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).

References

[1]	E. Cartan, "Sur la théorie des systèmes en involution et ses applications à la relativité" Bull. Soc. Math. France , 59 (1931) pp. 88–118 MR1504975 Zbl 0002.26401 Zbl 57.0551.02
[2]	E. Cartan, "Leçons sur les invariants intégraux" , Hermann (1922) MR0355764 Zbl 48.0538.02
[3]	P.K. Rashevskii, "Geometric theory of partial differential equations" , Moscow-Leningrad (1947) (In Russian)
[4]	S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964) MR0193578 Zbl 0129.13102
[5]	P.A. Griffiths, "Exterior differential systems and the calculus of variations" , Birkhäuser (1983) MR0684663 Zbl 0512.49003

Pfaffian problems and partial differential equations.

Let

$$ \tag{a1 } F _ {h} \left ( x ^ {i} , u ^ {j} , \frac{\partial ^ \alpha }{\partial x ^ \alpha } u ^ {k} \right ) = 0 , $$

$$ h = 1 \dots p,\ i = 1 \dots n,\ j = 1 \dots m, $$

$$ \alpha = ( a _ {1} \dots a _ {n} ) , $$

$$ | \alpha | = a _ {1} + \dots + a _ {n} \leq r,\ a _ {i} \in \{ 0, 1, . . . \} , $$

be a system $ p $ partial differential equations for $ m $ functions in $ n $ variables of order $ \leq r $. Introduce the variables

$$ p ^ {\alpha , k } ,\ \ 1 \leq | \alpha | \leq r,\ \ k = 1 \dots m . $$

Replacing the equations (a1) with the equations

$$ \tag{a2 } \widetilde{F} {} _ {h} ( x ^ {i} , u ^ {j} , p ^ {\alpha ,k } ) = 0 $$

and adding to this the Pfaffian system

$$ \tag{a3 } dp ^ {\alpha , k } - \sum _ { i=1} ^ { n } p ^ {\alpha ( i), k } d x ^ {i} = 0 ,\ \ 0 \leq | \alpha | \leq r- 1 , $$

where $ p ^ {0,k} = u ^ {k} $ and $ \alpha ^ {(} i) = ( a _ {1} \dots a _ {i-1} , a _ {i} + 1 , a _ {i+1} \dots a _ {n} ) $ if $ \alpha = ( a _ {1} \dots a _ {n} ) $ for $ i = 1 \dots n $, one finds a system (a2)–(a3) of equations which are equivalent to equations (a1) in a suitable sense. Thus, if (locally) (a2) defines a subvariety $ M $ in $ ( x ^ {i} , u ^ {j} , p ^ {\alpha , k } ) $- space, then a solution of the Pfaffian problem (a3) on $ M $ defines a solution of (a1) in the sense that the projection onto $ \mathbf R ^ {n} \times \mathbf R ^ {m} $( or $ \mathbf C ^ {n} \times \mathbf C ^ {m} $ as the case may be) gives the graph of a solution of (a1).

For instance, in the case of a single second-order equation

$$ F \left ( x ^ {1} , x ^ {2} , u ,\ \frac{\partial u }{\partial x ^ {1} } , \frac{\partial u }{\partial x ^ {2} } ,\ \frac{\partial ^ {2} u }{\partial x ^ {1} \partial x ^ {1} } ,\ \frac{\partial ^ {2} u }{\partial x ^ {1} \partial x ^ {2} } ,\ \frac{\partial ^ {2} u }{\partial x ^ {2} \partial x ^ {2} } \right ) = 0 $$

one has for (a2) and (a3), respectively,

$$ \tag{a2\prime } \widetilde{F} ( x ^ {1} , x ^ {2} , u , p ^ {1} , p ^ {2} ,\ p ^ {11} , p ^ {12} , p ^ {22} ) = 0, $$

$$ \tag{a3\prime } \left . \begin{array}{c} du = p ^ {1} dx ^ {1} + p ^ {2} dx ^ {2} , \\ dp ^ {1} = p ^ {11} dx ^ {1} + p ^ {12} dx ^ {2} , \\ dp ^ {2} = p ^ {12} dx ^ {1} + p ^ {22} dx ^ {2} \end{array} \right \} . $$

The main equations are (a2); the remaining equations (a3) express that the solutions of (a2) of interest are $ r $- jets (cf. Jet and Partial differential equations on a manifold) of functions $ \mathbf R ^ {n} \rightarrow \mathbf R ^ {m} $. This leads to the idea of a system of partial differential equations on a manifold of order $ r $ as being determined by a set of functions on the $ r $- th jet bundle; cf. Partial differential equations on a manifold for more details.

