Page 67 - Curvature and Homology

EXERCISES 49
Consider a completely integrable pfaffian system. The manifold (9, M')
is an integral manifold, if on every neighborhood U of M such that U n M' # 0
the pfafKan forms el, ..., OQvanish. If P E M', the tangent space to M' at P
is the r-plane defined by the pfak system.
9. The Frobenius theorem is a generalization of well-known theorems on total
differential equations. Consider, for example, the case n = 3, r = 2 with the
fom 8 considered above given in the local coordinates x, y, a by

By Frobenius' theorem, a necessary and sufficient condition for complete
integrability is given by
de A 8 = 0,

E. Local flatness 1231
1. If the curvature and torsion of an affineiy connected manifold M are both
zero, show that the manifold is locally flat.
Hint: By means of the equations (1.7.5) it suffices to show the existence of a
local coordinate system (G) such that

dzii = p: du'
and
dp: = 9: w:.
Use Frobenius' theorem.
This may also be seen as follows: From the structural equations it is seen that
zero curvature implies that the distribution of horizontal planes in B given
by Bf = 0 is completely integrable. An integral manifold is thus a covering of M.
Since the torsion is also zero the other structural equation gives d@ = 0,
i = 1, ..., n on the integral mmifold. Consequently, P = dui, where (ul, ..., un)
is a flat coordinate system.

F. Development of frames along a parametrized curve into An [23]
1. In the notation of 9 1.9 show that the frames {Xl(t), ..., Xn(t)} can be mapped
into An in such a way that the pfaffian forms Bi, 8,, are dual images of cor-
responding forms in An :
Let X;(t), ..., X;(t) denote the images of the frame vectors under the mapping.
In the notation of 5 1.9 a typical frame along C is denoted by p,{e1, ..., en}
and its image vectors in An by P',{e;, ..., ei}. If the 9, and 8, are the dual imagea