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lecture33 - Lecture 33 6.5 Differential Forms on Manifolds...

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Unformatted text preview: Lecture 33 6.5 Differential Forms on Manifolds Let U R n be open. By definition, a k-form on U is a function which assigns to each point p U an element p k ( T R n ). p We now define the notion of a k-form on a manifold. Let X R N be an n - dimensional manifold. Then, for p X , the tangent space T p X T p R N . Definition 6.14. A k-form on X is a function on X which assigns to each point p X an element p k (( T p X ) ). Suppose that f : X R is a C map, and let f ( p ) = a . Then df p is of the form df p : T p X T a R R . (6.47) = We can think of df p ( T p X ) = 1 (( T p X ) ). So, we get a one-form df on X which maps each p X to df p . Now, suppose is a k-form on X , and (6.48) is an -form on X . (6.49) For p X , we have p k ( T X ) and (6.50) p p ( T X ) . (6.51) p Taking the wedge product, p p k + ( T X ) . (6.52) p The wedge product is the ( k + )-form mapping p X to p p . Now we consider the pullback operation. Let X R N and Y R be manifolds, and let f : X Y be a C map. Let p X and a = f ( p ). We have the map df p : T p X T a Y. (6.53) From this we get the pullback ( df p ) : k ( T a Y ) k ( T X ) . (6.54) p Let be a k-form on Y . Then f is defined by ( f ) p = ( df p ) q . (6.55) 1 Let f : X Y and g : Y Z be C maps on manifolds X, Y, Z . Let be a k-form. Then ( g f ) = f ( g ) , (6.56) where g f : X Z . So far, the treatment of k-forms for manifolds has been basically the same as our...
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This note was uploaded on 10/02/2010 for the course MAT unknown taught by Professor Unknown during the Fall '04 term at MIT.

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lecture33 - Lecture 33 6.5 Differential Forms on Manifolds...

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