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A natural operation on a [[Differentiable manifold|differentiable manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l0585801.png" /> that associates with a differentiable vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l0585802.png" /> and a differentiable geometric object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l0585803.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l0585804.png" /> (cf. [[Geometric objects, theory of|Geometric objects, theory of]]) a new geometric object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l0585805.png" />, which describes the rate of change of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l0585806.png" /> with respect to the one-parameter (local) transformation group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l0585807.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l0585808.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l0585809.png" />. The geometric object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858010.png" /> is called the Lie derivative of the geometric object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858011.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858012.png" /> (cf. also [[Lie derivative|Lie derivative]]). Here it is assumed that transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858013.png" /> induce transformations in the space of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858014.png" /> in a natural way.
+
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In the special case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858015.png" /> is a vector-valued function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858016.png" />, its Lie derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858017.png" /> coincides with the derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858018.png" /> of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858019.png" /> in the direction of the vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858020.png" /> and is given by the formula
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858021.png" /></td> </tr></table>
+
A natural operation on a [[Differentiable manifold|differentiable manifold]]  $  M $
 +
that associates with a differentiable vector field  $  X $
 +
and a differentiable geometric object  $  Q $
 +
on  $  M $(
 +
cf. [[Geometric objects, theory of|Geometric objects, theory of]]) a new geometric object  $  {\mathcal L} _ {X} Q $,
 +
which describes the rate of change of  $  Q $
 +
with respect to the one-parameter (local) transformation group  $  \phi _ {t} $
 +
of  $  M $
 +
generated by  $  X $.
 +
The geometric object  $  {\mathcal L} _ {X} Q $
 +
is called the Lie derivative of the geometric object  $  Q $
 +
with respect to  $  X $(
 +
cf. also [[Lie derivative|Lie derivative]]). Here it is assumed that transformations of  $  M $
 +
induce transformations in the space of objects  $  Q $
 +
in a natural way.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858022.png" /> is the one-parameter local transformation group on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858023.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858024.png" />, or, in the local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858025.png" />, by the formula
+
In the special case when  $  Q $
 +
is a vector-valued function on $  M $,  
 +
its Lie derivative  $  {\mathcal L} _ {X} Q $
 +
coincides with the derivative  $  \partial  _ {X} Q $
 +
of the function  $  Q $
 +
in the direction of the vector field  $  X $
 +
and is given by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858026.png" /></td> </tr></table>
+
$$
 +
\left . ( {\mathcal L} _ {X} Q ) ( x)  = \
 +
 
 +
\frac{d}{dt}
 +
Q \circ \phi _ {t} ( x)
 +
\right | _ {t=} 0 ,\  x \in M ,
 +
$$
 +
 
 +
where  $  \phi _ {t} $
 +
is the one-parameter local transformation group on  $  M $
 +
generated by  $  X $,
 +
or, in the local coordinates  $  x  ^ {i} $,
 +
by the formula
 +
 
 +
$$
 +
{\mathcal L} _ {X} Q ( x  ^ {i} )  = \
 +
\sum _ { j } X  ^ {j}
 +
 
 +
\frac \partial {\partial  x  ^ {j} }
 +
Q ( x  ^ {i} ) ,
 +
$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858027.png" /></td> </tr></table>
+
$$
 +
= \sum _ { j } X  ^ {j} ( x)
  
In the general case the definition of Lie differentiation consists in the following. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858028.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858030.png" />-space, that is, a manifold with a fixed action of the general differential group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858032.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858033.png" /> (the group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858034.png" />-jets at the origin of diffeomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858036.png" />). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858037.png" /> be a geometric object of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858038.png" /> and type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858039.png" /> on an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858040.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858041.png" />, regarded as a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858042.png" />-equivariant mapping of the principal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858043.png" />-bundle of coframes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858044.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858045.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858046.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858047.png" />. The one-parameter local transformation group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858048.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858049.png" /> generated by a vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858050.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858051.png" /> induces a one-parameter local transformation group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858052.png" /> on the manifold of coframes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858053.png" />. Its velocity field
+
\frac \partial {\partial  x  ^ {j} }
 +
.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858054.png" /></td> </tr></table>
+
In the general case the definition of Lie differentiation consists in the following. Let  $  W $
 +
be a  $  \mathop{\rm GL}  ^ {k} ( n) $-
 +
space, that is, a manifold with a fixed action of the general differential group  $  \mathop{\rm GL}  ^ {k} ( n) $
 +
of order  $  k $(
 +
the group of  $  k $-
 +
jets at the origin of diffeomorphisms  $  \phi : \mathbf R  ^ {n} \rightarrow \mathbf R  ^ {n} $,
 +
$  \phi ( 0) = 0 $).
 +
Let  $  Q : P  ^ {k} M \rightarrow W $
 +
be a geometric object of order  $  k $
 +
and type  $  W $
 +
on an  $  n $-
 +
dimensional manifold  $  M $,
 +
regarded as a  $  \mathop{\rm GL}  ^ {k} ( n) $-
 +
equivariant mapping of the principal  $  \mathop{\rm GL}  ^ {k} ( n) $-
 +
bundle of coframes  $  P  ^ {k} M $
 +
of order  $  k $
 +
on  $  M $
 +
into  $  W $.  
 +
The one-parameter local transformation group  $  \phi _ {t} $
 +
on  $  M $
 +
generated by a vector field  $  X $
 +
on  $  M $
 +
induces a one-parameter local transformation group  $  \phi _ {t}  ^ {(} k) $
 +
on the manifold of coframes  $  P  ^ {k} M $.  
 +
Its velocity field
  
