differential geometry w notes from teacher_Part_53

# differential geometry w notes from teacher_Part_53 - ´ 4.4...

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Unformatted text preview: ´ 4.4. POINCARE LEMMA • 105 Theorem 4.4.2 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 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 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 : 1. Let α be a closed p-form 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 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 : 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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## This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.

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differential geometry w notes from teacher_Part_53 - ´ 4.4...

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