Conformally Invariant Variational Problems. - SAM

Let us take locally about p a normal frame : a smooth map
(⃗n 1 ,··· ,⃗n m−2 ) from a neighborhood U ⊂ Σ 2 into (S m−1 ) m−2
such that for any point q in U (⃗n 1 (q),··· ,⃗n m−2 (q)) realizes a
positive orthonormal basis of ( ⃗ Φ ∗ T q Σ 2 ) ⊥ . Then
m−2
∑
π ⃗n (d 2 Φ(X,Y)) ⃗ =
α=1
〈d 2 ⃗ Φ(X,Y),⃗nα
〉
⃗n α ,
from which we deduce the following expression - which is the
natural extension of (X.1) -
⃗ Ip (X,Y) = −
m−2
∑
α=1
〈
d⃗n α ·X, Y ⃗ 〉
⃗n α ,
(X.15)
where we denote ⃗ Y := d ⃗ Φ · Y. Let (e 1 ,e 2 ) be an orthonormal
basisofT p Σ 2 , thepreviousexpressionofthesecond fundamental
form implies
| ⃗ I p | 2 =
Observe that
2∑
m−2
∑
i,j=1 α=1
| < d⃗n α ·e i ,⃗e j > | 2 2∑
i=1
m−2
∑
d⃗n = (−1) α−1 d⃗n α ∧ β≠α ⃗n β = d⃗n
=
α=1
m−2
m−2
∑
α=1
2∑ ∑
(−1) α−1 < d⃗n α ,⃗e i > ⃗e i ∧ β≠α ⃗n β
i=1
α=1
| < d⃗n α ,⃗e i > | 2
(X.16)
(X.17)
(⃗e i ∧ β≠α ⃗n β ) for α = 1···m−2 and i = 1,2 realizes a free family
of 2(m−2) orthonormal vectors in ∧ m−2 R m . Hence
|d⃗n| 2 g =
2∑
i=1
m−2
∑
α=1
| < d⃗n α ,⃗e i > | 2 = | ⃗ I p | 2 . (X.18)
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