Page 65 - Curvature and Homology

EXERCISES 47

r-dimen3ional subspace V,. of Tp. In order that the systems (BL), (Bi) give rise
to the same r-dimensional subspace it is necessary and sufficient that there
exist a matrix (a:) E GL(q, R) satisfying

3. Let D be a domain of M. A pfa#an system of rank q and class l(2 I; 1 S k - 1)
is defined, if, for every covering of D by coordinate neighborhoods {u} and
every point P of U a system of q linearly independent pfaffian forms is given
such that for P E U n 0

where the matrix (a;) E GL(q, R) is of class 1.
A pfaffian system of rank q(= n - r) on D defines an r-dimensional subspace
of the tangent space Tp at each point P E D, that is, afield of r-planes of class 1.
A manifold may not possess pfaffian systems of a given rank. For example,
the existence of a pfaffian system of rank n - 1 is equivalent to the existence
of a field of directions. This is not possible on an even-dimensional sphere.
4. Suppose a pfafin system of rank q and class 1 is defined on the coordinate
neighborhood U by the 1-forms P, i = 1, ..., q. This system is said to be
completely integrable if there are q functions of class 1 + 1 such that
0' = a; df '. (a;) E GL(q,R).

The pfaffian system may then be defined by the q differentials dfi. Under the
circumstances the functions f form afirst integral of the system.
The following result is due to Frobenius:
In order that a pfam system (BL) be completely integrable it is necessary and
sumt that dBL A O1 A ... A iB = 0 for every i = 1, ..., q.
The necessity is clear. The sufficiency may be proved by employihg a result
on the existence of a 'canonical pfhn system' in P and then proceeding by
induction on r [23]. Since a pfaffian system of rank q on U defines and can be
defined by a non-zero decomposable form 8 of degree q determined up to a
non-zero factor this result may be stated as follows:
If a pfaflan system of rank q hos the propty that at every point P E M there
is a local coordinate system such that the form 63 can be chosen to involve only q
of these coordinates, the system is completely integrable.
5. If 8 = 8' A ... A Oq, the condition