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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v0964301.png" />''
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A section of the [[Tangent bundle|tangent bundle]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v0964302.png" />. The set of differentiable vector fields forms a module over the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v0964303.png" /> of differentiable functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v0964304.png" />.
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 +
'' $  M $''
 +
 
 +
A section of the [[Tangent bundle|tangent bundle]] $  \tau ( M) $.  
 +
The set of differentiable vector fields forms a module over the ring $  F $
 +
of differentiable functions on $  M $.
  
 
===Example 1.===
 
===Example 1.===
For a chart <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v0964305.png" /> of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v0964306.png" /> one defines the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v0964307.png" />-th basic vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v0964308.png" /> according to the formula
+
For a chart $  x _ {U} $
 +
of the manifold $  M $
 +
one defines the $  i $-
 +
th basic vector field $  \partial  / \partial  x  ^ {i} $
 +
according to 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/v/v096/v096430/v0964309.png" /></td> </tr></table>
+
\frac \partial {\partial  x  ^ {i} }
 +
( p)  = \
 +
\left .  
 +
\frac \partial {\partial  x  ^ {i} }
 +
\right | _ {p} ,\ \
 +
p \in U ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643010.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643011.png" />-th basic tangent vector to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643012.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643013.png" />. Any vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643014.png" /> can be uniquely represented in the form
+
where $  \partial  / \partial  x _ {i} \mid  _ {p} $
 +
is the $  i $-
 +
th basic tangent vector to $  M $
 +
at the point $  p $.  
 +
Any vector field $  X $
 +
can be uniquely represented in the form
  
<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/v/v096/v096430/v09643015.png" /></td> </tr></table>
+
$$
 +
= \sum _ { i } \xi  ^ {i} ( p)
 +
\frac \partial {\partial  x  ^ {i} }
 +
( p),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643016.png" /> are the components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643017.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643018.png" />. Since a vector field can be regarded as a derivation of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643019.png" /> (see example 2), the set of vector fields forms a Lie algebra with respect to the commutation operation (the Lie bracket).
+
where $  \xi  ^ {i} ( p) $
 +
are the components of $  X $
 +
in $  x _ {U} $.  
 +
Since a vector field can be regarded as a derivation of the ring $  F $(
 +
see example 2), the set of vector fields forms a Lie algebra with respect to the commutation operation (the Lie bracket).
  
 
===Example 2.===
 
===Example 2.===
For the chart <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643021.png" />, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643022.png" /> is defined by the formula
+
For the chart $  x _ {U} $
 +
and $  f \in F $,  
 +
the function $  Xf $
 +
is defined 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/v/v096/v096430/v09643023.png" /></td> </tr></table>
+
$$
 +
( Xf) ( p)  = \sum _ { i } \xi  ^ {i} ( p)
 +
\left . D _ {i} ( f( x _ {U}  ^ {-1})) \right | _ {x _ {U}  ( p) } =
 +
$$
  
<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/v/v096/v096430/v09643024.png" /></td> </tr></table>
+
$$
 +
= \
 +
\sum _ { i } \left . \xi  ^ {i} ( p)
 +
\frac \partial {\partial  x  ^ {i} }
 +
\right | _ {p} ( f  ),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643025.png" /> is the partial derivative with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643026.png" />. Note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643027.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643028.png" /> is called the derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643029.png" /> in the direction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643030.png" />.
+
where $  D _ {i} $
 +
is the partial derivative with respect to $  x  ^ {i} $.  
 +
Note that $  \xi  ^ {i} ( p)= ( X x  ^ {i} ) ( p) $;  
 +
$  Xf $
 +
is called the derivative of $  f $
 +
in the direction $  X $.
  
 
===Example 3.===
 
===Example 3.===
For the chart <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643032.png" />, the commutator (Lie bracket) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643033.png" /> of the vector fields
+
For the chart $  x _ {U} $
 +
and $  f \in F $,  
 +
the commutator (Lie bracket) $  [ X, Y] $
 +
of the vector fields
  
<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/v/v096/v096430/v09643034.png" /></td> </tr></table>
+
$$
 +
= \sum _ { i } \xi  ^ {i}
 +
\frac \partial {\partial  x  ^ {i} }
 +
\  \textrm{ and } \ \
 +
= \sum _ { i } \eta  ^ {i}
 +
\frac \partial {\partial  x  ^ {i} }
 +
 
 +
$$
  
 
is defined by the formula
 
is defined 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/v/v096/v096430/v09643035.png" /></td> </tr></table>
+
$$
 +
([ X, Y ] f  )( p)  = ( X( Yf  )) ( p) - ( Y( Xf  ))( p) =
 +
$$
  
