338 K. IMPLICIT FUNCTION THEOREM
As a special case, if M = N and f is the identity map and if we choose the
coordinates x and y to b e the same, then £f2 oxk = 8Z so that
(K.21)
is the trace of the difference of the two Christoffel symbols. Note that although a
connection is not a tensor, the difference of two connections is a tensor. One way
to rewrite formula (K.21) is
(uA 9 ,h i "d)k --^1 ij (h-l)k£ ("gh. '79h · - '79h· ·)
2
g v i 1 e + v j ie v e , 1= (h-^1 )ke ( div^9 (h)e - ~\leH),where H ~ tr 9 (h) = gij hij.
If (Pn, k) and (Nm, h) are Riemannian manifolds, f : P --7 N is a map and
r.p : M --7 P is a diffeomorphism, then (see for example (2.56) in [77])
(K.22)In p articular , i f f : M --7 N is a diffeomorphism, then we may rewrite its map-
Laplacian as
(K.23) (6. 9 ,hf) (x) = (6.u-1r 9 ,h idN) (f (x)).
We also h ave the formula
(f-^1 )*(6.g,hf) = 6.g,J•hidM ·
DEFINITION K.12 (Harmonic m ap). A map f : (Mn, g) --7 (Nm, h) is called a
harmonic map if(K.24)As we shall see below, harmonic maps are critical points of the map energy.
If N = ~m, then a map is h armonic if and only if each of its components is a
harmonic function. If M is 1-dimensional, then a ha rmonic map is the same as a
constant speed geodesic.
EXAMPLE K.13. Let f be a smooth immersion of a differentiable manifold Mn
into a Riemannian manifold (Nm, h). Then the mean curvature vector ii off (M)
is equal to 6. 1 • (h),hf. In particular, an isometric immersion is a minimal immersion
(i.e., ii= 0) if and only if it is a h armonic map.
3.2. First variation formula for the map energy.
Let f: Mn --7 Nm b e a map between Riem annian ma nifolds (with or without
boundary) and let UsLE(-e,e) b e a 1-parameter family of maps with Jo= f. Define
the variation field V ~ ~ls=o' which is a section of the vector bundle f* (TN).
We have^2 ts I s=O ld!s 1 ~0h = 2 ('V r h v, df) g0h. We verify this equation in local
(^2) Since dfs is a section of T* M@ J; (TN) and this bundle d ep ends on s , t h e expression
8
(^81) dfs in and of itself does not make sense.
s s=O