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4.4. POINCARE LEMMA • 105 Theorem 4.4.2 Let M be a manifold with ﬁrst Betti number equal to
zero. Then every closed 1form on M is exact. That is, if α is a 1form
such that dα = 0, then there is a function f such that α = d f .
Proof :
1. Let α be a closed 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 4.4.3 Let α be a closed pform 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 :
1. Let α be a closed pform in Rn .
2. We deﬁne a ( p − 1)form β by
1 βi1 ...i p−1 ( x) = dτ τ p−1 x j α ji1 ...i p−1 (τ x)
0 3. We can show that
dβ = α . • Corollary 4.4.1 Let M be a manifold and α be a closed pform 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 :
diﬀgeom.tex; April 12, 2006; 17:59; p. 106 106 CHAPTER 4. INTEGRATION OF DIFFERENTIAL FORMS
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. diﬀgeom.tex; April 12, 2006; 17:59; p. 107 ...
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 Spring '10
 Wong
 Geometry

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