Page 66 - Curvature and Homology

48 I. RIEMANNIAN MANIFOLDS

is equivalent to the condition
d@=wAB
for some 1-form w.
6. The linear subspaces of dimension r of Tp are in 1-1 correspondence
with the classes of non-zero decomposable r-vectors-each class consisting
of r-vectors differing from one another by a scalar factor. The set of r-vectors
can be given a topology by means of the components relative to some basis.
This defines a topology and, in fact, a differentiable structure in the set of
subspaces denoted by Gr(Tp) of dimension r of Tp The manifold so obtained
is called the GYUSS~~UM manifold over Tp The Grassman manifold Gr(Tp*) over
the dual space may be similarly defined. There is a 1-1 correspondence

This map is independent of the choice of a basis in A"(Tp*). Evidently then,
it is a homeomorphism.
Define the fibre bundle

over M and show that it can be given a topology and a differentiable structure
of class k - 1.
7. A cross section
F : M -+ Gr(M)

of this bundle is a pfaffian system of rank q sometimes called a diflerGntial system
of dimension r or r-distribution. A differential system of dimension r therefore
associates with every point P of M a linear subspace of dimension r of Tp
By means of the correspondence Gr(Tp) -* GQ(T:), F defines (up to a non-
zero factor) a decomposable form of degree q.
8. A submanifold (Q, M') is called an integral manifold of F if, for every P' E M',

The dimension of an integral manifold is therefore I; r. Show that F is com-
pletely integrable if every P E M has a coordinate neighborhood with the local
coordinates ul, ..., un such that the 'coordinate slices'
u1 = const., ..., u9 = const.

are integral manifolds of F.