In the setting of equations like (a2), (a3) the following generalization of Frobenius' theorem on complete integrability is of interest. Let $ \omega ^ {1} , \dots , \omega ^ {r} $ be a set of differential forms on a manifold $ M $ and $ f ^ { 1 } , \dots , f ^ { s } $ a set of functions on $ M $. Let $ m \in M $ be such that $ f ^ { i } ( m) = 0 $, $ i = 1 \dots s $. Suppose that

i) $ d \omega ^ {i} $ and $ df ^ { j } $ are in the ideal of differential forms generated by $ \omega ^ {1} , \dots , \omega ^ {r} ; f ^ { 1 } \dots f ^ { s } $;

ii) the $ \omega ^ {i} $ are linearly independent at $ m $.

(Recall that the linearly independent $ 1 $- forms $ \omega ^ {1} \dots \omega ^ {r} $ form an involutive system if $ d \omega ^ {i} $ is in the ideal generated by the $ \omega ^ {i} $, cf. Involutive distribution.) Then there is a unique germ of a submanifold $ N $ at $ m $ of dimension $ n $, $ r+ n = \mathop{\rm dim} M $, such that the differential forms $ \omega ^ {i} $ and functions $ f ^ { j } $ restricted to $ N $ are zero. Further if $ x ^ {1} \dots x ^ {n} $ are functions on $ M $ near $ m $ such that $ \omega ^ {1} \dots \omega ^ {r} , dx ^ {1} \dots dx ^ {n} $ are linearly independent at $ m $, then the $ x ^ {1} \dots x ^ {n} $ give a coordinate chart of $ N $ near $ m $.

Cartan–Kähler theorem for differential systems defined by ideals.

Let $ \theta ^ {a} = 0 $, $ a = 1 \dots q $, be a Pfaffian system on $ M $ and let $ N $ be an integral manifold of this system. Then obviously the $ d \theta ^ {a} $ and $ \theta ^ {a} \wedge \omega $, where $ \omega $ is any differential form on $ M $, are also zero on $ N $. Thus all the elements of the differential ideal generated by $ \theta ^ {1} \dots \theta ^ {q} $ in the differential algebra of exterior differential forms $ F( M) $( cf. Differential form; Differential ring) are zero on $ N $. This leads to the idea of a differential system (of equations) on $ M $ as being defined by such an ideal. From now on let $ M $ be a real analytic manifold. Let $ {\mathcal F} ( M) $ be the associated sheaf to $ F( M) $, i.e. $ {\mathcal F} ( M) $ is the sheaf of germs of rings of differential forms on $ M $. Let $ {\mathcal O} ( M) $ be the sheaf of analytic functions on $ M $ and let $ {\mathcal F} _ {p} ( M) $ be the $ {\mathcal O} ( M) $- module of $ p $- forms on $ M $. A differential system on $ M $ is a graded differential subsheaf $ {\mathcal G} $ of ideals of $ {\mathcal F} ( M) = {\mathcal F} $, i.e. $ {\mathcal F} {\mathcal G} = {\mathcal G} = {\mathcal G} {\mathcal F} $( the ideal property), $ {\mathcal G} $ is generated by the $ {\mathcal G} _ {p} = {\mathcal F} _ {p} \cap {\mathcal G} $( the graded property) and $ d {\mathcal G} \subset {\mathcal G} $( the differential property). A $ p $- dimensional integral manifold for $ {\mathcal G} $ is a submanifold $ N $ of $ M $ on which $ {\mathcal G} $ is zero. For each $ m \in M $ let $ \mathop{\rm Gr} _ {p} ( m) $ be the Grassmann manifold of $ p $- dimensional subspaces of the tangent space $ T _ {m} M $. The union of the $ \mathop{\rm Gr} _ {p} ( m) $ for $ m \in M $ has a natural structure of a real-analytic manifold and the projection $ \mathop{\rm Gr} _ {p} ( m) \ni E _ {p} \rightarrow m $ then defines a locally trivial fibre bundle $ \mathop{\rm Gr} _ {p} ( M) \rightarrow M $. An element $ E _ {p} \in \mathop{\rm Gr} _ {p} ( m) $ is called a contact element at $ m $. Such an element is an integral element of $ {\mathcal G} _ {p} $ if $ \omega ( E _ {p} ) = 0 $ for all $ \omega \in {\mathcal G} _ {p} $; it is an integral element of a differential system $ {\mathcal G} $ if for all $ E _ {q} \subset E _ {p} $, $ 0 \leq q \leq p $, $ E _ {q} $ is an integral element of $ {\mathcal G} _ {p} $. An integral element of dimension zero (i.e. a point of $ M $) is an integral point (which is simply a solution of the equations $ f( m) = 0 $ for the functions $ f \in {\mathcal G} _ {0} $). The polar element of an integral element $ E _ {p} $ for $ {\mathcal G} $ is the element $ P( E _ {p} ) \supset E _ {p} $ consisting of all vectors $ v \in T _ {m} M $ such that the span of $ v , E _ {p} $ is an integral element of $ {\mathcal G} $. Let $ z ^ {i _ {1} \dots i _ {p} } $, $ 1 \leq i _ {1} < \dots < i _ {p} \leq n $, be the Grassmann coordinates of $ E _ {p} $( cf. Exterior algebra; these are only defined up to a common scalar multiple). Now associate to $ {\mathcal G} _ {p} $ the sheaf $ {\mathcal G} _ {p} ^ {0} $ of $ {\mathcal O} ( M) $- modules in $ {\mathcal O} ( \mathop{\rm Gr} _ {p} ( M)) $ consisting of all the functions $ \sum _ {i \leq i _ {1} < \dots < i _ {p} \leq n } a _ {i _ {1} \dots i _ {p} } z ^ {i _ {1} \dots i _ {p} } $ for all $ p $- forms $ \sum _ {1 \leq i _ {1} < \dots < i _ {p} \leq n } a _ {i _ {1} \dots i _ {p} } dx ^ {i _ {1} } \wedge \dots \wedge dx ^ {i _ {p} } \in {\mathcal G} _ {p} $. Let $ {\mathcal I} ( {\mathcal G} _ {p} ) $ be the set of integral elements of $ {\mathcal G} _ {p} $( so that $ {\mathcal I} ( {\mathcal G} _ {p} ) $ is a certain subset of the Grassmann bundle $ \mathop{\rm Gr} _ {p} ( M) $). The element $ E _ {p} $ is called a regular integral element if $ {\mathcal G} _ {p} ^ {0} $ is a regular local equation for $ {\mathcal I} ( {\mathcal G} _ {p} ) $ at $ E _ {p} $ and $ \mathop{\rm dim} ( P( E _ {p} )) $ is constant near $ E _ {p} $ on $ {\mathcal I} ( {\mathcal G} _ {p} ) $. Recall that a subsheaf $ {\mathcal A} \subset {\mathcal O} ( X) $, where $ X $ is a manifold, is a regular local equation for (its set of zeros) $ N \subset X $ at $ m \in N \subset X $ if locally around $ m $ there exist sections $ s _ {1} \dots s _ {t} \in \Gamma ( U, {\mathcal O} ( X)) $ such that the $ ds _ {1} \dots ds _ {t} $ are linearly independent on $ U $ and $ m ^ \prime \in N \cap U $ if and only if $ s _ {1} ( m ^ \prime ) = \dots = s _ {t} ( m ^ \prime ) = 0 $.

