Flow of a Vector Field

Posted on 2025-06-19 In Math

Outline

Section 3: Flow of a Vector Field

3 Flow of a Vector Field

A geometric way of looking upon dierentiation in R

is the following. Suppose given

a vector a = (a

, . . . , a

). For any t ∈ R, consider the transformation φ

: (x

, . . . , x

) 7→

+ ta

, . . . , x

+ ta

) of R

into itself. This is a homomorphism of the additive group

R into the group of dieomorphisms of R

. For any dierentiable function f on R

dene

(f))(x) = lim

t→0

f(φ

(x)) −f(x)

This exists and in fact denes the dierential operator



∂

∂x

. We may generalise this

idea to an arbitrary dierential manifold. Let (φ

), t ∈ R, be a one-parameter group of

dieomorphisms of M in the sense that

i) the map R ×M → M given by (t, x) 7→ φ

(x) is dierentiable.

ii) the map φ

: M → M is the identity.

iii) φ

◦ φ

′

= φ

t+t

′

for all t, t

′

∈ R.

Then we may dene dierentiation of functions f with respect to the above data by

setting

f)(x) = lim

t→0

f(φ

(x)) −f(x)

It is easy to see that X

is indeed a homogeneous rst order operator (i.e. a vector

eld). In fact, the linearity of X

is obvious, while we have for every f, g ∈ A(M ),

(fg))(x) = lim

→

f(φ

(x))

g(φ

(x)) −g(x)

+ lim

→

g(x)

f(φ

(x)) −f(x)

= f(x)(X

g)(x) + g(x)(X

f)(x).

3.1. Examples.

1) Take M = R and φ

(x) = x + t for all t, x ∈ R.

2) Take M = R and φ

(x) = e

x, for all t, x ∈ R.

In these two cases, we see that the associated operators are respectively

and

3) Again take M = R and consider the function φ

(x) =

1−tx

. It is easy to see

formally that φ

◦φ

′

= φ

t+t

′

. But, for any given x 6= 0, φ

(x) is only dened for t <

1/x. We are dealing here with a local 1-parameter group of local automorphisms.

In other words, for any x there exist a neighbourhood U and ϵ > 0, such that

(y) is dened for all |t| < ϵ and y ∈ U . The group condition iii) above is

Chapter II: Differential Operators

satised to the extent it makes sense. But notice that the associated vector eld

is still meaningful.

3.2. Exercises.

1) Compute the vector eld given in Example 3) above.

2) Determine the vector eld given by the one-parameter group (φ

) whose action on

is given by

(v, w) = (cos(t)v + sin(t)w, −sin(t)v + cos(t)w).

With this extended notion we have a converse.

3.3. Theorem. Let X be a vector eld on a dierential manifold M. Then for every

x ∈ M , there exist an open neighbourhood U of x, ϵ > 0, and maps φ

: U → M for all

|t| < ϵ, satisfying:

i) the map (−ϵ, ϵ) ×U → M given by (t, y) 7→ φ

(y) is dierentiable.

ii) the map φ

: U → M is the inclusion.

iii) (φ

◦ φ

′

)(y) = φ

t+t

′

(y) for all |t|, |t

′

| < ϵ and y ∈ M such that |t + t

′

| < ϵ and

y, φ

′

(y) are in U .

iv) X

= X.

Proof. Since X

and X are both vector elds, the assertion is purely local and so we may

replace M by an open set in R

and prove the existence of maps φ

such that X

= Xx

for all i. This reduces our task to showing the following. Given dierentiable functions

, we need to nd functions

φ(t, x) : (−ϵ, ϵ) × U → R

for some neighbourhood U of a given point such that

lim

t→0

(t, x) −x

= a

(x), for all i.

This implies that

∂

∂t

(t, x) = lim

′

→0

′

+ t, x) − φ

(t, x)

′

= lim

′

→0

′

, φ(t, x)) −φ

(t, x)

′

= a

(φ(t, x)).

Section 3: Flow of a Vector Field

We also have the initial condition φ

(0, x) = x

. So we start with this equation and

note that it has a unique solution in a neighbourhood of (0, x) in R × R

. To prove

that iii) is satised, we use the uniqueness of the solution. In fact, both φ(t + t

′

, x)

and φ(t, φ(t

′

, x)) are solutions of the equation

(t, x) = a

(ψ(t, x)) with the initial

condition ψ

(0, x) = φ

′

, x). Finally equation iv) is obvious from the construction.