is called the complete lift of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858055.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858056.png" />. The Lie derivative of a geometric object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858057.png" /> of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858058.png" /> with respect to a vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858059.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858060.png" /> is defined as the geometric object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858061.png" /> of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858062.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858063.png" /> is the tangent bundle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858064.png" />, regarded in a natural way as a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858065.png" />-space), given by the formula
+
$$
 +
X  ^ {(} k)  = \left .  
 +
\frac{d}{dt}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858066.png" /></td> </tr></table>
+
\phi _ {t}  ^ {(} k) \right | _ {t=} 0
 +
$$
  
The value of the Lie derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858067.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858068.png" /> depends only on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858069.png" />-jet of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858070.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858071.png" />, and does so linearly, and on the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858072.png" /> at this point (or, equivalently, on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858073.png" />-jet of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858074.png" /> at the corresponding point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858075.png" />).
+
is called the complete lift of  $  X $
 +
to  $  P  ^ {k} M $.
 +
The Lie derivative of a geometric object  $  Q $
 +
of type  $  W $
 +
with respect to a vector field  $  X $
 +
on $  M $
 +
is defined as the geometric object  $  {\mathcal L} _ {X} Q $
 +
of type  $  TW $(
 +
where  $  TW $
 +
is the tangent bundle of $  W $,
 +
regarded in a natural way as a  $  \mathop{\rm GL}  ^ {k} ( n) $-
 +
space), given by the formula
  
If the geometric object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858076.png" /> is linear, that is, the corresponding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858077.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858078.png" /> is a vector space with linear action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858079.png" />, then the tangent manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858080.png" /> can in a natural way be identified with the direct product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858081.png" />, and so the Lie derivative
+
$$
 +
{\mathcal L} _ {X} Q  = \left .  
 +
\frac{d}{dt}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858082.png" /></td> </tr></table>
+
Q \circ \phi _ {t}  ^ {(} k) \right | _ {t=} 0 .
 +
$$
  
can be regarded as a pair of geometric objects of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858083.png" />. The first of these is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858084.png" /> itself, and the second, which is usually identified with the Lie derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858085.png" />, is equal to the derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858086.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858087.png" /> in the direction of the vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858088.png" />:
+
The value of the Lie derivative $  {\mathcal L} _ {X} Q $
 +
at a point  $  p _ {k} \in P  ^ {k} M $
 +
depends only on the  $  1 $-
 +
jet of $  Q $
 +
at  $  p _ {k} $,
 +
and does so linearly, and on the value of $  X  ^ {(} k) $
 +
at this point (or, equivalently, on the $  k $-
 +
jet of $  X $
 +
at the corresponding point  $  x \in M $).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858089.png" /></td> </tr></table>
+
If the geometric object  $  Q $
 +
is linear, that is, the corresponding  $  \mathop{\rm GL}  ^ {k} ( n) $-
 +
space  $  W $
 +
is a vector space with linear action of  $  \mathop{\rm GL}  ^ {k} ( n) $,
 +
then the tangent manifold  $  TW $
 +
can in a natural way be identified with the direct product  $  W \times W $,
 +
and so the Lie derivative
  
Thus, the Lie derivative of a linear geometric object can be regarded as a geometric object of the same type as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858090.png" />.
+
$$
 +
{\mathcal L} _ {X} Q :  P  ^ {k} M  \rightarrow  T W  = W \times W
 +
$$
  
