Constructions of harmonic maps between Hadamard manifolds

2.3. CONSTRUCTIONS OF EQUIVARIANT HARMONIC MAPS 53
2.3 Constructions of equivariant harmonic maps
2.3.1 Join parameterization
We introduce the join parameterization of real hyperbolic spaces, which is an analogue of
that for standard unit spheres due to Smith([34]).
We consider the hyperboloid model of the real hyperbolic space RH m+p+1 , that is,
RH m+p+1 = {(x 0 ,x 1 , ··· ,x m+p+1 ) ∈ R m+p+2 | −x 2 0 + x 2 1 + ···+ x 2 m+p+1 = −1, x 0 > 0}.
For any z ∈ RH m+p+1 there exist x ∈ RH p ,y ∈ S m and t ∈ [0, ∞) such that
z = ((cosh t)x, (sinh t)y).
Note that x and y are uniquely determined for t>0. Define a map f :[0, ∞)×RH p ×S m →
RH m+p+1 by
f(t, x, y) = ((cosh t)x, (sinh t)y).
Then the pull-back metric of the standard metric ˜g on RH m+p+1 via f is given by
f ∗˜g = dt 2 + (cosh 2 t)g 1 + (sinh 2 t)g 2 ,
where g 1 (resp. g 2 ) is the standard metric on RH p (resp. S m ).
Let ψ : RH p → RH q ,ϕ : S m → S n be eigenmaps, and 2e(ψ) =µ 2 , 2e(ϕ) =ν 2 . Then
the map
u : RH m+p+1 ∋ (t, x, y) → (r(t),ψ(x),ϕ(y)) ∈ RH n+q+1
is a harmonic map if and only if r = r(t) is a solution to the following ordinary differential
equation:
{
¨r(t)+ p sinh t
cosh t + m cosh t }
ṙ(t)
sinh t
(2.2.3)
{ }
µ
2
−
cosh 2 t +
ν2
sinh 2 sinh r(t) cosh r(t) =0.
t
In order for u to be continuous, we require that r = r(t) satisfies
lim
t→0
r(t) =0.