Page 68 - Curvature and Homology

50 I. RIEMANNIAN MANIFOLDS
of corresponding forms in An the position vector P' together with the vectors
e; satisfy the pfaffian system

(cf. equations (1.9.23)). The variables of this system are t, x,! and the components
of the vectors P', e;, ..., ei. Since the curvature forms Bij are quadratic in the
differentials of the local coordinates, they vanish along a parametrized curve.
It follows that there exists a local differentiable homeomorphism f from the
bundle of frames over the submanifold C(t) to the bundle of frames over
C'(t)-the submanifold defined by the image of C(t) in A", such that

where 4, dij denote the forms in. An corresponding to ei, 8,. Show that the
conditions in Frobenius' theorem are satisfied by this system and hence that
it is completely integrable. As a consequence of this, show that there is exactly
one set of vectors P', e;, ..., ei satisfying (*) and taking arbitrary initial values
for t = to and x,! = 8;. If e;, ..., ei are linearly independent for t = to show that
they are independent for all values of t, that is, for all t,{e;, ..., e:} is a frame
on Ct(t).

1. Denote the fine transformation defined by equation (1.9.22) by Ttatl

Show that Ttat1 is not, in general, a linear map. Define the linear map

TP(t1) -+ TP(t,)

sending the vector xiAi(t3 E Tp(tl) into the vector ZiA4tg) E Tp(ta, by means
of the equation
xiAi(tl) = SiA;(t,).

Show that Ttat1 is independent of (a) the choice of initial frame 4 = 8: for t = to
and.@) the choice of the family {Al(t), ..., A;(t)} of frames along C(t).