differential geometry w notes from teacher_Part_55

differential geometry w notes from teacher_Part_55 - 109...

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

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 109 5.1. LIE DERIVATIVE OF A VECTOR FIELD Definition 5.1.2 The Lie derivative of the vector field Y with respect to the vector field X is the vector field LX Y defined at a point x by • (LX Y) x = lim t→0 1 Y ϕt ( x ) − ϕ t ∗ Y x t • Remark. Notice that this can also be written as (LX Y) x = lim t →0 or (LX Y) x = • 1 ϕ−t∗ Yϕt ( x) − Y x t d ϕ−t∗ Yϕt ( x) dt t =0 . Proposition 5.1.1 LX Y = [X, Y] . Proof : 1. We compute in local coordinates i d ϕ−t∗ Yϕt ( x) t =0 dt d (ϕ−t∗ ))i j Y j (ϕt ( x)) = t =0 dt d d (ϕ−t∗ ))i j Y j ( x) + δij Y j (ϕt ( x)) = dt dt t =0 t =0 (LX Y)i = ∂X i j ∂Y i Y ( x) + j X j ( x) ∂x j ∂x = [X, Y]i . =− • Definition 5.1.3 The Lie bracket of two vector fields X and Y is a vector field [X, Y] such that for any smooth function f on M [X, Y] = X(Y( f )) − Y(X( f )) . • Notice that [X, Y] = −[Y, X] . diffgeom.tex; April 12, 2006; 17:59; p. 109 110 CHAPTER 5. LIE DERIVATIVE • In particular, LX X = 0 . • In local coordinates the Lie bracket is given by [X, Y]i = X j ∂ j Y i − Y j ∂ j X i . • The linear ordinary differential equation [X, Y]i = dY i (t) − (∂ j X i )(ϕt ( x))Y j (t) = 0 dt for Y i (t) is called the Jacobi equation. For a given vector field X and given initial conditions for Y i it defines a unique Jacobi field along the flow ϕt . • In particular, L∂i ∂ j = 0 . 5.1.2 Flow generated by the Lie Bracket Theorem 5.1.1 Let M be a manifold. Let X and Y be vector fields on M and ϕtX and ϕY be the flows generated by X and Y respectively. Let t σt : M → M be a diffeomorphism defined by X σt = ϕY t ◦ ϕ−t ◦ ϕY ◦ ϕtX . − t Let f be a smooth function on M. Then • [X, Y] x ( f ) = lim t→0 = 1 f (σt ( x)) − f ( x) t2 d f (σ √t ) . dt t =0 which means [X, Y] = d√ σ . dt t t=0 Proof : 1. Diagram. diffgeom.tex; April 12, 2006; 17:59; p. 110 ...
View Full Document

Page1 / 2

differential geometry w notes from teacher_Part_55 - 109...

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

View Full Document Right Arrow Icon
Ask a homework question - tutors are online