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differential geometry w notes from teacher_Part_99

# differential geometry w notes from teacher_Part_99 - 7.6...

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7.6. RELATIVE HOMOLOGY AND MORSE THEORY 197 7.6 Relative Homology and Morse Theory 7.6.1 Relative Homology Let M be a compact Riemannian n -dimensional manifold with smooth bound- ary M and i : M M be the inclusion map. Let ˆ x i , i = 1 , . . . , ( n - 1), be the local coordinates on the boundary M and x μ , μ = 1 , . . . , n , be the local coordinates on M in a patch U close to the boundary. Then the inclusion map is defined locally by x μ = x μ x ) . Close to the boundary there exists a system of local coordinates ( x μ ) = x i , r ) so that the boundary is described by the equation r = 0 . The coordinate r can be chosen to be the normal geodesic distance from the boundary so that the vector r is normal to the boundary. The inclusion map in this case is given by x i = ˆ x i , r = 0 . Therefore, x k ˆ x j = δ k j , r ˆ x j = 0 . A p -form α on M is called normal to M if i * α = 0 , that is, x μ 1 ˆ x j 1 · · · x μ p ˆ x j p α μ 1 ...μ p ( x x )) = 0 .

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differential geometry w notes from teacher_Part_99 - 7.6...

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