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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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 Spring '10
 Wong
 Geometry, Derivative

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