<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/v/v096/v096430/v09643036.png" /></td> </tr></table>
+
$$
 +
= \
 +
\sum _ { i,k } \left . \left ( \xi  ^ {k}
 +
\frac{\partial  \eta  ^ {i} }{\partial
 +
x  ^ {k} }
 +
- \eta  ^ {k}
 +
\frac{\partial  \xi  ^ {i} }{\partial  x  ^ {k} }
 +
\right )
 +
\partial 
 +
\frac{f}{\partial  x  ^ {i} }
 +
\right | _ {p} .
 +
$$
  
 
It satisfies the relations
 
It satisfies the relations
  
<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/v/v096/v096430/v09643037.png" /></td> </tr></table>
+
$$
 +
[ X, Y]  = - [ Y, X],
 +
$$
  
<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/v/v096/v096430/v09643038.png" /></td> </tr></table>
+
$$
 +
[[ X, Y] , Z] + [[ Y, Z], X] + [[ Z, X], Y]  = 0;
 +
$$
  
 
in particular,
 
in particular,
  
<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/v/v096/v096430/v09643039.png" /></td> </tr></table>
+
$$
 +
\left [
 +
\frac \partial {\partial  x  ^ {i} }
 +
,
 +
 +
\frac \partial {\partial  x  ^ {j} }
 +
\right ]  = 0.
 +
$$
  
Each vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643040.png" /> induces a local flow on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643041.png" /> — a family of diffeomorphisms of a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643042.png" />,
+
Each vector field $  X $
 +
induces a local flow on $  M $—  
 +
a family of diffeomorphisms of a neighbourhood $  U $,
  
<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/v/v096/v096430/v09643043.png" /></td> </tr></table>
+
$$
 +
\Phi : (- \epsilon , + \epsilon ) \times U  \rightarrow  M,
 +
$$
  
such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643044.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643045.png" /> and
+
such that $  \Phi ( 0, p)= p $
 +
for $  p \in U $
 +
and
  
<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/v/v096/v096430/v09643046.png" /></td> </tr></table>
+
$$
 +
\Phi ( t, p) = \Phi _ {p} ( t): (- \epsilon , \epsilon )  \rightarrow  M
 +
$$
  
is the integral curve of the vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643047.png" /> through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643048.png" />, i.e.
+
is the integral curve of the vector field $  X $
 +
through $  p $,  
 +
i.e.
  
<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/v/v096/v096430/v09643049.png" /></td> </tr></table>
+
$$
 +
\Phi  ^  \star  \left (
 +
\frac \partial {\partial  t }
 +
\right ) ( t)  = \
 +
X ( \Phi ( t, p) ),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643050.png" /> is the tangent vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643051.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643052.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643053.png" />. Conversely, a vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643054.png" /> is associated with a local flow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643055.png" />, which is a variation of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643056.png" />; here
+
where $  \Phi  ^ {*} ( \partial  / \partial  t ) ( t) $
 +
is the tangent vector $  d \Phi _ {p} ( t) $
 +
to $  M $
 +
at $  \Phi _ {p} ( t) $.  
 +
Conversely, a vector field $  X $
 +
is associated with a local flow $  \Phi ( t, p)= \Phi _ {t} ( p) $,  
 +
which is a variation of the mapping $  \Phi _ {0} ( p) $;  
 +
here
  
<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/v/v096/v096430/v09643057.png" /></td> </tr></table>
+
$$
 +
( Xf)( p)  = \lim\limits _ {t \rightarrow 0 } 
 +
\frac{f( \Phi _ {t} ( p) )- f( p) }{t}
 +
.
 +
$$
  
Each vector field defines a Lie derivation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643058.png" /> of a tensor field of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643060.png" /> with values in a vector space (infinitesimal transformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643061.png" />), corresponding to the local flow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643062.png" />; its special cases include the action of the vector field on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643063.png" />,
+
Each vector field defines a Lie derivation $  L _ {X} $
 +
of a tensor field of type $  \lambda $
 +
with values in a vector space (infinitesimal transformation of $  \lambda $),  
 +
corresponding to the local flow $  \Phi ( t, p) $;  
 +
its special cases include the action of the vector field on $  f \in F $,
  
<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/v/v096/v096430/v09643064.png" /></td> </tr></table>
+
$$
 +
L _ {X} f  = X f,
 +
$$
  
 
and the Lie bracket
 
and the Lie bracket
  
<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/v/v096/v096430/v09643065.png" /></td> </tr></table>
+
$$
 
+
L _ {X} = [ X, Y] = \lim\limits _ {t \rightarrow 0 }
A vector field without singularities generates an integrable one-dimensional differential system as well as a [[Pfaffian system|Pfaffian system]] associated with it on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643066.png" />.
+
 
+
\frac{Y- \Phi _ {t} ^  \star  Y \Phi _ {-t} }{t}
A generalization of the concept of a vector field on a manifold is that of a vector field along a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643067.png" />, which is a section of the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643068.png" /> induced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643069.png" />, as well as a tensor field of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643070.png" />, which is a section of the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643071.png" /> associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643072.png" /> with the aid of the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096430/v09643073.png" />.
+
.
 