Local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858091.png" /> in the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858092.png" /> determine local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858093.png" /> in the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858094.png" /> of coframes of order 1: for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858095.png" /> one has
+
can be regarded as a pair of geometric objects of type  $  W $.  
 +
The first of these is  $  Q $
 +
itself, and the second, which is usually identified with the Lie derivative of  $  Q $,
 +
is equal to the derivative  $  \partial  _ {X  ^ {(}  k) } Q $
 +
of  $  Q $
 +
in the direction of the vector field  $  X  ^ {(} k) $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858096.png" /></td> </tr></table>
+
$$
 +
{\mathcal L} _ {X} Q  = ( Q , \partial  _ {X  ^ {(}  k) } Q ) .
 +
$$
  
In these coordinates the Lie derivative of any geometric object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858097.png" /> of order 1 (for example, a tensor field) in the direction of the vector field
+
Thus, the Lie derivative of a linear geometric object can be regarded as a geometric object of the same type as  $  Q $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858098.png" /></td> </tr></table>
+
Local coordinates  $  x  ^ {i} $
 +
in the manifold  $  M $
 +
determine local coordinates  $  x  ^ {i} , y _ {j}  ^ {i} $
 +
in the manifold  $  P  ^ {1} M $
 +
of coframes of order 1: for  $  \theta \in P  ^ {1} M $
 +
one has
 +
 
 +
$$
 +
\theta  = \sum _ { j } y _ {j}  ^ {i}  d x  ^ {j} .
 +
$$
 +
 
 +
In these coordinates the Lie derivative of any geometric object  $  Q = Q ( x  ^ {i} , y _ {j}  ^ {i)} $
 +
of order 1 (for example, a tensor field) in the direction of the vector field
 +
 
 +
$$
 +
= \sum _ { j } X  ^ {j}
 +
\frac \partial {\partial  x  ^ {j} }
 +
 
 +
$$
  
 
is given by the formula
 
is given by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l05858099.png" /></td> </tr></table>
+
$$
 +
( {\mathcal L} _ {X} Q ) ( x  ^ {i} , y _ {j}  ^ {i} )  = \
 +
\sum _ { j }
 +
\frac \partial {\partial  x  ^ {i} }
 +
 
 +
Q - \sum _ { i,j,l }
 +
y _ {l}  ^ {i} X _ {j}  ^ {l}
 +
\frac \partial {\partial  y _ {j}  ^ {i} }
 +
Q ,
 +
$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l058580100.png" /></td> </tr></table>
+
$$
 +
X _ {j}  ^ {l}  =
 +
\frac \partial {\partial  x  ^ {j} }
 +
X  ^ {l} .
 +
$$
  
 
A similar formula holds for the Lie derivative of a geometric object of arbitrary order.
 
A similar formula holds for the Lie derivative of a geometric object of arbitrary order.
  
The Lie derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l058580101.png" /> in the space of differential forms on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l058580102.png" /> can be expressed in terms of the operator of exterior differentiation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l058580103.png" /> and the operator of interior multiplication <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l058580104.png" /> (defined as the contraction of a vector field with a differential form) by means of the following homotopy formula:
+
The Lie derivative $  {\mathcal L} _ {X} $
 +
in the space of differential forms on a manifold $  M $
 +
can be expressed in terms of the operator of exterior differentiation $  d $
 +
and the operator of interior multiplication $  i _ {X} $(
 +
defined as the contraction of a vector field with a differential form) by means of the following homotopy formula:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l058580105.png" /></td> </tr></table>
+
$$
 +
{\mathcal L} _ {X}  = d \circ i _ {X} + i _ {X} \circ d .
 +
$$
  
Conversely, the operator of exterior differentiation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l058580106.png" />, acting on a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l058580107.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l058580108.png" />, can be expressed in terms of the Lie derivative by the formula
+
Conversely, the operator of exterior differentiation $  d $,  
 +
acting on a $  p $-
 +
form $  \omega $,  
 +
can be expressed in terms of the Lie derivative by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l058580109.png" /></td> </tr></table>
+
$$
 +
d \omega ( X _ {1} \dots X _ {p+} 1 ) =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l058580110.png" /></td> </tr></table>
+
$$
 +
= \
 +
\sum _ { i= } 1 ^ { p+ }  1 (- 1)  ^ {i+} 1 {\mathcal L} _ {X _ {i}  } \omega
 +
( X _ {1} \dots \widehat{X}  _ {i} \dots X _ {p+} 1 ) +
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l058580111.png" /></td> </tr></table>
+
$$
 +
+
 +
\sum _ {i < j } (- 1)  ^ {i+} j \omega ( {\mathcal L} _ {X _ {i}  } X _ {j} , X _ {1} \dots \widehat{X}  _ {i} \dots \widehat{X}  _ {j} \dots X _ {p+} 1 ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l058580112.png" /> means that the corresponding symbol must be omitted, and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l058580113.png" /> are vector fields.
+
where $  \widehat{ {}}  $
 +
means that the corresponding symbol must be omitted, and the $  X _ {1} \dots X _ {p+} 1 $
 +
are vector fields.
  
