differential geometry w notes from teacher_Part_42

differential geometry w notes from teacher_Part_42 - F * =...

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3.4. PULLBACK OF FORMS 83 3.4 Pullback of Forms Let M be a n -dimensional manifold and W be a r -dimensional manifold. Let F : M W be a smooth map of a manifold M to a manifold W . Let p M be a point in M and q = F ( p ) W be the image of p in W . Let x i , ( i = 1 , . . . , n ), be a local coordinate system about p and y μ , ( μ = 1 , . . . , r ), be a local coordinate system about q so that y μ = y μ ( x ) . Let f : W R be a smooth function on W . The pullback of f to M is a function F * f : M R on M defined by F * f = f F , that is, for any x ( F * f )( x ) = f ( y ( x )) . Suppose that n = r and the map F is bijective. Let h : M R be a smooth function on M . The pushforward of h to W is a function F * h : W R on W defined by F * h = h F - 1 , that is, for any y ( F * h )( y ) = h ( x ( y )) . Remarks. The pullback is well defined for an arbitrary map F . The pushforward is only defined for bijections! The pullback is the map F * : Λ p W Λ p M defined as follows. Let α Λ p W be a p -form on W . The pullback of α is a p -form F * α on M defined by: for any vectors v 1 , . . . , v p ( F * α )( v 1 , . . . , v p ) = α ( F * v 1 , . . . , F * v p ) . di geom.tex; April 12, 2006; 17:59; p. 85
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84 CHAPTER 3. DIFFERENTIAL FORMS In local cordinates
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Unformatted text preview: F * = 1 p ! 1 ... p ( y ( x )) dy 1 dy p = 1 p ! y 1 x i 1 y p x i p 1 ... p ( y ( x )) dx i 1 dx i p In components ( F * ) i 1 ... i p = y 1 x i 1 y p x i p 1 ... p ( y ( x )) Remark. The pullback is well-dened only for covariant tensors and the pushforward is well dened only for contravariant tensors. Theorem 3.4.1 Let F : M W. The pullback F * : p W p M has the properties: 1. F * is linear. 2. For any two forms and F * ( ) = ( F * ) ( F * ) 3. F * commutes with exterior derivative. That is, for any p-form F * ( d ) = d ( F * ) . Proof : Direct calculation. di geom.tex; April 12, 2006; 17:59; p. 86...
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differential geometry w notes from teacher_Part_42 - F * =...

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