# Pfaffian structure

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=48175
This article was adapted from an original article by D.V. Alekseevskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article