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CHAPTER 3. POINCARÉ, INTEGRABILITY, DEGREE
3.2
3.2.1
Frobenius Theorem
Distributions
Definition 3.2.1 Let M be a n-dimensional manifold. A k-dimensional
distribution (or a tangent subbundle) Δ : M → Δ x ⊂ T x M is a smooth
assignment to each point x ∈ M a k dimensional subspace Δ x of the
tangent space T x M.
An submanifold V of M that is everywhere tangent to the distribution is
called an integral manifold of the distribution.
•
A k-didmensional distribution Δ is called integrable if at each point
x ∈ M there is a k-dimensional integral submanifold of Δ.
In other words, the distribution Δ is integrable if everywhere in M there
exist local coordinates (x1 , . . . , xk , y1 , . . . , yn−k ) such that the coordinate
surfaces ya = ca , a = 1, . . . , n − k, ca being some constants, are integral
manifolds of the distribution Δ. Such a coordinate system is called a
Frobenius chart.
• Examples in R3 .
• A one-dimensional distribution in R3 is a family of lines at every point in
R3 (that is, a vector field). It is always integrable. Integral manifolds of the
distribution are families of integral curves of the vector field.
• A two-dimensional distribution is a smooth family of planes at every point
in R3 . It is integrable if there exist nonintersecting surfaces that are everywhere tangent to the planes (and fill up a region in R3 ). Not every twodimensional distribution is integrable!
• Let α be a 1-form in R3
α = αi (x)dxi ,
where i = 1, 2, 3. A two-dimensional distribution can be described by
α = 0.
• It is integrable if there exists a diffeomorphism xi = xi (t, u), u = (uµ ), µ =
1, 2, such that for a fixed t, xi = xi (t, u) describes a smooth one-parameter
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3.2. FROBENIUS THEOREM
63
family of surfaces S t and
(F ∗ α)µ = αi (x(t, u))
∂xi
=0
∂uµ
and for fixed u = (u1 , u2 ), xi = xi (t, u) describes a smooth two-parameter
family of curves Cu transversal to the surfaces S t .
• Let t = t(x) and uµ = uµ (x) be the inverse diffeomorphism. Then the level
surfaces t(x) = C are the integral surfaces S t of the distribution, so that the
coordinate system (t, u1 , u2 ) is the Frobenius chart.
• Therefore, we have
∂xi µ
∂xi
dt
du
+
α
i
∂uµ
∂t
∂xi
= αi dt
∂t
= f dt ,
α = αi
where
f = αi
• Therefore,
and
∂xi
.
∂t
dα = d f ∧ dt .
α ∧ dα = 0 .
• Thus the distribution is integrable if
α ∧ dα = 0 .
This is called the Euler’s integrability condition.
• In local coordinates
α[i ∂ j αk] = 0 .
In R3 this means
α · curl α = 0 .
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CHAPTER 3. POINCARÉ, INTEGRABILITY, DEGREE
•
Definition 3.2.2 Let Δ be a distribution on M. It is said to be in
involution if it is closed under Lie brackets, that is, for any two vector
field X and Y in Δ the Lie bracket [X, Y] is also in the distribution, or
[Δ, Δ] ⊂ Δ .
• Let Δ be an integrable distribution. Let X and Y be two vector fields in Δ.
Then X and Y are tangent to the integral manifold of Δ. Therefore, the Lie
bracket [X, Y] is also tangent to the integral manifold and is in Δ.
•
•
Proposition 3.2.1 Every integral distribution is in involution.
Definition 3.2.3 Let α be a 1-form on M. Let x ∈ M be a point such
that α x � 0. The null space of the form α at x is the (n − 1)-dimensional
subspace of T x M spanned by the vectors X ∈ T x M such that
α(X) = 0 .
• Remark. A 1-form is also called a Pfaffian.
• Let α1 , . . . , αn−k be (n − k) linearly independent 1-forms such that
α1 ∧ · · · ∧ αn−k � 0
• Let N1 , . . . Nn−k be their null spaces. Then the intersection of the null spaces
forms a k-dimensional distribution Δ
Δ=
n−k
�
Nµ .
µ=1
• Locally this distribution is described by (n − k) Pfaffian equations
α1 = · · · = αn−k = 0 .
• Remarks.
• Let Δ be in involution. Then dαµ |Δ = 0, µ = 1, . . . , (n − k), that is, for any
X, Y ∈ Δ
(dαµ )(X, Y) = 0 .
