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.