86 3. SHORT TIME EXISTENCE
be a diffeomorphism. Then if "' is a metric on Nn, we have
HenceSince
a./3 - ( * )ij ac.pa. ac.p/3
K, - i.pK, ux>:i"i uxJ >:i " '(3.41)
Putting c.p = f and "' = (f-^1 )* g in (3.41) and substituting into (3.39), we
get(3.42) (tif)^1 = ( u-^1 ) g) a.f3 [-r ( u-^1 ) g) :/3 + I'(h):/3] '
whence the lemma follows. DCOROLLARY 3.19. Taking Mn= Nn and f to be the identity, we have(ti 9 ,h id)^1 = ga.^13 (-r (g): 13 + r(h): 13 ).
A particular case of this result is of some independent interest:COROLLARY 3.20. If (Mn, g) is a manifold of strictly positive Ricci cur-
vature, thenis a harmonic map.PROOF. Note that h ~Re (g) is a metric, and let A denote the globally-
defined tensor field
A = r ( h) - r ( 9 ).