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3.2. FROBENIUS THEOREM
65
• Now, suppose that dαµ |Δ = 0, µ = 1, . . . , (n − k). Then the distribution Δ is
in involution.
Theorem 3.2.1 Let M be a n-dimensional manifold and α1 , . . . , αn−k
be (n − k) one-forms and ω be a (n − k)-form defined by
ω = α1 ∧ · · · ∧ αn−k .
Suppose that the forms α1 , . . . , αn−k are linearly independent and ω � 0.
Let Δ be the k-dimensional distribution defined by the intersection of the
null spaces of the 1-forms α1 , . . . , αn−k . Then the following conditions
are locally equivalent
•
1. the distribution Δ is in involution,
2. dαµ |Δ = 0, that is, for any X, Y ∈ Δ
(dαµ )(X, Y) = 0 .
3. dαµ ∧ ω = 0,
4. there exist 1-forms γµν such that
dαµ =
n−k
�
ν=1
Proof :
γµν ∧ αν .
1. (1) ⇔ (2). Use the formula
dα(X, Y) = X(α(Y)) − Y(α(X)) − α([X, Y]) .
2. (4) =⇒ (2) and (4) =⇒ (3). Trivial.
3. (2) =⇒ (4). Suppose dαµ |Δ = 0.
4. Let β1 , . . . , βk be 1-forms such that α1 , . . . , αn−k , β1 , . . . , βk is a basis in
T x∗ M.
5. Let e1 , . . . en−k , v1 , . . . , vk be the dual basis in T x M.
6. Then for µ = 1, . . . , (n − k), i = 1, . . . , k
αµ (vi ) = 0 .
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66
CHAPTER 3. POINCARÉ, INTEGRABILITY, DEGREE
Thus, span {vi } = Δ.
7. We have
dαµ =
�
µ<ν
=
�
ν
8. Thus,
Cµλν αλ ∧ αν +
γµν ∧ αν +
�
i< j
�
i,ν
Aµλi βi ∧ αν +
Bµi j βi ∧ β j .
�
i< j
Bµi j βi ∧ β j
Bµi j = (dαµ )(vi , v j ) = 0 .
9. (3) =⇒ (4). Suppose that dαµ ∧ ω = 0.
10. Then we have
0 = dαµ ∧ ω =
=
�
ν
�
i< j
11. Thus, Bµi j = 0.
γµν ∧ αν ∧ ω +
�
i< j
Bµi j βi ∧ β j ∧ ω .
Bµi j βi ∧ β j ∧ α1 ∧ · · · ∧ αn−k .
�
• Remarks.
• A k-dimensional distribution Δ can be described locally by either k linearly
independent vector fields that span Δ or by (n − k) linearly independent
1-forms whose common null space is Δ.
�
• If dαµ = n−k
ν=1 γµν ∧ αν for some 1-forms γµν , then we write
dαµ = 0 (mod α) .
3.2.2
Frobenius Theorem
Definition 3.2.4 Let M be an n-dimensional manifold, W be a kdimensional manifold and F : W → M be a smooth map.
•
Then F is an immersion if for each x ∈ W the differential F∗ : T x W →
T F(x) M is injective, that is, Ker F∗ = 0.
The image F(W) of the manifold W is called an immersed submanifold.
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3.2. FROBENIUS THEOREM
67
• Let M be an n-dimensional manifold.
• Let Δ be a k-dimensional distribution on M.
• Let X1 , . . . , Xk be vector fields that span Δ and ϕ1t , . . . , ϕkt be the corresponding flows.
• Let x ∈ M be a point in M.
• Let B ∈ Rk be a sufficiently small open ball in Rk around the origin.
• We define
F:B→M
for any t = (t1 , . . . , tk ) ∈ B by
• Note that F(0) = x.
�
�
F(t) = ϕktk ◦ · · · ◦ ϕ1t1 (x) .
• Then for the differential at the origin
F∗ T 0 B = Rk → T x M
we have
F∗
where µ = 1, . . . , k.
• Thus,
��
∂ ���
=
X
�
µ�x ,
∂tµ t=0
F∗ T 0 B = Δ x .
• So, F∗ is injective at the origin and, hence, in some neighborhood of the
origin.
• Therefore, F(B) is tangent to Δ at the single point x.
• Thus, F(B) is an immersed submanifold of M.
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68
CHAPTER 3. POINCARÉ, INTEGRABILITY, DEGREE
•
Theorem 3.2.2 Frobenius Theorem. Let M be an n-dimensional manifold, Δ be a k-dimensional distribution in involution on M, X1 , . . . , Xk
be vector fields that span Δ and ϕ1t , . . . , ϕkt be the corresponding flows.
