Page 147 - Curvature and Homology
P. 147
EXERCISES
Indeed,
Differentiating the equations (3.D.3) we obtain
dA' = dt A dai.
On the other hand,
It follows that
a
i (T) daj = akcirar
and
i (=) a dAj = cirak A wr + akQr - akc~ac;,wt A wa.
Thus,
a
i (T) dt~ = akckp.
On the other hand, by setting
afik - arci p
-- (3.D.4)
at ra jk*
Since hX0, a', -, an) = 0, it follows that fL(0, a', -, an) = 0. Consequently,
by (3.D.4) the flk vanish for all t, and so the V vanish identically. Hence,
K = A wk.
Now, consider the map -
+R"-cR"+'
defined by
+(XI, ***, X") = (1, Xl, ***, X")
and set
d = d*w'.
