Conformally Invariant Variational Problems. - SAM

R m is given by ∇ X σ := π T (dσ ·X) where π T is the orthogonal
projection onto the tangent plane. The Levi-Civita connection
moreover is symmetric 57 (see [doC2] theorem 3.6 chap. 2) hence
we have in particular
[⃗e 1 ,⃗e 2 ] = ∇ e1 ⃗e 2 −∇ e2 ⃗e 1 = π T (d⃗e 2 ·e 1 −d⃗e 1 ·e 2 )
(X.136)
Since⃗e 1 and⃗e 2 have unit length, the tangentialprojectionof d⃗e 1
(resp. d⃗e 2 ) are oriented along ⃗e 2 (resp. ⃗e 1 ). So we have
⎧
⎨ π T (d⃗e 2 ·e 1 ) =< d⃗e 2 ,⃗e 1 > ·e 1 ⃗e 1
(X.137)
⎩
π T (d⃗e 1 ·e 2 ) =< d⃗e 1 ,⃗e 2 > ·e 2 ⃗e 2
Combining (X.135), (X.136) and (X.137) gives then
⎧
⎨ de ∗ 1 (e 1,e 2 ) = − < d⃗e 2 ,⃗e 1 > ·e 1
⎩
de ∗ 2(e 1 ,e 2 ) =< d⃗e 1 ,⃗e 2 > ·e 2
(X.138)
Equation (X.133) gives
⎧
⎨ − < d⃗e 2 ,⃗e 1 > ·e 1 = (∗ g dλ,e 1 ) = −dλ·e 2
(X.139)
⎩
< d⃗e 1 ,⃗e 2 > ·e 2 = (∗ g dλ,e 2 ) = dλ·e 1
Thus combining (X.138) and (X.139) gives then
⎧
⎨ de ∗ 1 = −dλ·e 2 e ∗ 1 ∧e∗ 2 = dλ∧e∗ 1
⎩
de ∗ 2 = dλ·e 1 e ∗ 1 ∧e ∗ 2 = dλ∧e ∗ 2 .
(X.140)
We have thus proved at the end
⎧
⎨ d ( e −λ e1) ∗ = 0
⎩
d ( ) (X.141)
e −λ e ∗ 2 = 0
57 We recall that a connection ∇ on the tangent bundle of a manifold M m is symmetric
if for any pair of tangent fields X and Y one has
T(X,Y) := ∇ X Y −∇ Y X −[X,Y] = 0 .
161