differential geometry w notes from teacher_Part_59

Differential - 5.2 LIE DERIVATIVE OF FORMS AND TENSORS 117 • Recall that a linear map A Λ p → Λ p r is a derivation if r is even and for any

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Unformatted text preview: 5.2. LIE DERIVATIVE OF FORMS AND TENSORS 117 • Recall that a linear map A : Λ p → Λ p + r is a derivation if r is even and for any α ∈ Λ p and β ∈ Λ q A ( α ∧ β ) = ( A α ) ∧ β + α ∧ A β . and an anti-derivation if r is odd and A ( α ∧ β ) = ( A α ) ∧ β + (- 1) p α ∧ A β . • Examples. • The Lie derivative L X Λ p → Λ p is a derivation. • The exterior derivative d : Λ p → Λ p + 1 and the interior product i X : Λ p → Λ p- 1 are anti-derivations. • Recall the following theorem about exterior derivative Theorem 5.2.7 Let Y 1 , . . . , Y p + 1 be vector fields on a manifold M and α ∈ Λ p be a p-form. Then ( d α )( Y 1 , . . . , Y p + 1 ) = p + 1 X i = 1 (- 1) i + 1 Y i α ( Y 1 , . . . , ˆ Y i , . . . , Y p + 1 ) + X 1 ≤ i < j ≤ p + 1 (- 1) i + j α ([ Y i , Y j ] , . . . , ˆ Y i , . . . , ˆ Y j , . . . , Y p + 1 ) . • In particular, for p = 1 this takes the form Theorem 5.2.8 Let X and Y be vector fields on a manifold M and...
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This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.

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Differential - 5.2 LIE DERIVATIVE OF FORMS AND TENSORS 117 • Recall that a linear map A Λ p → Λ p r is a derivation if r is even and for any

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