From this point of view, the term ‘innitesimal transformation’ is an appropriate

alternative to that of a ‘vector eld’.

3.4. Denition. The one-parameter group associated to a vector eld is called the

ow of the vector eld.

3.5. Remarks.

1) Although a vector eld gives rise in general only to a local 1-parameter group,

a limited globalization is possible. Indeed, given a compact set K ⊂ M , we can

dene φ

in a neighbourhood of K for all |t| < ϵ. For, by Theorem 3.3 this can be

done in a neighbourhood of every point of K. Since K can be covered by nitely

many of these neighbourhoods, φ

(y) are dened in the same open set |t| < ϵ for

small enough ϵ, and for y in a neighbourhood of K. In particular, if M is itself

compact, then φ

is dened as an automorphism of M for all |t| < ϵ, and hence

by iteration, we get in this case, a global ow.

2) It is obvious that if X depends dierentiably on some parameters s, then the

one-parameter group is dened for small values of the parameters and depends

dierentiably on them.

3.6. Denition. A vector eld which gives rise to a global ow is said to be complete.

We have seen above that any vector eld on a compact manifold is complete. It is

easily seen that the vector eld x

on R is not complete.

Chapter II: Differential Operators

3.7. Exercise. Let M be a compact manifold and X a vector eld. If m ∈ M,

determine when the restriction of X to the open submanifold M \ {m} is complete.

3.8. Denition. If X is a vector eld and (φ

) the ow corresponding to it, the orbit

of a point m ∈ M under φ

, namely the map t 7→ φ

(m), is called an integral curve for

3.9. Remark. The integral curve of a vector eld X has the property that the dier-

ential of this map at t maps

to X

(m)

. This characterises the curve. In particular,

the curve degenerates to a constant map if and only if X

= 0. If X is 0 at a point m, we

say that m is a singularity of X. Our remark amounts to saying that the one-parameter

group φ

xes a point m if and only if m is a singularity of X.

Suppose that m is not a singularity. Then by continuity, we see that X

6= 0 for all x

in a neighbourhood of m. Now the integral curve has injective dierential at all points

near 0 and hence it is an immersed manifold of dimension 1. Actually there is a local

coordinate system (U, x) in which X is given by

∂

∂x

. Indeed, suppose X =



∂

∂x

where by our assumption one of the a

’s, say a

, is nonzero. Consider the coordinate

system given by (y

, . . . , y

) where y

= φ

, 0, x

, . . . , x

). We now compute the

partial derivatives

∂y

∂x

at m given by x

= 0 for all i:

∂y

∂x

= a

(0);

∂y

∂x

= δ

i,j

for j ≥ 2

This shows that (y

, . . . , y

) is a coordinate system in a neighbourhood of m. It is easy

to see that this coordinate system serves the purpose.

Invariant vector elds. We wish to study now vector elds on a Lie group.

3.10. Denition. Suppose M is dierential manifold and a Lie group G acts on it.

Then a vector eld X on M is said to be invariant under the action if the transform of

X by any element of G is the same as X, that is to say, for every m ∈ M and g ∈ G,

is the image of the vector X

under the dierential at m of the map x 7→ gx of M

into itself.

From the uniqueness of the ow corresponding to a vector eld, we deduce that if

the ow of X is φ

, then the ow corresponding to gX is given by ψ

(m) = gφ

−1

m).

Therefore, if X is invariant under G, then the ows ψ

and φ

are the same, so that φ

commutes with the action of G for all t.

Notice also that from the denition of Lie brackets of vector elds it follows that if

X and Y are G-invariant, then [X, Y ] is also invariant. In particular, the vector space

of invariant vector elds is actually a Lie algebra. We have already remarked that G

acts on itself by left translations and so left invariant vector elds of a Lie group form

a Lie algebra over R.

Section 3: Flow of a Vector Field

3.11. Denition. Let G be a connected Lie group. The Lie algebra of vector elds

which are left invariant (i.e. invariant under left translation by elements of G) is called

the Lie algebra of G, and is often denoted by Lie(G) or g.

3.12. Remark. There is a natural linear map of g into the tangent space T

(G) at

1, given by X 7→ X

. It is an isomorphism and the inverse associates to any vector

v ∈ T

(G), the vector eld X given by X

= L

(v), where L

denotes the dierential at

1 of the left translation by g. So g is a Lie algebra of dimension n = dim(G).

