differential geometry w notes from teacher_Part_56

# differential geometry w notes from teacher_Part_56 - X is...

This preview shows pages 1–2. Sign up to view the full content.

5.1. LIE DERIVATIVE OF A VECTOR FIELD 111 2. Use Taylor expansion in local coordinates. ± Corollary 5.1.1 Let M be a manifold and W be a submanifold of M. Let X and Y be vector ﬁelds on M tangent to W. Then the Lie bracket [ X , Y ] is also tangent to W. di geom.tex; April 12, 2006; 17:59; p. 111

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
112 CHAPTER 5. LIE DERIVATIVE 5.2 Lie Derivative of Forms and Tensors Let X be a vector ﬁeld on a manifold M . Let ϕ t : M M be the ﬂow generated by X and ϕ * t : T ϕ t ( x ) M T x M be the corresponding pullback. Deﬁnition 5.2.1 Let f be a function ( 0 -form) on M. Then the Lie derivative of f with respect to X is a function L X f deﬁned by ( L X f ) x = d dt ( ϕ * t f ) x ± ± ± ± t = 0 = d dt f ( ϕ t ( x )) ± ± ± ± t = 0 . Proposition 5.2.1 The Lie derivative of a function f with respect to a vector ﬁeld
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: X is equal to L X f = X ( f ) . • In local coordinates L X f = X i ∂ i f . • Deﬁnition 5.2.2 Let α be a 1-form on M. The Lie derivative of α with respect to X is a 1-form L X α deﬁned by ( L X α ) x = lim t → 1 t ² ϕ * t α ϕ t ( x )-α x ³ = d dt ( ϕ * t α ) x ± ± ± ± t = • We can immediately generalize this to p-forms. • Deﬁnition 5.2.3 Let α be a p-form on M. The Lie derivative of α with respect to X is a p-form L X α deﬁned by ( L X α ) x = lim t → 1 t ² ϕ * t α ϕ t ( x )-α x ³ = d dt ( ϕ * t α ) x ± ± ± ± t = di ﬀ geom.tex; April 12, 2006; 17:59; p. 112...
View Full Document

### Page1 / 2

differential geometry w notes from teacher_Part_56 - X is...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online