Constructions of harmonic maps between Hadamard manifolds

64 CHAPTER 3. DAMEK-RICCI SPACES
where y, ȳ ∈ R + and n = v + z (resp. n ′ = v ′ + z ′ ) with dim v = n (resp. dim v ′ = n ′ ),
dim z = m (resp. dim z ′ = m ′ ). Take an orthonormal basis {v 1 ,... ,v n ,z 1 ,... ,z m } (resp.
{¯v 1 ,... ,¯v n ′, ¯z 1 ,... ,¯z m ′}) ofn (resp. n ′ ) as above, and denote their left invariant extensions
on S by the same letters. Let
⎧
e 0 = ∂ ⎧
∂y ,
f 0 = ∂ ∂ȳ ,
⎪⎨
⎪⎨
e i = v i (1 ≤ i ≤ n),
f α =¯v α (1 ≤ α ≤ n ′ ),
⎪⎩ e i = z i−n (n +1≤ i ≤ n + m),
⎪⎩ f α =¯z α−n ′ (n ′ +1≤ α ≤ n ′ + m ′ ),
and
u α j = f ∗ α (du(e i)), u α ij = e j · u α i , τα (u) =f ∗ α (τ(u)),
where f ∗ α is the dual frame of f α . Then, since the tension field of u is given by
we get
(3.2.2)
⎧
⎪⎨
⎪⎩
τ 0 (u) =
τ α (u) =
τ α (u) =
τ(u) :=
n+m
∑
i=0
n+m
∑
i=0
n∑
( ˜∇ ei du(e i ) − du(∇ ei e i ))=0,
i=1
n+m
∑
g ii u 0 ii +(1− n − 2m)u 0 0y − (u 0 ) −1
n+m
∑
+(u 0 ) −1
i=0
g ii n ′
i=0
∑
n+m
∑
(u β i )2 +2(u 0 ) −3
β=1
i=0
g ii (u 0 i ) 2
∑
g ii n ′ +m ′
β=n ′ +1
n+m
g ii u α ii +(1− n − 2m)uα 0 y − ∑
2(u0 ) −1 g ii u 0 i uα i
n+m
∑ ∑n ′
+(u 0 ) −2 g ii
n+m
∑
i=0
i=0
n
∑
′ +m ′
β=1 γ=n ′ +1
i=0
(u β i )2 ,
u γ i uβ i Γ γ−n′
αβ
(1 ≤ α ≤ n ′ ),
n+m
g ii u α ii +(1− n − 2m)u0 0 y − ∑
4(u0 ) −1 g ii u 0 i uα i
i=0
(n ′ +1≤ α ≤ n ′ + m ′ ),
where u 0 =ȳ(u), and (g ij ) denotes the matrix component of the metric g M , (g ij ) its inverse
matrix. Note that u α ij ≠ uα ji because [v i,v j ] ≠0.