Page 146 - Curvature and Homology

128 111. RIEMANNIAN MANIFOLDS: CURVATURE, HOMOLOGY

D. The third fundamental theorem of Lie
By differentiating the equations (3.5.3), the relations

are obtained. Conversely, assuming ~ constants ch are given with the property
that
c:k + = 0,

show that the conditions (3.D.1) are sufficient for the existence of n linear
differential forms, linearly independent at each point of a region in R", and
which satisfy the relations (3.5.3).
This may be shown in the following way:
Consider the system

of n% linear partial differential equations in ns variables hi in the space R"+l of
independent variables t, al, -, @-the al, go-, @ being treated as parameters.
Given the initial conditions

the equations (3.D.2) have unique (analytic) solutions hxt, a', -*, @) valid
throughout R"+l.
Observe that

Hence,

In particular,

Now, define n linear differential forms of by

In terms of the of, 2-forms hi and 1-forms af (both sets independent of dt) are
defined by the equations
dof = hi + dt A a*. (3.D.3)