The first Cartan–Kähler existence theorem is now as follows. Let $ N $ be a $ p $- dimensional integral manifold of $ G $ which defines a regular element $ T _ {m} N \subset T _ {m} M $ at $ m \in N \subset M $. Suppose that there is a submanifold $ M ^ \prime $ of $ M $ containing $ N $ and of dimension $ n+ p+ 1 - \mathop{\rm dim} P( T _ {m} N) $ such that $ \mathop{\rm dim} ( T _ {m} M ^ \prime \cap P( T _ {m} N)) = p+ 1 $. Then locally around $ m $ there exists a unique integral manifold $ N ^ \prime $ of dimension $ p+ 1 $ contained in $ M ^ \prime $.

If $ \mathop{\rm dim} P( T _ {m} N) = p+ 1 $, the only possible choice (locally) for $ M ^ \prime $ is $ M $ itself, and there is a unique integral manifold of dimension $ p+ 1 $ extending $ N $. If $ \mathop{\rm dim} P( T _ {m} N) = p+ 2 $ there is "one arbitrary function worth" freedom in choosing $ M ^ \prime $ and one re-encounters the phenomenon that the solution of a partial differential equation may depend on arbitrary functions (such as $ u _ {x} = u _ {t} $ with as solutions any function of the form $ \phi ( x+ t) $). The second Cartan–Kähler existence theorem, which is obtained by repeated application of the first, details the dependence on initial conditions and arbitrary functions.

An immediate corollary of the first Cartan–Kähler existence theorem is as follows. Suppose one is given an integral element $ E _ {p+1} $ of dimension $ p+ 1 $ of the differential system $ {\mathcal G} $ at $ m \in M $ which contains a regular integral element $ E _ {p} $. Then there exists (locally) an integral manifold $ N $ of dimension $ p+ 1 $ such that $ T _ {m} N = E _ {p+1} $.

References

[a1]	P. Libermann, C.-M. Marle, "Symplectic geometry and analytical mechanics" , Reidel (1987) pp. Chapt. V, Appendix 3 (Translated from French) MR0882548 Zbl 0643.53002
[a2]	E. Cartan, "Les systèmes différentiels extérieurs et leurs applications géométriques" , Hermann (1945)
[a3]	E. Cartan, "Sur l'intégration des systèmes d'équations aux différentielles totales" Ann. Sci. Ec. Norm. Sup. , 18 (1901) pp. 241–311 Zbl 32.0351.04
[a4]	E. Kähler, "Einführung in die Theorie der Systeme von Differentialgleichungen" , Teubner (1934) Zbl 0011.16103 Zbl 60.0401.08
[a5]	M. Kuranishi, "Lectures on exterior differential systems" , Tata Inst. (1962)
[a6]	J. Dieudonné, "Eléments d'analyse" , 4 , Gauthier-Villars (1977) pp. Chapt. XVIII, Sect. 13 MR0467780