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''distribution''
 
''distribution''
  
A vector subbundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p0725401.png" /> of the [[Tangent bundle|tangent bundle]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p0725402.png" /> of a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p0725403.png" />. The dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p0725404.png" /> of the fibres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p0725405.png" /> is called the dimension of the Pfaffian structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p0725406.png" />, and the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p0725407.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p0725408.png" />) is called the rank or codimension. A Pfaffian structure of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p0725409.png" /> can be considered as a field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254010.png" />-dimensional subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254011.png" /> on the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254012.png" />.
+
A vector subbundle $  \pi : P \rightarrow M $
 +
of the [[Tangent bundle|tangent bundle]] $  T M \rightarrow M $
 +
of a manifold $  M $.  
 +
The dimension p $
 +
of the fibres $  P _ {x} = \pi  ^ {-} 1 ( x) $
 +
is called the dimension of the Pfaffian structure $  \pi $,  
 +
and the number $  q = n - p $(
 +
where $  n = \mathop{\rm dim}  M $)  
 +
is called the rank or codimension. A Pfaffian structure of dimension p $
 +
can be considered as a field of p $-
 +
dimensional subspaces $  x \rightarrow P _ {x} $
 +
on the manifold $  M $.
  
Usually a Pfaffian structure is given by a system of Pfaffian equations (cf. [[Pfaffian equation|Pfaffian equation]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254013.png" /> or, dually, by indicating vector fields whose values at an arbitrary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254014.png" /> form a basis of the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254015.png" />.
+
Usually a Pfaffian structure is given by a system of Pfaffian equations (cf. [[Pfaffian equation|Pfaffian equation]]) $  \theta  ^ {1} = \dots = \theta  ^ {q} = 0 $
 +
or, dually, by indicating vector fields whose values at an arbitrary point $  x \in M $
 +
form a basis of the subspace $  P _ {x} $.
  
A submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254016.png" /> is called an integral manifold of the Pfaffian structure if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254017.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254018.png" />. A Pfaffian structure is said to be completely integrable if through each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254019.png" /> there passes a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254020.png" />-dimensional integral manifold or, what is equivalent, if it can be locally given by a system of Pfaffian equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254022.png" /> are local coordinates in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254023.png" />. This concept corresponds to the concept of a completely-integrable system of Pfaffian equations. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254024.png" /> be the space of sections of the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254025.png" /> (cf. also [[Section of a mapping|Section of a mapping]]) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254026.png" /> be the space of differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254027.png" />-forms (cf. [[Differential form|Differential form]]) which vanish on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254028.png" />. According to Frobenius' theorem the Pfaffian structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254029.png" /> is completely integrable if and only if the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254030.png" /> is a subalgebra of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254031.png" /> of vector fields on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254032.png" /> or, what is equivalent, if the ideal generated by the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254033.png" /> in the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254034.png" /> of differential forms is closed with respect to exterior differentiation.
+
A submanifold $  N \subset  M $
 +
is called an integral manifold of the Pfaffian structure if $  T _ {x} N \subset  P _ {x} $
 +
for all $  x \in N $.  
 +
A Pfaffian structure is said to be completely integrable if through each point $  x \in M $
 +
there passes a p $-
 +
dimensional integral manifold or, what is equivalent, if it can be locally given by a system of Pfaffian equations $  d y  ^ {1} = \dots = d y  ^ {q} = 0 $,  
 +
where $  y  ^ {1} \dots y  ^ {n} $
 +
are local coordinates in $  M $.  
 +
This concept corresponds to the concept of a completely-integrable system of Pfaffian equations. Let $  \Gamma ( \pi ) $
 +
be the space of sections of the bundle $  \pi : P \rightarrow M $(
 +
cf. also [[Section of a mapping|Section of a mapping]]) and let $  L ( \pi ) $
 +
be the space of differential $  1 $-
 +
forms (cf. [[Differential form|Differential form]]) which vanish on $  P $.  
 +
According to Frobenius' theorem the Pfaffian structure $  \pi $
 +
is completely integrable if and only if the space $  \Gamma ( \pi ) $
 +
is a subalgebra of the Lie algebra $  D ( M) $
 +
of vector fields on $  M $
 +
or, what is equivalent, if the ideal generated by the space $  L ( \pi ) $
 +
in the algebra $  \Omega ( M) $
 +
of differential forms is closed with respect to exterior differentiation.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254035.png" /> be the Lie algebra of infinitesimal automorphisms of the Pfaffian structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254036.png" />, that is, the set of vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254037.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254038.png" />. The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254039.png" /> is a subalgebra of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254040.png" /> and at the same time a module over the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254041.png" /> of smooth functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254042.png" />. The quotient module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254043.png" /> characterizes the degree of non-integrability of the Pfaffian structure.
+
Let $  A ( \pi ) $
 +
be the Lie algebra of infinitesimal automorphisms of the Pfaffian structure $  \pi $,  
 +
that is, the set of vector fields $  X \in \Gamma ( \pi ) $
 +
for which $  [ X , \Gamma ( \pi ) ] \subset  \Gamma ( \pi ) $.  
 +
The algebra $  A ( \pi ) $
 +
is a subalgebra of the Lie algebra $  D ( M) $
 +
and at the same time a module over the ring $  F ( M) $
 +
of smooth functions on $  M $.  
 +
The quotient module $  \Gamma ( \pi ) / A ( \pi ) $
 +
characterizes the degree of non-integrability of the Pfaffian structure.
  
