- HARMONIC MAPS AND THEIR LINEARIZATION 337
where C ~ C o f , for C E C^00 (TN). In local coordinates, this is
(K.17) ("Vrhcrr = ~c' + ar (r (h)' a ncf3
i o x ' ox' Oi{3 )
where the r (h):f3 denote the Christoffel symbols of h.
Let \J9-i denote the connection on T* M dual to \J9. Let \J9^181 h ~ \J9-^1 0 \Jf*h
be the unique linear connection on £ satisfying(K.18)for A E C^00 (T* M), B E C^00 (f* (TN)). This connection is compatible with the
bundle metric g ~ h ; that is ,X (a, (3) gl81h = ("V9x.181h a, f3) + (a, "V9x,181h f3)
gl81h gl81h
for any sections a , (3 E C^00 (£) and X ET M.
Now the map-Hessian \J9^181 hdj is a section of the bundle T* M 0 s T* M 0
f*TN. The map-Laplacian off is the trace with respect tog of \J9^181 hdf; i.e.,(K.19)In local coordinates,where(\J
(^9181) hd!)' = ("V^9181 h · (()Jf3 dxk 0 __§__))'
ij 8/ax' oxk oyf3.
J
a^2 f' k of' of°' off3
=a x' a. xJ -f(g)iJ' u x !:l k +(r(h):f3of)~~ ux' uxJ
- \J (^9) i \J (^9) j (') f + r ( ()' h Oi{3 0 f )of°'off3 ()xi 8x1.
(Note that the above displayed expression is symmetric in i and j.^1 ) Hence, by
tracing the Hessian off with respect to g, we obtain the following.
LEMMA K.11 (Map-Laplacian in local coordinates). If f: M---+ N, then
(K.20a) (6. 9 ,hf)' = 6. 9 (!') + gi1 (r (h):f3 of):;
i.(a^2 r kof' , of°'off3)
(K.^2 0b) = g J 8xiox1 - r (g)ij oxk + (r (h)Oi{3 ° f) ()xi ox1.
(^1) See also p. 5 of [99], where this symmetry is expressed invariantly. In particular, given vector
fields X , Y E C^00 (M), we have
\Jrh (df (Y)) -\JC" (df (X)) = df ([X, Y]).
Since df ([X, Y]) = (df) (\J x Y) -(df) (\Jy X), another way to write this, using the product rule,
is
( \JIJcri?lh (df)) (Y) = ( \J~ri?Jh (df)) (X).
That is , the Hessian of f is symmetric.