View source for Pfaffian structure

''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" />.

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" />.

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.

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.

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

<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>

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.

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" />.

For references see [[Pfaffian problem|Pfaffian problem]].