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Unformatted text preview: Lecture 24 We review the pullback operation from last lecture. Let U be open in R m and let V be open in R n . Let f : U → V be a C ∞ map, and let f ( p ) = q . From the map df p : T p R m T q R n , (4.212) → we obtain the pullback map ( df p ) ∗ : Λ k ( T q ∗ ) → Λ k ( T p ∗ ) ω ∈ Ω k ( V ) → f ∗ ω ∈ Ω k ( U ) . (4.213) We define, f ∗ ω p = ( df p ) ∗ ω q , when ω q ∈ Λ k ( T q ∗ ). The pullback operation has some useful properties: 1. If ω i ∈ Ω k i ( V ) , i = 1 , 2, then f ∗ ( ω 1 ∧ ω 2 ) = f ∗ ω 1 ∧ f ∗ ω 2 . (4.214) 2. If ω ∈ Ω k ( V ), then df ∗ ω = f ∗ dω. (4.215) We prove some other useful properties of the pullback operation. Claim. For all ω ∈ Ω k ( W ) , f ∗ g ∗ ω = ( g ◦ f ) ∗ ω. (4.216) Proof. Let f ( p ) = q and g ( q ) = w . We have the pullback maps ( df p ) ∗ :Λ k ( T ∗ q ) → Λ k ( T ∗ p ) (4.217) ( dg q ) ∗ :Λ k ( T ∗ w ) → Λ k ( T ∗ q ) (4.218) ( g ◦ f ) ∗ :Λ k ( T ∗ w ) → Λ k ( T ∗ p ) . (4.219) The chain rule says that ( dg ◦ f ) p = ( dg ) q ◦ ( df ) p , (4.220) so d ( g ◦ f ) ∗ p = ( df p ) ∗ ( dg q ) ∗ . (4.221) Let U, V be open sets in R n , and let f : U → V be a C ∞ map. pullback operation on nforms ω ∈ Ω n ( V ). Let f (0) = q . Then We consider the ( dx i ) p , i = 1 , . . . , n, is a basis of T ∗ p , and (4.222) ( dx i ) q , i = 1 , . . . , n, is a basis of T ∗ q . ....
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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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