3.13. Examples.

1) The left invariant vector elds on the additive Lie group R are of the form a

a ∈ R. Hence the Lie algebra of R is canonically isomorphic to the abelian Lie

algebra R. In the same way, if we take the additive Lie group underlying a vector

space V , then its Lie algebra is identied with the abelian Lie algebra V .

2) On the other hand, if we take the multiplicative group R

or its connected compo-

nent R

containing 1, then the invariant vector elds are scalar multiples of t

Thus its Lie algebra is also the abelian Lie algebra R.

3) The Lie algebra of the group GL(n, R) or the connected component GL(n, R)

containing 1, can be identied with M(n, R). It is clear that the tangent space at

1 of this open submanifold is canonically

(

)

as a vector space. It only remains

to compute the Lie algebra structure. Denote by x

the function which associates

to any matrix A its (i, j)-th coecient. Let A ∈ M (n) and X be the left invariant

vector eld on GL(n) such that X

) = A

. For any s ∈ GL(n, R)

, we have

) = X

◦L

), where L

is left translation by s. Hence the function Xx

is given by s 7→



(s)X



. If Y is any left invariant vector eld

with Y

) = B

, then we have

(XY − Y X)

) = X

(Y x

) − Y

(Xx

)

= X

(



) − Y

(



)



−



= (AB − BA)

In other words, if we identify left invariant vector elds on GL(n) with M(n),

then the Lie bracket is given by the bracket associated with the multiplication in

the matrix algebra.

3.14. Exercises.

1) Show that the left invariant vector elds on GL(n) are generated by E

p,q



i,p

∂

∂x

i,q

and that the right invariant vector elds by F

p,q



q,j

∂

∂x

p,j

Chapter II: Differential Operators

2) Deduce that E

p,q

and F

r,s

commute. Explain this in terms of their ows.

The integral curve through 1 of a left invariant vector eld, namely t 7→ φ

(1), is

actually a homomorphism of the group R into G. In fact, left invariance of X implies

that for every g ∈ G, the automorphism x 7→ gφ

(x) is the same as x 7→ φ

(gx). Taking

g = φ

(1), we get φ

′

(1)φ

(1) = φ

(φ

′

(1)) = φ

t+t

′

(1).

3.15. Denition. A dierentiable group homomorphism of the Lie group R into a

Lie group G is called a one-parameter group.

If ρ : R → G is a 1-parameter group, then ρ





gives a vector at 1 which in turn

denes a left invariant vector eld X. It is clear that ρ gives an integral curve for X.

3.16. Remarks.

1) If the vector eld X is 0, the corresponding one-parameter group is the constant

homomorphism t 7→ 1. Even if X is not 0, the map t 7→ φ

(1) mentioned above,

may not be injective. If we take G = S

= {(x, y) ∈ R

: x

+ y

= 1}, then

the left invariant vector elds form a 1-dimensional vector space generated by

X = x

∂

∂y

− y

∂

∂x

. The image of

under the map t 7→ (cos at, sin at) is easily

computed to be aX. Hence it is the one-parameter group of the vector eld aX.

Its kernel is the subgroup

2π

Z of R.

2) Also the image of a one-parameter group is not in general closed. For example,

consider the case when G = S

× S

and X =



, a



. Then the induced

one-parameter group is given by

t 7→ ((cos t, sin t), ( cos at, sin at)).

This is a closed submanifold if and only if a is rational. See the gure in [Ch. 1,

3.17].

3) If g ∈ G, then (the dierential of) the inner automorphism x 7→ gxg

−1

of G

takes the vector eld X to another left invariant vector eld which we may denote

gXg

−1

. If t 7→ c(t) is the ow of X, then the ow of gXg

−1

is given by t 7→

gc(t)g

−1

3.17. Denition. The representation of a Lie group G into GL(g) which associates

to g ∈ G the automorphism Ad(g) = X 7→ gXg

−1

is called the adjoint representation

of G.

3.18. Remark. The linear automorphism Ad(g) is actually an automorphism of the

Lie algebra g.

3.19. Exercise. Show that if the image of the one-parameter group is closed, then it

is actually a closed submanifold.