Let x ∈ M, B be a sufficiently small ball around the origin in Rk and
F : B → M be defined by
�
�
F(t) = ϕktk ◦ · · · ◦ ϕ1t1 (x) .
Then:
1. F(B) is an immersed submanifold of M,
2. F(B) is an integral manifold of Δ,
3. the distribution Δ is integrable.
Proof :
1. (I) done earlier.
2. (II). We need to show that Δ is tangent to F(B) at each point of F(B).
3. We have for µ = 1, . . . , k
F∗
� ��
� �
�
∂
µ
k
k−1
1
X
ϕ
=
ϕ
◦
·
·
·
◦
ϕ
◦
·
·
·
◦
ϕ
µ
tµ ∗
tk ∗
tk−1
t1 (x) ,
∂tµ
4. Thus, the tangent space T F(t) F(B) has a basis
�
� �
�
ϕktk ∗ ◦ · · · ◦ ϕ2t2 ∗ X1 ϕ1t1 (x)
�
� ��
� �
ϕktk ∗ ◦ · · · ◦ ϕ3t3 ∗ X2 ϕ2t2 ◦ ϕ1t1 (x)
...
��
� �
1
◦
·
·
·
◦
ϕ
ϕktk ∗ Xk−1 ϕk−1
tk−1
t1 (x)
��
� �
Xk ϕktk ◦ · · · ◦ ϕ1t1 (x) .
5. Therefore, we need to show that for each µ = 1, . . . , k, the differentials
ϕµt∗ map the distribution Δ into itself, that is,
ϕµt∗ (Δ) ⊂ Δ .
6. Claim: This follows from the fact that Δ is in involution.
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3.2. FROBENIUS THEOREM
69
7. Let y ∈ F(B) and Y ∈ Δy .
8. Let Yµt = ϕµt∗ Y, µ = 1, . . . , k.
9. We will show that Yµt ∈ Δϕµt (y) for µ = 1, . . . , k.
10. Let Δ be defined by the 1-forms
α1 = · · · = αn−k = 0 .
11. The vector field Yµt is invariant under the flow of Xµ , so along the
orbits ϕµt (y) we have
LXµ Yµt = 0 .
µ
12. Let f1µ , . . . , fn−k
be real valued functions defined by
fiµ (t) = αi (Yµt ) ,
(i = 1, . . . , n − k) .
13. Then at t = 0 we have the initial conditions
fi (0) = αi (Y) = 0 .
14. Further,
d µ
f (t) = iYµt iXµ dαi .
dt i
�
15. Since Δ is in involution, by using dαi = k βik ∧ αk , we obtain
n−k
�
d µ
fi (t) =
γi j (Xµ ) f jµ (t) .
dt
j=1
16. Thus, the above system of the differential equations has the unique
solution
fiµ (t) = 0 .
17. Thus, Yµt is in Δ for all t and Δ is tangent to the immersed ball F(B)
at each point of F(B).
18. (III). By constructing Frobenius chart.
19. That is, we construct coordinates x1 , . . . , xk , y1 , . . . , yn−k so that the immersed balls F(B) are described locally by
y1 = c1 , . . . , yn−k = cn−k ,
where ci are constants.
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CHAPTER 3. POINCARÉ, INTEGRABILITY, DEGREE
20. We define a transversal to F(B) (n − k)-dimensional submanifold W
with local coordinates y1 , . . . , yn−k .
21. If the integral balls are sufficiently small, then for different points of
W the integral balls at those points are disjoint.
�
3.2.3
Foliations
• Let M be an n-dimensional manifold and Δ be a k-dimensional distribution
on M.
• Let Δ be in involution, and, therefore, integrable.
• Then, at each point x of M there exists an integral manifold of Δ.
• The integral manifold may return to the Frobenius coordinate patch around
the point x infinitely many times.
Definition 3.2.5 The integral manifolds of an integrable distribution
define a foliation of M.
•
Each connected integral manifold is called a leaf of the foliation.
A leaf that is not properly contained in another leaf is called a maximal
leaf.
• A maximal leaf is not necessarily an embedded submanifold.
• An immersed submanifold does not have to be an embedded submanifold.
•
Theorem 3.2.3
submanifold.
A maximal leaf of a foliated manifold is an immersed
More precisely, for each maximal leaf V there is an injective immersion
F : V → M that realizes V globally.
• Examples.
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