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appendixD

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

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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, Φ) of an open subset U of X and a continuous map Φ of U into R n such that Φ( U ) is open in R n and Φ is a homeomorphism of U onto Φ( U ) . A C n - atlas for X is a collection { ( U α , Φ α ) } α I of n -charts for X such that (1) the collection of sets { U α } α I is an open covering of X , (2) the maps Φ β Φ - 1 α : Φ α ( U α U β ) d47 Φ β ( U α U β ) are of class C for all α,β I . Examples 1. Let X = R n and take U = R n and Φ the identity map. Then ( U, Φ) is an n -chart for X and { ( U, Φ) } 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 > 0 } and S i, - = { ( x 1 ,...,x n +1 ) S n : x i < 0 } for i = 1 ,...,n + 1 . Define Φ i, ± ( x ) = ( x 1 ,...,x i - 1 ,x i +1 ,...,x n +1 ) 669

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670 APPENDIX D. MANIFOLDS AND LIE GROUPS for x S n i, ± (the projection of S i, ± onto the hyperplane { x i = 0 } ). Then Φ i, ± ( S n i, ± ) = B n = { x R n : x 2 1 + · · · + x 2 n < 1 } and Φ - 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, Φ i, ± ) is a chart. The sets S n i, ± cover S n . From (D.1) it is clear that { ( S n i,epsilon1 , Φ 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 } . For y R n we define u q ( y ) = f i q ( y ) for 1 q k and u k + q ( y ) = y p q for 1 q n - k . Then det bracketleftbigg ∂u i ∂x j ( y ) bracketrightbigg negationslash = 0 for y U i 1 ,i 2 ,...,i k . Set Ψ( y ) = ( u 1 ( y ) ,...,u n ( y )) . The inverse function theorem (see Lang [1993]) implies that then there exists an open subset V x R n containing x such that W x = Ψ( V x ) is open in R n and Ψ is a bijection from V x onto W x with C inverse map.
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