Constructions of harmonic maps between Hadamard manifolds

3.2. HARMONIC MAPS BETWEEN DAMEK-RICCI SPACES 63
3.2.1 Harmonic map equation
We compute explicitly the tension field of a harmonic map between Damek-Ricci spaces. Let
(S, g S ) be a Damek-Ricci space with left invariant metric g S , and set s = a ⊕ n = a ⊕ v ⊕ z.
Take the unit vector h ∈ a such that ad h(v) =v, ad h(z) =2z for any v ∈ v,z ∈ z and
an orthonormal basis {h, v 1 ,... ,v n ,z 1 ,... ,z m } of s, {v i } (resp. {z j }) being an orthonormal
basis of v (resp. z). We define a map ϕ : R × N → S by
ϕ(t, n) :=n(exp(th)),
where N is the simply connected Lie group associated with n and exp is the exponential
map on a. Then the induced metric ϕ ∗ g S is given by
ϕ ∗ g S = dt 2 + e −2t g Ú
+ e −4t g Þ
,
where g Ú
+ g Þ
is a left invariant metric on N. Setting y = e t , one can verify that (S, g S )is
isometric to M := R + × N with the Riemannian metric
g M := y −2 dy 2 + y −2 g Ú
+ y −4 g Þ
,
where R + = {r ∈ R | r>0}. A straightforward calculation then yields that the Levi-Civita
connection ∇ on (S, g S ) is given by
⎧
∇ η η = −y −1 η, ∇ η v i = −y −1 v i , ∇ η z k = −2y −1 z k ,
(3.2.1)
⎪⎨
⎪⎩
∇ vi v j = y −1 δ ij η + 1 2
∇ vi z k = 1 n∑
2 y−2
j=1
m∑
k=1
a k
ijz k ,
where a k
ij are the structure constants, [v i ,v j ]=
regarded as left invariant vector fields on S.
a k
j iv j , ∇ zk z l =2y −3 δ kl η,
m∑
k=1
a k
ijz k ,η= ∂/∂y and, v i and z k
Now we compute the tension field. Let (S, g S ) and (S ′ ,g S ′) be Damek-Ricci spaces and
u ∈ C 2 (S, S ′ ). We represent (S, g S ) (resp. (S ′ ,g S ′)) as
(R + × N,y −2 dy 2 + y −2 g Ú
+ y −4 g Þ
) (resp. (R + × N ′ , ȳ −2 dȳ 2 +ȳ −2 g Ú ′ +ȳ −4 g Þ
′)),
are