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differential geometry w notes from teacher_Part_100

# differential geometry w notes from teacher_Part_100 - 7.6...

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7.6. RELATIVE HOMOLOGY AND MORSE THEORY 199 Proposition 7.6.1 Let M be a compact Riemannian manifold with smooth boundary. Let α and β be two p-forms on M, which are either both normal to the boundary or both tangent to the boundary. Then ( d α, β ) = - ( α, δβ ) . That is, on normal or tangent p-forms d * = - δ . Proof : Obvious. ± Theorem 7.6.1 Let M be a compact Riemannian manifold with smooth boundary. Let k = B p ( M ) be the p-th Betti number. Let z (1) p , . . . , z ( k ) p , be a basis of real p-cycles in the real homology groups H p ( M ) and π 1 , . . . , π k be arbitrary real numbers. Then there is a unique tangent harmonic p-form h p such that dh = δ h = 0 and D h p , z ( i ) p E = π i , i = 1 , 2 , . . . , k . Proof : 1. ± More generally, di geom.tex; April 12, 2006; 17:59; p. 197

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200 CHAPTER 7. HOMOLOGY THEORY Theorem 7.6.2 Let M be a compact Riemannian manifold with smooth boundary. Let k = B p ( M ) be the p-th Betti number. Let z (1) p , . . . , z ( k ) p , be a basis of real p-cycles in the real homology groups H p ( M ) and π 1 , . . . , π k be arbitrary real numbers. Let γ be a closed ( n - p ) -form on M such that for every
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differential geometry w notes from teacher_Part_100 - 7.6...

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