The Pfaffian structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254044.png" /> is regular if the dimension of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254045.png" /> does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254046.png" />. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254047.png" /> is the space of sections of a completely-integrable Pfaffian structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254048.png" />, called the characteristic system of the Pfaffian structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254049.png" />. The rank of the structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254050.png" /> is called the class of the Pfaffian structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254051.png" />, and it is equal to the smallest possible number of coordinates of a local coordinate system in which all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254052.png" />-forms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254053.png" /> can be expressed. The class of a regular Pfaffian structure of rank 1 (that is, a field of hyperplanes) is odd and forms a complete system of local invariants: In a local coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254054.png" /> a Pfaffian structure of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254055.png" /> is locally given by a Pfaffian equation
+
The Pfaffian structure $  \pi $
 +
is regular if the dimension of the space $  A _ {p} ( \pi ) = \{ {X _ {p} } : {X \in A ( \pi ) } \} $
 +
does not depend on p \in M $.  
 +
In this case $  A ( \pi ) $
 +
is the space of sections of a completely-integrable Pfaffian structure $  \pi  ^  \prime  :  P  ^  \prime  = \cup _ {p \in M }  A _ {p} ( \pi ) \rightarrow M $,  
 +
called the characteristic system of the Pfaffian structure $  \pi $.  
 +
The rank of the structure $  \pi  ^  \prime  $
 +
is called the class of the Pfaffian structure $  \pi $,  
 +
and it is equal to the smallest possible number of coordinates of a local coordinate system in which all $  1 $-
 +
forms in $  L ( \pi ) $
 +
can be expressed. The class of a regular Pfaffian structure of rank 1 (that is, a field of hyperplanes) is odd and forms a complete system of local invariants: In a local coordinate system $  y  ^ {i} $
 +
a Pfaffian structure of class $  2 k + 1 $
 +
is locally given by a Pfaffian equation
  
<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/p/p072/p072540/p07254056.png" /></td> </tr></table>
+
$$
 +
d y  ^ {1} + y  ^ {2} d y  ^ {3} + \dots + y  ^ {2k} d y  ^ {2k+} 1  = 0 .
 +
$$
  
Another important local invariant of the Pfaffian structure is its genus, which indicates the dimension of the maximal integral non-singular manifolds (see [[Pfaffian problem|Pfaffian problem]]). A complete system of local invariants of a Pfaffian structure of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254057.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254058.png" /> is unknown.
+
Another important local invariant of the Pfaffian structure is its genus, which indicates the dimension of the maximal integral non-singular manifolds (see [[Pfaffian problem|Pfaffian problem]]). A complete system of local invariants of a Pfaffian structure of dimension p $
 +
when $  1 < p < n - 1 $
 +
is unknown.
  
A Pfaffian structure can be considered as a [[G-structure|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254059.png" />-structure]] of infinite type, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254060.png" /> is the group of linear transformations of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254061.png" /> leaving invariant a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254062.png" />-dimensional coordinate plane. Its first-order structure function corresponds to an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254063.png" />-bilinear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254064.png" />, defined by the commutation of vector fields. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254065.png" /> coincides with the kernel of the vector-valued bilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072540/p07254066.png" />.
+
A Pfaffian structure can be considered as a [[G-structure| $  G $-
 +
structure]] of infinite type, where $  G $
 +
is the group of linear transformations of the space $  \mathbf R  ^ {n} $
 +
leaving invariant a p $-
 +
dimensional coordinate plane. Its first-order structure function corresponds to an $  F ( M ) $-
 +
bilinear mapping $  c : \Gamma ( \pi ) \times \Gamma ( \pi ) \rightarrow D ( M) / \Gamma ( \pi ) $,  
 +
defined by the commutation of vector fields. The space $  A ( \pi ) $
 +
coincides with the kernel of the vector-valued bilinear form $  c $.
  
 
For references see [[Pfaffian problem|Pfaffian problem]].
 
For references see [[Pfaffian problem|Pfaffian problem]].

