Constructions of harmonic maps between Hadamard manifolds

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that for any point z ∈ S m+n−1 , there exist x ∈ S m−1 ,y ∈ S n−1 and t ∈ [0,π/2] such that z
is expressed as
z = ((cos t)x, (sin t)y),
the join u ∗ r v of these maps is defined by
u ∗ r v : S m+n−1 ∋ ((cos t)x, (sin t)y) ↦→ ((cos r(t))u(x), (sin r(t))v(y)) ∈ S p+q−1 .
Consequently, the harmonic map equation for the join map is reduced to an ordinary differential
equation in r = r(t) with a suitable boundary condition. By solving this boundary
value problem, Smith proved that there exists a harmonic representative in each homotopy
class π n (S n ) ≃ Z for n ≤ 7. Subsequently, his method was extended by several authors.
In particular, Ding([11]) utilized a variational method to clarify the meaning of the dumping
condition, Eells and Ratto([15]) generalized the method to the case of ellipsoids, and
Xin([45],[46]) reformulated Smith’s method from the view point of Riemannian submersions
(see also [16]).
2) We can generalize the notion of a join map to the case of real hyperbolic spaces RH m
as follows. First, note that for any point z ∈ RH m+n−1 , there exist x ∈ RH m−1 ,y ∈ S n−1
and t ∈ [0, ∞) such that
z = ((cosh t)x, (sinh t)y).
Let u : RH m−1 → RH p−1 and v : S n−1 → S q−1 be eigenmaps, and r :[0, ∞) → [0, ∞) a
smooth function. Then the join map u ∗ r v of u, v and r is defined to be
u ∗ r v : RH m+n−1 ∋ ((cosh t)x, (sinh t)y) ↦→ ((cosh r(t))u(x), (sinh r(t))v(y)) ∈ RH p+q−1 .
Then we see that u ∗ r v is a harmonic map if and only if the function r = r(t) satisfies the
following ordinary differential equation defined on [0, ∞):
{
¨r(t)+ p sinh t
cosh t + m cosh t } { }
µ
2
ṙ(t) −
sinh t cosh 2 t +
ν2
sinh 2 sinh r(t) cosh r(t) =0,
t
where e(u) =µ 2 /2 and e(v) =ν 2 /2 denote the energy density functions of u and v, respectively.