appendixD - Appendix D Manifolds and Lie Groups The purpose...

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Unformatted text preview: Appendix D Manifolds and Lie Groups The purpose of this appendix is to collect the essential parts of manifold and Lie group theory in a convenient form for the body of the book. The philosophy of this appendix is to give the main definitions and to prove many of the basic theorems. Some of the more difficult results are stated with appropriate references that the careful reader who is unfamiliar with differential geometry can study. D.1 C Manifolds D.1.1 Basic Definitions Let X be a Hausdorff topological space with a countable basis for its topology. Then an n- chart for X is a pair ( U, ) of an open subset U of X and a continuous map of U into R n such that ( U ) is open in R n and is a homeomorphism of U onto ( U ) . A C n- atlas for X is a collection { ( U , ) } I of n-charts for X such that (1) the collection of sets { U } I is an open covering of X , (2) the maps - 1 : ( U U ) d47 ( U U ) are of class C for all , I . Examples 1. Let X = R n and take U = R n and the identity map. Then ( U, ) is an n-chart for X and { ( U, ) } is a C atlas. 2. Let X = S n = { ( x 1 , . . ., x n +1 ) R n +1 : x 2 1 + + x 2 n +1 = 1 } with the topology as a closed subset of R n +1 . Let S n i, + = { ( x 1 , . . ., x n +1 ) S n : x i > } and S i,- = { ( x 1 , . . ., x n +1 ) S n : x i < } for i = 1 , . . ., n + 1 . Define i, ( x ) = ( x 1 , . . ., x i- 1 , x i +1 , . . ., x n +1 ) 669 670 APPENDIX D. MANIFOLDS AND LIE GROUPS for x S n i, (the projectionof S i, ontothe hyperplane { x i = 0 } ). Then i, ( S n i, ) = B n = { x R n : x 2 1 + + x 2 n < 1 } and - 1 i, ( x 1 , . . ., x n ) = ( x 1 , . . ., x i- 1 , radicalBig 1- x 2 1- - x 2 n , x i , . . ., x n ) . (D.1) Thus each ( U, i, ) is a chart. The sets S n i, cover S n . From (D.1) it is clear that { ( S n i,epsilon1 , n,epsilon1 ) : 1 i n + 1 , epsilon1 = } is a C n-atlas for X . 3. Let f 1 , . . ., f k be C real-valued functions on R n with k n . Let X be the set of points x R n so that f i ( x ) = 0 for all i = 1 , . . ., k and some k k minor of the k n matrix D ( x ) = bracketleftbigg f i x j ( x ) bracketrightbigg is nonzero (note that X might be empty). Give X the subspace topology in R n . For 1 i 1 < < i k n let D i 1 ,i 2 ,...,i k ( x ) be the k k matrix formed by rows i 1 , . . ., i k of D ( x ) . Define U i 1 ,i 2 ,...,i k = { x X : det D i 1 ,i 2 ,...,i k ( x ) negationslash = 0 } . Then these sets constitute an open covering of X . We now construct an ( n- k )-atlas for X as follows: Given x X , choose 1 i 1 < < i k n so that x U i 1 ,i 2 ,...,i k . Let 1 p 1 < < p n- k n be the complementary set of indices: { i 1 , . . ., i k } { p 1 , . . ., p n- k } = { 1 , . . ., n } ....
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This note was uploaded on 02/28/2012 for the course MATH 207C taught by Professor N.r.wallach during the Spring '10 term at Colorado.

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appendixD - Appendix D Manifolds and Lie Groups The purpose...

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