This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 antiderivation 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 antiderivations. • 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 pform. 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...
View
Full
Document
This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.
 Spring '10
 Wong
 Geometry, Derivative

Click to edit the document details