+
$$
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> C. Godbillon, "Géométrie différentielle et mécanique analytique" , Hermann (1969) {{MR|0242081}} {{ZBL|0653.53001}} {{ZBL|0284.53018}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968) {{MR|0229177}} {{ZBL|0155.30701}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S. Lang, "Introduction to differentiable manifolds" , Interscience (1967) pp. App. III {{MR|1931083}} {{MR|1532744}} {{MR|0155257}} {{ZBL|1008.57001}} {{ZBL|0103.15101}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> K. Nomizu, "Lie groups and differential geometry" , Math. Soc. Japan (1956) {{MR|0084166}} {{ZBL|0071.15402}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> M.M. Postnikov, "Introduction to Morse theory" , Moscow (1971) (In Russian) {{MR|0315739}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> S. Helgason, "Differential geometry and symmetric spaces" , Acad. Press (1962) {{MR|0145455}} {{ZBL|0111.18101}} </TD></TR></table>
 
 
 
 
 
  
====Comments====
+
A vector field without singularities generates an integrable one-dimensional differential system as well as a [[Pfaffian system]] associated with it on  $  M $.
  
 +
A generalization of the concept of a vector field on a manifold is that of a vector field along a mapping  $  \phi :  N \rightarrow M $,
 +
which is a section of the bundle  $  \tau _  \phi  ( N) $
 +
induced by  $  \phi $,
 +
as well as a tensor field of type  $  \lambda $,
 +
which is a section of the bundle  $  \lambda [ \tau ] $
 +
associated with  $  \tau ( M) $
 +
with the aid of the functor  $  \lambda $.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German) {{MR|0666697}} {{ZBL|0495.53036}} </TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top"> C. Godbillon, "Géométrie différentielle et mécanique analytique" , Hermann (1969) {{MR|0242081}} {{ZBL|0653.53001}} {{ZBL|0284.53018}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968) {{MR|0229177}} {{ZBL|0155.30701}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S. Lang, "Introduction to differentiable manifolds" , Interscience (1967) pp. App. III {{MR|1931083}} {{MR|1532744}} {{MR|0155257}} {{ZBL|1008.57001}} {{ZBL|0103.15101}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> K. Nomizu, "Lie groups and differential geometry" , Math. Soc. Japan (1956) {{MR|0084166}} {{ZBL|0071.15402}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> M.M. Postnikov, "Introduction to Morse theory" , Moscow (1971) (In Russian) {{MR|0315739}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> S. Helgason, "Differential geometry and symmetric spaces" , Acad. Press (1962) {{MR|0145455}} {{ZBL|0111.18101}} </TD></TR>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German) {{MR|0666697}} {{ZBL|0495.53036}} </TD></TR>
 +
</table>

Latest revision as of 18:48, 18 July 2024


$ M $

A section of the tangent bundle $ \tau ( M) $. The set of differentiable vector fields forms a module over the ring $ F $ of differentiable functions on $ M $.

Example 1.

For a chart $ x _ {U} $ of the manifold $ M $ one defines the $ i $- th basic vector field $ \partial / \partial x ^ {i} $ according to the formula

$$ \frac \partial {\partial x ^ {i} } ( p) = \ \left . \frac \partial {\partial x ^ {i} } \right | _ {p} ,\ \ p \in U , $$

where $ \partial / \partial x _ {i} \mid _ {p} $ is the $ i $- th basic tangent vector to $ M $ at the point $ p $. Any vector field $ X $ can be uniquely represented in the form

$$ X = \sum _ { i } \xi ^ {i} ( p) \frac \partial {\partial x ^ {i} } ( p), $$

where $ \xi ^ {i} ( p) $ are the components of $ X $ in $ x _ {U} $. Since a vector field can be regarded as a derivation of the ring $ F $( see example 2), the set of vector fields forms a Lie algebra with respect to the commutation operation (the Lie bracket).

Example 2.

For the chart $ x _ {U} $ and $ f \in F $, the function $ Xf $ is defined by the formula

$$ ( Xf) ( p) = \sum _ { i } \xi ^ {i} ( p) \left . D _ {i} ( f( x _ {U} ^ {-1})) \right | _ {x _ {U} ( p) } = $$

$$ = \ \sum _ { i } \left . \xi ^ {i} ( p) \frac \partial {\partial x ^ {i} } \right | _ {p} ( f ), $$

where $ D _ {i} $ is the partial derivative with respect to $ x ^ {i} $. Note that $ \xi ^ {i} ( p)= ( X x ^ {i} ) ( p) $; $ Xf $ is called the derivative of $ f $ in the direction $ X $.

