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 dened by the Hodge star operator * : p n-p , that maps any p-form to a ( n-p )-form * dual to dened as follows. For each p-form the form * is the unique ( n-p )-form such that for any p-form * = h , i vol . In components, this means that ( * ) i p + 1 ... i n = 1 p ! i 1 ... i p i p + 1 ... i n p | g | g i 1 j 1 g i p j p j 1 ... j p = 1 p ! 1 p | g | g i p + 1 j p + 1 g i n j n j 1 ... j p j p + 1 ... j n j 1 ... j p . Recall that the Hodge star maps forms to pseudo-forms and vice-versa. Recall also that for any p-form , * 2 = (-1) p ( n-p ) , meaning that *-1 = (-1) p ( n-p ) * . di geom.tex; April 12, 2006; 17:59; p. 188...
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