# Pfaffian problem

The problem of describing the integral manifolds of maximal dimension for a Pfaffian system of Pfaffian equations

$$\tag{* } \theta ^ \alpha = 0 ,\ \ \alpha = 1 \dots q ,$$

given by a collection of $q$ differential $1$- forms which are linearly independent at each point in a certain domain $M \subset \mathbf R ^ {n}$( or on a certain manifold). A submanifold $N \subset M$ is called an integral manifold of the system (*) if the restrictions of the forms $\theta ^ \alpha$ to $N$ are identically zero. The problem was posed by J. Pfaff (1814).

From a geometric point of view the system (*) determines an $( n - q )$- dimensional distribution (a Pfaffian structure) on $M$, that is, a field

$$x \mapsto P _ {x} = \ \{ {y \in \mathbf R ^ {n} } : {\theta _ {x} ^ \alpha ( y) = 0 } \} ,\ \ x \in M ,$$

of $( n - q )$- dimensional subspaces, and the Pfaffian problem consists of describing the submanifolds of maximum possible dimension tangent to this field. The importance of the Pfaffian problem lies in the fact that the integration of an arbitrary partial differential equation can be reduced to a Pfaffian problem. For example, the integration of a first-order equation

$$F \left ( x ^ {i} , u , \frac{\partial u }{\partial x ^ {i} } \right ) = 0$$

reduces to the Pfaffian problem for the Pfaffian equation $\theta = d u - p _ {i} d x ^ {i} = 0$ on the submanifold (generally speaking with singularities) of the space $\mathbf R ^ {2n+} 1$ defined by the equation

$$F ( x ^ {i} , u , p _ {i} ) = 0 .$$

A completely-integrable Pfaffian system (and also a single Pfaffian equation of constant class) can be locally reduced to a simple canonical form. In these cases the solution of the Pfaffian problem reduces to the solution of ordinary differential equations. In the general case (in the class of smooth functions) the Pfaffian problem has not yet been solved (1989). The Pfaffian problem was solved by E. Cartan in the analytic case in his theory of systems in involution (cf. Involutional system). The formulation of the basic theorem of Cartan is based on the concept of a regular integral element. A $k$- dimensional subspace $E _ {k}$ of the tangent space $T _ {x} M$ is called a $k$- dimensional integral element of the system (*) if

$$\theta ^ \alpha ( E _ {k} ) = 0 ,\ \ d \theta ^ \alpha ( E _ {k} \wedge E _ {k} ) = 0 ,\ \alpha = 1 \dots q .$$

The subspace $S ( E _ {k} )$ of the cotangent space $T _ {r} ^ {*} M$ generated by the $1$- forms $\theta ^ \alpha \mid _ {x}$, $( v \llcorner d \theta ^ \alpha ) \mid _ {x}$, where $v \in E _ {k}$ and $\llcorner$ is the operation of interior multiplication (contraction), is called the polar system of the integral element $E _ {k}$. The integral element $E _ {k}$ is regular if there exists a flag $E _ {k} \supset {} \dots \supset E _ {1} \supset 0$ for which

$$\mathop{\rm dim} E _ {i} = i ,\ \ \mathop{\rm dim} S ( E _ {i} ) = {\max \mathop{\rm dim} } S ( E _ {i} ^ \prime ) ,$$

where the maximum is taken over all $i$- dimensional integral elements $E _ {i} ^ \prime$ containing $E _ {i-} 1$. Cartan's theorem asserts the following: Let $N$ be a $k$- dimensional integral manifold of a Pfaffian system with analytic coefficients and let, for a certain $x \in N$, the tangent space $T _ {x} N$ be a regular integral element. Then for any integral element $E _ {k+} 1 \supset T _ {x} N$ of dimension $k + 1$ there exists in a certain neighbourhood of the point $x$ an integral manifold $\widetilde{N}$, locally containing $N$, for which $E _ {k+} 1 = T _ {x} \widetilde{N}$. Cartan's theorem has been generalized to arbitrary differential systems given by ideals in the algebra of differential forms on a manifold (the Cartan–Kähler theorem).

How to Cite This Entry:
Pfaffian problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pfaffian_problem&oldid=48174
This article was adapted from an original article by D.V. Alekseevskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article