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
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