Tutorial 5—Global Analysis
26.10.2021
1. Suppose M = R3
with standard coordinates (x, y, z). Consider the vector field
ξ(x, y, z) = 2
∂
∂x
−
∂
∂y
+ 3
∂
∂z
.
How does this vector field look like in terms of the coordinate vector fields associated
to the cylindrical coordinates (r, φ, z), where x = r cos φ, y = r sin φ and z =
z? Or with respect to the spherical coordinates (r, φ, θ), where x = r sin θ cos φ,
y = r sin θ cos φ and z = r cos θ?
2. Consider R3
with coordinates (x, y, z) and the vector fields
ξ(x, y, z) = (x2
− 1)
∂
∂x
+ xy
∂
∂y
+ xz
∂
∂z
η(x, y, z) = x
∂
∂x
+ y
∂
∂y
+ 2xz2 ∂
∂z
.
Are they tangent to the cylinder M = {(x, y, z) ∈ R3
: x2
+ y2
= 1} ⊂ R3
with
radius 1 (i.e. do they restrict to vector fields on M)?
3. 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?
4. 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].
5. 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.
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(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).