Conformally Invariant Variational Problems. - SAM

Lemma II.2. Let A ∈ L(R m ,R n ) the space of linear maps
from R m into R n
Assume that A ≠ 0. Then
∀X,Y ∈ R m < X,Y > R m= 0 (II.3)
=⇒ < AX,AY > R n= 0
Where R
k denotes the canonical scalar product on the euclidian
space R k if and only if there exists λ ∈ R such that
∀X,Y ∈ R m < AX,AY > R n= e 2λ < X,Y > R
m (II.4)
Proof of Lemma II.2
We assume II.2
Let X ∈ R m , X ≠ 0 We introduce L x R m −→ R n given by
∀y ∈ R m | AX |2
L X Y : = < AX,AY > R
n − < X,Y >
| X | 2 R
m
Denote (X) ⊥ the subspace of R m made of Vectors Y orthgonal
to X. We have ⎧⎪ ⎨
⎪ ⎩
L X X = 0
L X (X) ⊥ = 0
RX ⊕(X) ⊥ = R m
This clearly implies that L X ≡ 0
For any X and Y in R m \{0} we then have
⎧
⎪⎨
⎪⎩
< AX,AY > R n=
< AX,AY > R n= |AY|2
|Y| 2
| AX |2
| X | 2 < X,Y > R
m
< X,Y > R
m
(II.4b)
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