) = (AB, AX) TBρA(B, X) = (BA, XA),
where (B, X) ∈ TBGL(n, R) = {(B, X) : X ∈ Mn(R)}.
(b) Show that
T(A,B)µ((A, B), (X, Y )) = TBλAY + TAρB
X = (AB, AY + XB)
where (A, B) ∈ GL(n, R) × GL(n, R) and (X, Y ) ∈ Mn(R) × Mn(R).
(c) For any X ∈ Mn(R) ∼= TIdGL(n, R) consider the maps
LX : GL(n, R) → TGL(n, R) LX(B) = TIdλB(Id, X) = (B, BX).
RX : GL(n, R) → TGL(n, R) RX(B) = TIdρB(Id, X) = (B, XB).
Show that LX and RX are smooth vector field and that λ∗
ALX = LX and
ρ∗
ARX = RX for any A ∈ GL(n, R). What are their flows? Are these vector
fields complete?
(d) Show that [LX, RY ] = 0 for any X, Y ∈ Mn(R).
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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.