1. Let f ∈ C ∞ (M ), X and Y be vector fields on M , (φt )t and (ψt )t the corresponding
(local) flows. Prove that
∂2 f (ψs (φt (p))) − f (φt (ψs (p))) |s=t=0 = L[X,Y ] (f )(p).
∂s∂t
Conclude that if (φt ◦ ψs )(p) = (ψs ◦ φt )(p) (for all s, t and p where it makes sense), then
[X, Y ] = 0. In fact, the converse is also true.
2. Let X be a vector field and (φt )t the corresponding flow. For a vector field Y define
a vector field LX (Y ) by
d
LX (Y ) = (φ−t )∗ (Y )|t=0 ,
dt
so
1
LX (Y )(p) = lim ((dφt (p) φ−t )(Y (φt (p))) − Y (p)).
t→0 t
LX is called the Lie (or fisherman) derivative. Prove that
(i) LX (f Y ) = LX (f )Y + f LX (Y ) for f ∈ C ∞ (M );
(ii) LX (Y ) = [X, Y ].
3. Let f : M → N be a smooth map. A vector field X on M and a vector field Y on
N are called f -related if (dp f )(X(p)) = Y (f (p)) for any p ∈ M . Prove that X and Y are
f -related if and only if for the corresponding flows (φt )t and (ψt )t we have f ◦ φt = ψt ◦ f .
4. Consider G = GLn (R). Since G is an open subset of Matn (R), for each g ∈ G we can
identify τg (G) with Matn (R). Let A ∈ Matn (R), and X be the corresponding right-invariant
vector field onPG, that is, X(g) = Ag for g ∈ G. Prove that
tn n
(i) etA = ∞
n=0 n! A is an integral curve of X;
(ii) the flow defined by X is given by G 3 g 7→ etA g;
∂
(iii) the vector fields X on G and Tr(A)x ∂x
on R are det-related; conclude that
det(eA ) = eTr(A) .