Latest revision as of 08:06, 6 June 2020


distribution

A vector subbundle $ \pi : P \rightarrow M $ of the tangent bundle $ T M \rightarrow M $ of a manifold $ M $. The dimension $ p $ of the fibres $ P _ {x} = \pi ^ {-} 1 ( x) $ is called the dimension of the Pfaffian structure $ \pi $, and the number $ q = n - p $( where $ n = \mathop{\rm dim} M $) is called the rank or codimension. A Pfaffian structure of dimension $ p $ can be considered as a field of $ p $- dimensional subspaces $ x \rightarrow P _ {x} $ on the manifold $ M $.

Usually a Pfaffian structure is given by a system of Pfaffian equations (cf. Pfaffian equation) $ \theta ^ {1} = \dots = \theta ^ {q} = 0 $ or, dually, by indicating vector fields whose values at an arbitrary point $ x \in M $ form a basis of the subspace $ P _ {x} $.

A submanifold $ N \subset M $ is called an integral manifold of the Pfaffian structure if $ T _ {x} N \subset P _ {x} $ for all $ x \in N $. A Pfaffian structure is said to be completely integrable if through each point $ x \in M $ there passes a $ p $- dimensional integral manifold or, what is equivalent, if it can be locally given by a system of Pfaffian equations $ d y ^ {1} = \dots = d y ^ {q} = 0 $, where $ y ^ {1} \dots y ^ {n} $ are local coordinates in $ M $. This concept corresponds to the concept of a completely-integrable system of Pfaffian equations. Let $ \Gamma ( \pi ) $ be the space of sections of the bundle $ \pi : P \rightarrow M $( cf. also Section of a mapping) and let $ L ( \pi ) $ be the space of differential $ 1 $- forms (cf. Differential form) which vanish on $ P $. According to Frobenius' theorem the Pfaffian structure $ \pi $ is completely integrable if and only if the space $ \Gamma ( \pi ) $ is a subalgebra of the Lie algebra $ D ( M) $ of vector fields on $ M $ or, what is equivalent, if the ideal generated by the space $ L ( \pi ) $ in the algebra $ \Omega ( M) $ of differential forms is closed with respect to exterior differentiation.

Let $ A ( \pi ) $ be the Lie algebra of infinitesimal automorphisms of the Pfaffian structure $ \pi $, that is, the set of vector fields $ X \in \Gamma ( \pi ) $ for which $ [ X , \Gamma ( \pi ) ] \subset \Gamma ( \pi ) $. The algebra $ A ( \pi ) $ is a subalgebra of the Lie algebra $ D ( M) $ and at the same time a module over the ring $ F ( M) $ of smooth functions on $ M $. The quotient module $ \Gamma ( \pi ) / A ( \pi ) $ characterizes the degree of non-integrability of the Pfaffian structure.

The Pfaffian structure $ \pi $ is regular if the dimension of the space $ A _ {p} ( \pi ) = \{ {X _ {p} } : {X \in A ( \pi ) } \} $ does not depend on $ p \in M $. In this case $ A ( \pi ) $ is the space of sections of a completely-integrable Pfaffian structure $ \pi ^ \prime : P ^ \prime = \cup _ {p \in M } A _ {p} ( \pi ) \rightarrow M $, called the characteristic system of the Pfaffian structure $ \pi $. The rank of the structure $ \pi ^ \prime $ is called the class of the Pfaffian structure $ \pi $, and it is equal to the smallest possible number of coordinates of a local coordinate system in which all $ 1 $- forms in $ L ( \pi ) $ can be expressed. The class of a regular Pfaffian structure of rank 1 (that is, a field of hyperplanes) is odd and forms a complete system of local invariants: In a local coordinate system $ y ^ {i} $ a Pfaffian structure of class $ 2 k + 1 $ is locally given by a Pfaffian equation

$$ d y ^ {1} + y ^ {2} d y ^ {3} + \dots + y ^ {2k} d y ^ {2k+} 1 = 0 . $$

Another important local invariant of the Pfaffian structure is its genus, which indicates the dimension of the maximal integral non-singular manifolds (see Pfaffian problem). A complete system of local invariants of a Pfaffian structure of dimension $ p $ when $ 1 < p < n - 1 $ is unknown.

A Pfaffian structure can be considered as a $ G $- structure of infinite type, where $ G $ is the group of linear transformations of the space $ \mathbf R ^ {n} $ leaving invariant a $ p $- dimensional coordinate plane. Its first-order structure function corresponds to an $ F ( M ) $- bilinear mapping $ c : \Gamma ( \pi ) \times \Gamma ( \pi ) \rightarrow D ( M) / \Gamma ( \pi ) $, defined by the commutation of vector fields. The space $ A ( \pi ) $ coincides with the kernel of the vector-valued bilinear form $ c $.

For references see Pfaffian problem.

How to Cite This Entry:
Pfaffian structure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pfaffian_structure&oldid=33887
This article was adapted from an original article by D.V. Alekseevskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article