Constructions of harmonic maps between Hadamard manifolds

2.1. EIGENMAPS 49
exists a full orthogonal multiplication F : R 4 × R 4 → R r for r =8, 11. Therefore the Hopf
construction
ϕ(x, y) =(‖x‖ 2 −‖y‖ 2 , 2F (x, y)) (x, y ∈ R 4 )
gives rise to a full eigenmap ϕ : S 7 → S n of degree two for n =8, 11. As a consequence, we
have a full eigenmap ϕ : S 7 → S n of degree two for 8 ≤ n ≤ 34. Thus (i) is true for k =3.
(ii) Using Lemma 2.1.5 and the above result, we have a full eigenmap ϕ : S 8 → S n of
degree two for 17 ≤ n ≤ 43. Thus (ii) is also true for k =3.
Step 2. (i) We assume that the statements (i) and (ii) are true up to k = l(≥ 3). Since
there exists an eigenmap g : S 2l+2 → S q of degree two for l 2 +5l − 7 ≤ q ≤ 2l 2 +7l +4, it
follows from Lemma 2.1.5 that we have a new eigenmap
ϕ : S 2l+3 → S n for l 2 +7l − 3 ≤ n ≤ 2l 2 +9l +8.
On the other hand, from our assumption, there exist a full orthogonal multiplication F :
R 2 ×R 2l+2 → R r for 2l +2 ≤ r ≤ 4l +4 and a full eigenmap g : S 2l+1 → S q for l 2 +3l −10 ≤
q ≤ 2l 2 +5l +1. Thus, from Proposition 2.1.3, we have a new eigenmap
ϕ : S 2l+3 → S n for l 2 +5l − 6 ≤ n ≤ 2l 2 +6l +10.
Since
l 2 +5l − 6