- HARMONIC MAPS AND THEIR LINEARIZATION 339
coordinates as follows::S ls=O (f;h)ij (p)
= OS a I s=O ( (ho:(3 o fs) (p) of~ axi (p) off ()xi (p) )
(
a (vo: o !) aff3 ar a (vf3 o J) )
= (ho:(3 ° f) (p) 8xi (p) 8xi (p) + 8xi (p) axi (p)- V^1 (! (p)) ( ( 0 ~ 1 ho:f3) o f (p)). ~~~ (p) ~~; (p)
(K.25) -_ ( (Vi fh V) o:aff3 8xi +('VJ fh V) (3ar) 8xi (ho:f3 o !) (p).
The last equality above may be seen from (K.17) and the following identity:
of' aff3 o: of' afo: f3
~ --;;--:-ho:f3 r ( h) ~ 6 + --;;--:---;;--:-ho:f3 r ( h) ~ 6
ux' uxJ ' uxJ uxJ '
= ~a r a Jf3 ( ah6(3 + ah,(3 _ ah,6 )
2 8xi 8xi f)y'Y ay6 8yf3
+~of' ar (ahfo + ah,o: _ ah,6)
2 8xi 8xi ay'Y ay6 ()yo:
a r a Jf3 ah,(3
8xi 8xi ay6 ·In invariant notation we write (K.25) as
_§_I (f*h) = (vrhv,df) + (df, vrhv) ,
OS s=O s f*h f*hwhere the inner products contract the last components of the tensors.
Tracing (K.25), we have
(K.26) ~ I ws 1~0h = ~ I gij u: h )ij
us s=O uS s=O= 2gii(V{hv)o:~ff3 (ho:f3 o !)
uxJ=2(\i'f*hV,df).
g0h
In the following we shall assume that V has compact support in the interior of
M. From (K.26) we may deduce the first variation formula for the map energy:
:S ls=O Eg,h Us)= JM ( :S ls=O ldfsl~0h) dμg
= 2 r (vf*hv, df) dμg
}M g0h
=-2 JM (v,tr 9 (\7^90 hd!))
1
.hdμ 9= -2 JM (V, 6.g,hf) f*h dμg.