Example 3.

For the chart $ x _ {U} $ and $ f \in F $, the commutator (Lie bracket) $ [ X, Y] $ of the vector fields

$$ X = \sum _ { i } \xi ^ {i} \frac \partial {\partial x ^ {i} } \ \textrm{ and } \ \ Y = \sum _ { i } \eta ^ {i} \frac \partial {\partial x ^ {i} } $$

is defined by the formula

$$ ([ X, Y ] f )( p) = ( X( Yf )) ( p) - ( Y( Xf ))( p) = $$

$$ = \ \sum _ { i,k } \left . \left ( \xi ^ {k} \frac{\partial \eta ^ {i} }{\partial x ^ {k} } - \eta ^ {k} \frac{\partial \xi ^ {i} }{\partial x ^ {k} } \right ) \partial \frac{f}{\partial x ^ {i} } \right | _ {p} . $$

It satisfies the relations

$$ [ X, Y] = - [ Y, X], $$

$$ [[ X, Y] , Z] + [[ Y, Z], X] + [[ Z, X], Y] = 0; $$

in particular,

$$ \left [ \frac \partial {\partial x ^ {i} } , \frac \partial {\partial x ^ {j} } \right ] = 0. $$

Each vector field $ X $ induces a local flow on $ M $— a family of diffeomorphisms of a neighbourhood $ U $,

$$ \Phi : (- \epsilon , + \epsilon ) \times U \rightarrow M, $$

such that $ \Phi ( 0, p)= p $ for $ p \in U $ and

$$ \Phi ( t, p) = \Phi _ {p} ( t): (- \epsilon , \epsilon ) \rightarrow M $$

is the integral curve of the vector field $ X $ through $ p $, i.e.

$$ \Phi ^ \star \left ( \frac \partial {\partial t } \right ) ( t) = \ X ( \Phi ( t, p) ), $$

where $ \Phi ^ {*} ( \partial / \partial t ) ( t) $ is the tangent vector $ d \Phi _ {p} ( t) $ to $ M $ at $ \Phi _ {p} ( t) $. Conversely, a vector field $ X $ is associated with a local flow $ \Phi ( t, p)= \Phi _ {t} ( p) $, which is a variation of the mapping $ \Phi _ {0} ( p) $; here

$$ ( Xf)( p) = \lim\limits _ {t \rightarrow 0 } \frac{f( \Phi _ {t} ( p) )- f( p) }{t} . $$

Each vector field defines a Lie derivation $ L _ {X} $ of a tensor field of type $ \lambda $ with values in a vector space (infinitesimal transformation of $ \lambda $), corresponding to the local flow $ \Phi ( t, p) $; its special cases include the action of the vector field on $ f \in F $,

$$ L _ {X} f = X f, $$

and the Lie bracket

$$ L _ {X} Y = [ X, Y] = \lim\limits _ {t \rightarrow 0 } \frac{Y- \Phi _ {t} ^ \star Y \Phi _ {-t} }{t} . $$

A vector field without singularities generates an integrable one-dimensional differential system as well as a Pfaffian system associated with it on $ M $.

A generalization of the concept of a vector field on a manifold is that of a vector field along a mapping $ \phi : N \rightarrow M $, which is a section of the bundle $ \tau _ \phi ( N) $ induced by $ \phi $, as well as a tensor field of type $ \lambda $, which is a section of the bundle $ \lambda [ \tau ] $ associated with $ \tau ( M) $ with the aid of the functor $ \lambda $.

References

[1] C. Godbillon, "Géométrie différentielle et mécanique analytique" , Hermann (1969) MR0242081 Zbl 0653.53001 Zbl 0284.53018
[2] D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968) MR0229177 Zbl 0155.30701
[3] S. Lang, "Introduction to differentiable manifolds" , Interscience (1967) pp. App. III MR1931083 MR1532744 MR0155257 Zbl 1008.57001 Zbl 0103.15101
[4] K. Nomizu, "Lie groups and differential geometry" , Math. Soc. Japan (1956) MR0084166 Zbl 0071.15402
[5] M.M. Postnikov, "Introduction to Morse theory" , Moscow (1971) (In Russian) MR0315739
[6] S. Helgason, "Differential geometry and symmetric spaces" , Acad. Press (1962) MR0145455 Zbl 0111.18101
[a1] W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German) MR0666697 Zbl 0495.53036
How to Cite This Entry:
Vector field on a manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vector_field_on_a_manifold&oldid=24590
This article was adapted from an original article by A.F. Shchekut'ev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article