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''distribution''

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]])  $  \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|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  $  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|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| $  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]].