In contrast to [[Covariant differentiation|covariant differentiation]], which requires the introduction of a connection, the operation of Lie differentiation is determined by the structure of the differentiable manifold, and the Lie derivative of a geometric object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l058580114.png" /> in the direction of a vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l058580115.png" /> is a concomitant of the geometric objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l058580116.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058580/l058580117.png" />.
+
In contrast to [[Covariant differentiation|covariant differentiation]], which requires the introduction of a connection, the operation of Lie differentiation is determined by the structure of the differentiable manifold, and the Lie derivative of a geometric object $  Q $
 +
in the direction of a vector field $  X $
 +
is a concomitant of the geometric objects $  X $
 +
and $  Q $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Slebodziński,  "Sur les équations canonique de Hamilton"  ''Bull. Cl. Sci. Acad. Roy. Belgique'' , '''17'''  (1931)  pp. 864–870</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.L. Laptev,  "Lie differentiation"  ''Progress in Math.'' , '''6'''  (1970)  pp. 229–269  ''Itogi. Nauk. Algebra Topol. Geom. 1965''  (1967)  pp. 429–465</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  K. Yano,  "The theory of Lie derivatives and its applications" , North-Holland  (1957)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Sternberg,  "Lectures on differential geometry" , Prentice-Hall  (1964)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.V. Vagner,  "Theory of geometric objects and theory of finite and infinite continuous transformation groups"  ''Dokl. Akad. Nauk SSSR'' , '''46'''  (1945)  pp. 347–349  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  B.L. Laptev,  "Lie derivative in a space of supporting elements"  ''Trudy Sem. Vektor. Tenzor. Anal.'' , '''10'''  (1956)  pp. 227–248  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  L.E. Evtushik,  "The Lie derivative and differential field equations of a geometric object"  ''Soviet Math. Dokl.'' , '''1'''  (1960)  pp. 687–690  ''Dokl. Akad. Nauk SSSR'' , '''132'''  (1960)  pp. 998–1001</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  R.S. Palais,  "A definition of the exterior derivative in terms of Lie derivatives"  ''Proc. Amer. Math. Soc.'' , '''5'''  (1954)  pp. 902–908</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Slebodziński,  "Sur les équations canonique de Hamilton"  ''Bull. Cl. Sci. Acad. Roy. Belgique'' , '''17'''  (1931)  pp. 864–870</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.L. Laptev,  "Lie differentiation"  ''Progress in Math.'' , '''6'''  (1970)  pp. 229–269  ''Itogi. Nauk. Algebra Topol. Geom. 1965''  (1967)  pp. 429–465</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  K. Yano,  "The theory of Lie derivatives and its applications" , North-Holland  (1957)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Sternberg,  "Lectures on differential geometry" , Prentice-Hall  (1964)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.V. Vagner,  "Theory of geometric objects and theory of finite and infinite continuous transformation groups"  ''Dokl. Akad. Nauk SSSR'' , '''46'''  (1945)  pp. 347–349  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  B.L. Laptev,  "Lie derivative in a space of supporting elements"  ''Trudy Sem. Vektor. Tenzor. Anal.'' , '''10'''  (1956)  pp. 227–248  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  L.E. Evtushik,  "The Lie derivative and differential field equations of a geometric object"  ''Soviet Math. Dokl.'' , '''1'''  (1960)  pp. 687–690  ''Dokl. Akad. Nauk SSSR'' , '''132'''  (1960)  pp. 998–1001</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  R.S. Palais,  "A definition of the exterior derivative in terms of Lie derivatives"  ''Proc. Amer. Math. Soc.'' , '''5'''  (1954)  pp. 902–908</TD></TR></table>

Latest revision as of 22:16, 5 June 2020


A natural operation on a differentiable manifold $ M $ that associates with a differentiable vector field $ X $ and a differentiable geometric object $ Q $ on $ M $( cf. Geometric objects, theory of) a new geometric object $ {\mathcal L} _ {X} Q $, which describes the rate of change of $ Q $ with respect to the one-parameter (local) transformation group $ \phi _ {t} $ of $ M $ generated by $ X $. The geometric object $ {\mathcal L} _ {X} Q $ is called the Lie derivative of the geometric object $ Q $ with respect to $ X $( cf. also Lie derivative). Here it is assumed that transformations of $ M $ induce transformations in the space of objects $ Q $ in a natural way.

