Conformally Invariant Variational Problems. - SAM

vectors are denoted ⃗e i (t,p). The fact that this transport is parallel
w.r.t. ∇ implies that (⃗e 1 (t,p),⃗e 2 (t,p)) realizes a positive
orthonormal basis 48 of ⃗ Φ ∗ (t×T p Σ 2 ).
Next we extend ⃗e i (t,p) parallely and locally in {t}×Σ 2 with
respect again to ∇ along geodesics in {t} × Σ 2 starting from
(t,p) for the induced metric ⃗ Φ ∗ t g R m.
We will denote also e i := ( ⃗ Φ −1
t ) ∗ ⃗e i . By definition one has
⃗H(t,p) := 1 2
2∑
π ⃗n (d⃗e s ·e s ) = 1 2
s=1
2∑
D es ⃗e s ,
wherewehaveextendedtheuseofthenotationDforanysection
of T ⊕N in an obvious way 49 . Hence we have
D ∂
⃗H = 1
∂t 2
2∑
s=1
s=1
D ∂ (D es ⃗e s ) (X.69)
∂t
Since on ⃗ Φ([0,1] × {p}) one has ∇⃗e s ≡ 0 and since moreover
∇+D coincides with the differentiation 50 d in R m one has
D ∂(D es ⃗e s )(0,p) = D ∂ (d⃗e s ·e s )(0,p)
∂t
∂t
= π ⃗n (d(∂ t ⃗e s )·e s )+π ⃗n (d⃗e s ·∂ t e s )
(X.70)
Observe that for a fixed q ∈ Σ 2 e s (t,q) stays in T q Σ 2 as t varies
and then there isno need of a connectionto define ∂ t e s . Observe
moreover that ∂ t e s = [∂ t ,e s ]. Thus (X.69) together with (X.70)
48 Indeed
(
∇ ∂ ⃗e i = 0 ⇒ π T d⃗e i · ∂ )
= 0
∂t
∂t
〈 〉 ∂⃗ei
⇒ ∀i,j
∂t ,⃗e j = 0 ⇒ ∀i,j
∂
∂t = 0 .
49 D X σ := π ⃗n (dσ ·X). D does not realizes a connection on the whole T ⊕N.
50 d is the flat standard connection on TR m
134