differential geometry w notes from teacher_Part_99

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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.

Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online