differential geometry w notes from teacher_Part_95

differential geometry w notes from teacher_Part_95 -...

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7.4. DE RHAM COHOMOLOGY GROUPS 189 Proposition 7.4.2 . Let M be a closed manifold. Let α p Z p ( M ) be a closed p-form on M such that for any p-cycle z p Z p ( M ) D α p , z p E = 0 . Then the p-form α p is exact. Proof : Di cult. ± Theorem 7.4.1 de Rham Theorem . Let M be a closed manifold. Then the map H p ( M ) H * p ( M ) , that associates to each equivalence class [ α p ] of closed p-forms a linear functional H p ( M ) R on the homology group H p ( M ) deﬁned by [ z p ] [ α p ] 7-→ D α p , z p E , is an isomorphism. Proof : ± 7.4.1 Examples Torus T 2 . Closed Surfaces in R 2 . di geom.tex; April 12, 2006; 17:59; p. 187

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190 CHAPTER 7. HOMOLOGY THEORY 7.5 Harmonic Forms Let ( M , g ) be a closed oriented n -dimensional Riemannian manifold with a Riemannian metric g and the Riemannian volume n -form vol . Let Λ p ( M ) be the bundle of p -forms. Recall that there is a natural ﬁber inner product on Λ p deﬁned by h α, β i = 1 p ! g i 1 j 1 ··· g i p j p α i 1 ... i p β j 1 ... j p , and the corresponding ﬁber norm || α || = p h α, α i . Also, there is a duality between
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Unformatted text preview: p-forms and ( n-p )-forms deﬁned by the Hodge star operator * : Λ p → Λ n-p , that maps any p-form α to a ( n-p )-form * α dual to α deﬁned as follows. • For each p-form α the form * α is the unique ( n-p )-form such that for any p-form β β ∧ * α = h β, α i vol . • In components, this means that ( * α ) i p + 1 ... i n = 1 p ! ε i 1 ... i p i p + 1 ... i n p | g | g i 1 j 1 ··· g i p j p α j 1 ... j p = 1 p ! 1 p | g | g i p + 1 j p + 1 ··· g i n j n ε j 1 ... j p j p + 1 ... j n α j 1 ... j p . • Recall that the Hodge star maps forms to pseudo-forms and vice-versa. • Recall also that for any p-form α , * 2 α = (-1) p ( n-p ) α , meaning that *-1 = (-1) p ( n-p ) * . di ﬀ geom.tex; April 12, 2006; 17:59; p. 188...
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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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