Section 3: Flow of a Vector Field

If G and H are Lie groups and T : G → H is a homomorphism of Lie groups, then

the dierential at 1 is a linear map T

(G) → T

(H). As such it gives a linear map

t : g → h as well, using the isomorphisms T

(G) → g and T

(H) → h.

3.20. Proposition. Let X, Y be left invariant vector elds of a Lie group G. If

T : G → H is a homomorphism of Lie groups, then we have

t[X, Y ] = [t(X), t( Y )].

Proof. Let f be a dierentiable function in a neighbourhood of 1 in H. Then for any

Z ∈ h and any x ∈ G, we have (Zf)(T x) = Z

T x

f), since Z is an invariant vector

eld. Let us take Z = tX, that is to say, Z is the left invariant vector eld whose value

in T

(H) is the image of X

∈ T

(G) by the dierential at 1 of T . Then Z

T x

f) =

T x

f ◦ T ) = X

(f ◦ T )) = X

(f ◦ T ) since X is left invariant. In other words,

(tX)(f) ◦ T = X(f ◦ T ). Hence if Y ∈ g, we may replace f by (tY )f in this equation

and obtain (tX)(tY )(f) ◦ T = X(( tY )f ◦T ) = XY (f ◦T ). Interchanging X and Y and

subtracting, we get [tX, tY ]( f ) ◦ T = [X, Y ](f ◦ T ). Evaluating at 1, we see that the

image of [X, Y ] under the dierential of T at 1 is actually [t(X), t(Y )]. This proves our

assertion.

We have remarked that if a vector eld depends dierentiably on some parameters,

then its ow also depends dierentiably on the parameters. From this it also follows that

the one-parameter group associated to a left invariant vector eld depends dierentiably

on it, meaning that there exists a dierentiable map ρ : T

(G) × R → G such that

for any X ∈ T

(G) the map t 7→ ρ(X, t) is the corresponding one-parameter group.

The restriction of ρ to T

(G) × {1} is a dierentiable map g → G. This is called the

exponential mapping. Let v ∈ T

(G). Then we will evaluate at v, the dierential at 0

of the exponential. We might as well restrict ρ to Rv × R rst in order to compute this

dierential. In other words, we consider the map (sv, t) 7→ c(sv, t) but this is the same

as c(v, st). Setting t = 1, we need to compute the dierential of s 7→ c(v, s) at s = 0.

By denition it is v. In other words, we have shown

3.21. Proposition. The dierential at 0 of the exponential map from g to G is the

identity map. In particular, the exponential map gives a dieomorphism of a neigh-

bourhood of 0 in g onto a neighbourhood of 1 in G.

It follows from the uniqueness assertion regarding one-parameter groups, that if

T : G → H is a dierentiable homomorphism of connected Lie groups, the induced map

t commutes with the exponential map in the sense that exp

◦t = T ◦ exp

If T, T

′

: G → H are two homomorphisms of connected Lie groups such that the

induced homomorphisms t, t

′

are the same, then T = T

′

. For from Proposition 3.22, our

Chapter II: Differential Operators

assumption implies that T and T

′

coincide in a neighbourhood of 1 in G. But if G is

connected, any neighbourhood U of 1 generates G as a group. Hence T, T

′

coincide on

the whole of G.

Moreover, if a homomorphism T : G → H of connected Lie groups is injective, then

the induced homomorphism t is also injective. For if X is in the kernel of t, then the

image of the one-parameter group t 7→ T (exp(tX)) has tangent 0 at 1 and should be the

constant homomorphism. By our assumption, the map t 7→ exp(tX) is also constant,

and hence X = 0.

Conversely, if t is injective, the kernel of T cannot intersect the exponential neigh-

bourhood, and is therefore a discrete normal subgroup of G.

Finally, if t is an isomorphism of Lie algebras, then T has discrete kernel N and goes

down to a homomorphism of the Lie group G/N into H. This induces an isomorphism

at the Lie algebra level. Since the dierential of T at 1 and hence at any other point

is an isomorphism, it is a dieomorphism onto an open subgroup of H. Since H is

connected, the image coincides with H. In other words, T : G → H is an isomorphism.

3.22. Remark. We have associated to every Lie group, a Lie algebra and to every Lie

group homomorphism, a Lie algebra homomorphism of the corresponding Lie algebras.

This correspondence helps one to understand an analytical object such as a Lie group,

by a purely algebraic object, namely its Lie algebra.