Pfaffian structure
distribution
A vector subbundle of the tangent bundle
of a manifold
. The dimension
of the fibres
is called the dimension of the Pfaffian structure
, and the number
(where
) is called the rank or codimension. A Pfaffian structure of dimension
can be considered as a field of
-dimensional subspaces
on the manifold
.
Usually a Pfaffian structure is given by a system of Pfaffian equations (cf. Pfaffian equation) or, dually, by indicating vector fields whose values at an arbitrary point
form a basis of the subspace
.
A submanifold is called an integral manifold of the Pfaffian structure if
for all
. A Pfaffian structure is said to be completely integrable if through each point
there passes a
-dimensional integral manifold or, what is equivalent, if it can be locally given by a system of Pfaffian equations
, where
are local coordinates in
. This concept corresponds to the concept of a completely-integrable system of Pfaffian equations. Let
be the space of sections of the bundle
(cf. also Section of a mapping) and let
be the space of differential
-forms (cf. Differential form) which vanish on
. According to Frobenius' theorem the Pfaffian structure
is completely integrable if and only if the space
is a subalgebra of the Lie algebra
of vector fields on
or, what is equivalent, if the ideal generated by the space
in the algebra
of differential forms is closed with respect to exterior differentiation.
Let be the Lie algebra of infinitesimal automorphisms of the Pfaffian structure
, that is, the set of vector fields
for which
. The algebra
is a subalgebra of the Lie algebra
and at the same time a module over the ring
of smooth functions on
. The quotient module
characterizes the degree of non-integrability of the Pfaffian structure.
The Pfaffian structure is regular if the dimension of the space
does not depend on
. In this case
is the space of sections of a completely-integrable Pfaffian structure
, called the characteristic system of the Pfaffian structure
. The rank of the structure
is called the class of the Pfaffian structure
, and it is equal to the smallest possible number of coordinates of a local coordinate system in which all
-forms in
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
a Pfaffian structure of class
is locally given by a Pfaffian equation
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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 when
is unknown.
A Pfaffian structure can be considered as a -structure of infinite type, where
is the group of linear transformations of the space
leaving invariant a
-dimensional coordinate plane. Its first-order structure function corresponds to an
-bilinear mapping
, defined by the commutation of vector fields. The space
coincides with the kernel of the vector-valued bilinear form
.
For references see Pfaffian problem.
Pfaffian structure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pfaffian_structure&oldid=13963