differential geometry w notes from teacher_Part_55

# differential geometry w notes from teacher_Part_55 - 109...

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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 ...
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