Chapter 3
Poincaré, Integrability, Degree
3.1
•
Poincaré Lemma
Definition 3.1.1 Let M be a manifold.
A p-form α on M is called closed if dα = 0.
A p-form α on M is called exact if there is a (p − 1)-form β such that
α = dβ. The form β is called a potential of α.
57
58
CHAPTER 3. POINCARÉ, INTEGRABILITY, DEGREE
Theorem 3.1.1
1. Every exact form is closed (Poincaré Lemma).
That is, if α = dβ, then dα = 0.
2. The exterior product of closed forms is closed.
That is, if dα = 0 and dβ = 0, then d(α ∧ β) = 0.
3. The exterior product of a closed form and an exact form is exact.
That is, if dα = 0 and β = dγ, then there is σ such that α∧β = dσ.
•
4. Let M be an n-dimensional orientable compact manifold without
boundary and α be an exact n-form on M, that is, α = dβ for
some (n − 1)-form β. Then
�
α = 0.
M
5. Let M be an n-dimensional oriented compact manifold with
boundary ∂M and α be a closed (n−1)-form on M, that is, dα = 0.
Then
�
α = 0.
∂M
Proof : Use Stokes theorem.
�
• Example. On R2 − {0}
xdy − ydx
x 2 + y2
β=
Why is β � dθ?
Then
dβ = 0
�
C
dβ = 2π
if 0 is inside C and 0 if 0 is outside C.
topicsdiffgeom.tex; September 2, 2014; 10:25; p. 58
3.1. POINCARÉ LEMMA
•
Definition 3.1.2 Let M be a manifold. Suppose that for every closed
oriented smooth curve C there is a smooth oriented 2-dimensional surface S and a map F : S → M such that ∂F(S ) = C, that is, the curve C
is the boundary of the surface S . Then the manifold M is said to have
first Betti number equal to zero, B1 = 0.
• Example. For T 2
•
59
B1 � 0.
Theorem 3.1.2 Let M be a manifold with first Betti number equal to
zero. Then every closed 1-form on M is exact. That is, if α is a 1-form
such that dα = 0, then there is a function f such that α = d f .
Proof :
1. Let α be a closed one-form on M.
2. Let x and y be two points in M and C xy be an oriented curve with the
initial point y and the final point x.
3. Let f be defined by
f (x) =
�
α.
C xy
4. Then f is independent on the curve C xy and so is well defined.
5. Finally, we show that
df = α.
�
•
Theorem 3.1.3 Let α ∈ λ1 M. Suppose that for any closed curve C
�
α = 0.
C
Then α is exact.
•
Theorem 3.1.4 Let α be a closed p-form in Rn .
(p − 1)-form β in Rn such that α = dβ.
Then there is
That is, every closed form in Rn is exact.
Proof :
topicsdiffgeom.tex; September 2, 2014; 10:25; p. 59
60
CHAPTER 3. POINCARÉ, INTEGRABILITY, DEGREE
1. Let α be a closed p-form in Rn .
2. We define a (p − 1)-form β by
βi1 ...i p−1 (x) =
�
1
0
dτ τ p−1 x j α ji1 ...i p−1 (τx)
3. We can show that
dβ = α .
�
•
Corollary 3.1.1 Let M be a manifold and α be a closed p-form on
M. Then for every point x in M there is a neighborhood U of x and a
(p − 1)-form β on M such that α = dβ in U.
Proof :
1. Use the fact that a sufficiently small neighborhood of a point in M is
diffeomorphic to an open ball in Rn .
2. Pullback the form α from M to Rn by the pullback F ∗ of the diffeomorphism F : V → U, where U ⊂ M and V ⊂ Rn .
3. Use the previous theorem.
�
3.1.1
Complex Analysis
• Let M = C2 and
z = x + iy,
Then
dz = dx + idy,
and
�
1 ∂
∂
=
∂z 2 ∂x
�
1 ∂
∂
=
∂z̄ 2 ∂x
d̄z = dx − idy
�
∂
−i
∂y
�
∂
+i
∂y
dz ∧ d̄z = −2idx ∧ dy.
topicsdiffgeom.tex; September 2, 2014; 10:25; p. 60
3.1. POINCARÉ LEMMA
61
• Let f = u + iv be a function. Then
f dz = udx − vdy + i(udy + vdx)
and
d( f dz) = (−uy − v x )dx ∧ dy + i(u x − vy )dx ∧ dy =
• So,
�
C
f dz =
�
C
�
�
udx − vdy + i(udy + vdx) =
�
S
∂f
dz̄ ∧ dz
∂z̄
d( f dz)
• Thus, f dz is closed if and only if f is holomorphic (satisfies cauchy-Riemann
equations)
uy = −v x
u x = vy ,
or
∂f
=0
∂z̄
• The system of pde
∂i A j − ∂ j Ai = Fi j
(dA = F)
can be solved if and only if
∂[k Fi j] = 0
(dF = 0) .
topicsdiffgeom.tex; September 2, 2014; 10:25; p. 61