Constructions of harmonic maps between Hadamard manifolds

46 CHAPTER 2. EQUIVARIANT HARMONIC MAPS
Consequently, the multiplicity of the zero eigenvalue of the matrix G(F ) is even. Hence r
must be even.
Define orthogonal multiplications F 1 : R 2 × R 2 → R 2 and F 2 : R 2 × R 2 → R 4 by
⎧
⎪⎨ F 1 (x, y) =(x 1 y 1 + x 2 y 2 ,x 1 y 2 − x 2 y 1 ),
⎪⎩ F 2 (x, y) =(x 1 y 1 ,x 1 y 2 ,x 2 y 1 ,x 2 y 2 ),
respectively, where x =(x 1 ,x 2 ) and y =(y 1 ,y 2 ). Then, they are full and satisfy
‖F 1 (x, y)‖ = ‖F 2 (x, y)‖ = ‖x‖‖y‖.
In the case of n =2k, we decompose R n as the direct sum of k-copies of R 2 , that is,
R n = R 2 ⊕···⊕R 2 . Hence R 2 × R n =(R 2 × R 2 ) ⊕···⊕(R 2 × R 2 ) (direct sum of k-copies
of R 2 × R 2 ). Then, for i (0 ≤ i ≤ k),
i-copies (k − i)-copies
{ }} { { }} {
F = F 1 ⊕···⊕F 1 ⊕ F 2 ⊕···⊕F 2
defines an orthogonal multiplication F : R 2 × R n → R 2(n−i) . Thus for even r, where n ≤
r ≤ 2n, there exists an orthogonal multiplication R 2 × R 2k → R r .
In the case of n =2k +1, we have the direct sum R n
= R 2k ⊕ R. Associated with
the orthogonal multiplication F : R 2 × R 2k → R r , we obtain an orthogonal multiplication
˜F : R 2 × R 2k+1 → R r+2 defined by
˜F ((x 1 ,x 2 ), (y 1 ,... ,y 2k ,y 2k+1 ))=(F ((x 1 ,x 2 ), (y 1 ,... ,y 2k )), (x 1 y 2k+1 ,x 2 y 2k+1 )),
where (x 1 ,x 2 ) ∈ R 2 and (y 1 ,... ,y 2k ,y 2k+1 ) ∈ R 2k+1 . Hence for even r, where n+1 ≤ r ≤ 2n,
there exists an orthogonal multiplication R 2 × R 2k+1 → R r .
Now we introduce a method of constructing a new eigenmap of degree two from old
known ones.
Let f : S m → S p and g : S n → S q be eigenmaps of degree two, and F : R m+1 × R n+1 →
R r an orthogonal multiplication. Define a map ϕ : R m+1 × R n+1 → R p+q+r+2 by
(2.1.2) ϕ(x, y) :=(f(x),g(y), √ 2F (x, y)).