In the special case when $ Q $ is a vector-valued function on $ M $, its Lie derivative $ {\mathcal L} _ {X} Q $ coincides with the derivative $ \partial _ {X} Q $ of the function $ Q $ in the direction of the vector field $ X $ and is given by the formula

$$ \left . ( {\mathcal L} _ {X} Q ) ( x) = \ \frac{d}{dt} Q \circ \phi _ {t} ( x) \right | _ {t=} 0 ,\ x \in M , $$

where $ \phi _ {t} $ is the one-parameter local transformation group on $ M $ generated by $ X $, or, in the local coordinates $ x ^ {i} $, by the formula

$$ {\mathcal L} _ {X} Q ( x ^ {i} ) = \ \sum _ { j } X ^ {j} \frac \partial {\partial x ^ {j} } Q ( x ^ {i} ) , $$

where

$$ X = \sum _ { j } X ^ {j} ( x) \frac \partial {\partial x ^ {j} } . $$

In the general case the definition of Lie differentiation consists in the following. Let $ W $ be a $ \mathop{\rm GL} ^ {k} ( n) $- space, that is, a manifold with a fixed action of the general differential group $ \mathop{\rm GL} ^ {k} ( n) $ of order $ k $( the group of $ k $- jets at the origin of diffeomorphisms $ \phi : \mathbf R ^ {n} \rightarrow \mathbf R ^ {n} $, $ \phi ( 0) = 0 $). Let $ Q : P ^ {k} M \rightarrow W $ be a geometric object of order $ k $ and type $ W $ on an $ n $- dimensional manifold $ M $, regarded as a $ \mathop{\rm GL} ^ {k} ( n) $- equivariant mapping of the principal $ \mathop{\rm GL} ^ {k} ( n) $- bundle of coframes $ P ^ {k} M $ of order $ k $ on $ M $ into $ W $. The one-parameter local transformation group $ \phi _ {t} $ on $ M $ generated by a vector field $ X $ on $ M $ induces a one-parameter local transformation group $ \phi _ {t} ^ {(} k) $ on the manifold of coframes $ P ^ {k} M $. Its velocity field

$$ X ^ {(} k) = \left . \frac{d}{dt} \phi _ {t} ^ {(} k) \right | _ {t=} 0 $$

is called the complete lift of $ X $ to $ P ^ {k} M $. The Lie derivative of a geometric object $ Q $ of type $ W $ with respect to a vector field $ X $ on $ M $ is defined as the geometric object $ {\mathcal L} _ {X} Q $ of type $ TW $( where $ TW $ is the tangent bundle of $ W $, regarded in a natural way as a $ \mathop{\rm GL} ^ {k} ( n) $- space), given by the formula

$$ {\mathcal L} _ {X} Q = \left . \frac{d}{dt} Q \circ \phi _ {t} ^ {(} k) \right | _ {t=} 0 . $$

The value of the Lie derivative $ {\mathcal L} _ {X} Q $ at a point $ p _ {k} \in P ^ {k} M $ depends only on the $ 1 $- jet of $ Q $ at $ p _ {k} $, and does so linearly, and on the value of $ X ^ {(} k) $ at this point (or, equivalently, on the $ k $- jet of $ X $ at the corresponding point $ x \in M $).

If the geometric object $ Q $ is linear, that is, the corresponding $ \mathop{\rm GL} ^ {k} ( n) $- space $ W $ is a vector space with linear action of $ \mathop{\rm GL} ^ {k} ( n) $, then the tangent manifold $ TW $ can in a natural way be identified with the direct product $ W \times W $, and so the Lie derivative

$$ {\mathcal L} _ {X} Q : P ^ {k} M \rightarrow T W = W \times W $$

can be regarded as a pair of geometric objects of type $ W $. The first of these is $ Q $ itself, and the second, which is usually identified with the Lie derivative of $ Q $, is equal to the derivative $ \partial _ {X ^ {(} k) } Q $ of $ Q $ in the direction of the vector field $ X ^ {(} k) $:

$$ {\mathcal L} _ {X} Q = ( Q , \partial _ {X ^ {(} k) } Q ) . $$

Thus, the Lie derivative of a linear geometric object can be regarded as a geometric object of the same type as $ Q $.

