Constructions of harmonic maps between Hadamard manifolds

2.2. REDUCTION OF HARMONIC MAP EQUATIONS 51
⎧
Γj i
1 =Γ i f
1 j = ˙ 2 (t)
f 2 (t) δ ij
⎪⎨
(m 1 +2≤ j ≤ m 1 + m 2 +1),
Γj i
k = 2 Γj i
k (m 1 +2≤ j, k ≤ m 1 + m 2 +1), and
⎪⎩
Γ i
j k= 0
(otherwise),
(
)
∆ g = ∂2
∂t + f˙
1 (t)
m 2 1
f 1 (t) + m f˙
2 (t) ∂
2
f 2 (t) ∂t + 1
f 1 (t) ∆ 2 1 + 1
f 2 (t) ∆ 2.
2
Here x 1 = t ∈ I,(x 2 ,... ,x m1 +1) ∈ M 1 , (x m1 +2,... ,x m1 +m 2 +1) ∈ M 2 ,
m 1 = dimM 1 , and
m 2 = dimM 2 . Also, we denote by p Γj i
k (resp. ∆ p) the Christoffel symbols (resp. the Laplace-
Beltrami operator) of (M p ,g p ).
Let ( ˜M 1 , ˜g 1 ) and ( ˜M 2 , ˜g 2 ) be Riemannian manifolds, Ĩ(⊂ R) an interval, and h 1 and h 2
nonnegative smooth functions on Ĩ. We consider a product map u defined by
(2.2.1)
u :(I × M 1 × M 2 ,dt 2 + f 1 (t) 2 g 1 + f 2 (t) 2 g 2 ) ∋ (t, x, y)
↦→ (r(t),ϕ(x),ψ(y)) ∈ (Ĩ × ˜M 1 × ˜M 2 ,dr 2 + h 1 (r) 2˜g 1 + h 2 (r) 2˜g 2 ),
where r : I → Ĩ, ϕ : M 1 → ˜M 1 and ψ : M 2 → ˜M 2 are smooth maps. The tension field τ(u)
of the map u is computed as follows.
In local coordinates, x =(x i ) ∈ M,u(x) =(u α (x)) ∈ Ĩ × ˜M 1 × ˜M 2 , the components of
the tension field τ(u) is given by
τ(u) α =∆ g u α +
m 1 +m
∑ 2 +1
i,j=1
n 1 +n
∑ 2 +1
β,γ=1
g ij ˜Γ
α
β γ (u(x)) ∂uβ
∂x i
∂u γ
∂x j
.
Then, from Lemma 2.2.1, we have