Local coordinates $ x ^ {i} $ in the manifold $ M $ determine local coordinates $ x ^ {i} , y _ {j} ^ {i} $ in the manifold $ P ^ {1} M $ of coframes of order 1: for $ \theta \in P ^ {1} M $ one has

$$ \theta = \sum _ { j } y _ {j} ^ {i} d x ^ {j} . $$

In these coordinates the Lie derivative of any geometric object $ Q = Q ( x ^ {i} , y _ {j} ^ {i)} $ of order 1 (for example, a tensor field) in the direction of the vector field

$$ X = \sum _ { j } X ^ {j} \frac \partial {\partial x ^ {j} } $$

is given by the formula

$$ ( {\mathcal L} _ {X} Q ) ( x ^ {i} , y _ {j} ^ {i} ) = \ \sum _ { j } \frac \partial {\partial x ^ {i} } Q - \sum _ { i,j,l } y _ {l} ^ {i} X _ {j} ^ {l} \frac \partial {\partial y _ {j} ^ {i} } Q , $$

where

$$ X _ {j} ^ {l} = \frac \partial {\partial x ^ {j} } X ^ {l} . $$

A similar formula holds for the Lie derivative of a geometric object of arbitrary order.

The Lie derivative $ {\mathcal L} _ {X} $ in the space of differential forms on a manifold $ M $ can be expressed in terms of the operator of exterior differentiation $ d $ and the operator of interior multiplication $ i _ {X} $( defined as the contraction of a vector field with a differential form) by means of the following homotopy formula:

$$ {\mathcal L} _ {X} = d \circ i _ {X} + i _ {X} \circ d . $$

Conversely, the operator of exterior differentiation $ d $, acting on a $ p $- form $ \omega $, can be expressed in terms of the Lie derivative by the formula

$$ d \omega ( X _ {1} \dots X _ {p+} 1 ) = $$

$$ = \ \sum _ { i= } 1 ^ { p+ } 1 (- 1) ^ {i+} 1 {\mathcal L} _ {X _ {i} } \omega ( X _ {1} \dots \widehat{X} _ {i} \dots X _ {p+} 1 ) + $$

$$ + \sum _ {i < j } (- 1) ^ {i+} j \omega ( {\mathcal L} _ {X _ {i} } X _ {j} , X _ {1} \dots \widehat{X} _ {i} \dots \widehat{X} _ {j} \dots X _ {p+} 1 ) , $$

where $ \widehat{ {}} $ means that the corresponding symbol must be omitted, and the $ X _ {1} \dots X _ {p+} 1 $ are vector fields.

In contrast to covariant differentiation, which requires the introduction of a connection, the operation of Lie differentiation is determined by the structure of the differentiable manifold, and the Lie derivative of a geometric object $ Q $ in the direction of a vector field $ X $ is a concomitant of the geometric objects $ X $ and $ Q $.

References

[1] W. Slebodziński, "Sur les équations canonique de Hamilton" Bull. Cl. Sci. Acad. Roy. Belgique , 17 (1931) pp. 864–870
[2] B.L. Laptev, "Lie differentiation" Progress in Math. , 6 (1970) pp. 229–269 Itogi. Nauk. Algebra Topol. Geom. 1965 (1967) pp. 429–465
[3] K. Yano, "The theory of Lie derivatives and its applications" , North-Holland (1957)
[4] S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964)
[5] V.V. Vagner, "Theory of geometric objects and theory of finite and infinite continuous transformation groups" Dokl. Akad. Nauk SSSR , 46 (1945) pp. 347–349 (In Russian)
[6] B.L. Laptev, "Lie derivative in a space of supporting elements" Trudy Sem. Vektor. Tenzor. Anal. , 10 (1956) pp. 227–248 (In Russian)
[7] L.E. Evtushik, "The Lie derivative and differential field equations of a geometric object" Soviet Math. Dokl. , 1 (1960) pp. 687–690 Dokl. Akad. Nauk SSSR , 132 (1960) pp. 998–1001
[8] R.S. Palais, "A definition of the exterior derivative in terms of Lie derivatives" Proc. Amer. Math. Soc. , 5 (1954) pp. 902–908
How to Cite This Entry:
Lie differentiation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_differentiation&oldid=14570
This article was adapted from an original article by D.V. Alekseevskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article