Conformally Invariant Variational Problems. - SAM

where ⋆ is the Hodge operator 37 from ∧ 2 R m into ∧ m−2 R m and
⃗n 1···⃗n m−2 isapositiveorthonormalbasisoftheorientednormal
planeto ⃗ Φ ∗ T p Σ 2 (thisimpliesinparticularthat(⃗e 1 ,⃗e 2 ,⃗n 1 ,··· ,⃗n m−2 )
is an orthonormal basis of R m ).
We shall denote by π ⃗n the orthogonal projection onto the
m−2−plane at p given by ⃗n(p). Denote by
The second fundamental form associated to the immersion ⃗ Φ
is the following map
⃗ Ip : T p Σ 2 ×T p Σ 2 −→ ( ⃗ Φ ∗ T p Σ 2 ) ⊥
(X,Y)
−→ ⃗ I p (X,Y) := π ⃗n (d 2 ⃗ Φ(X,Y))
where X and Y are extended smoothlyintolocalsmoothvectorfields
around p. One easily verifies that, though d 2 ⃗ Φ(X,Y)
might depend on these extensions, π ⃗n (d 2 ⃗ Φ(X,Y)) does not depend
on these extensions and we have then defined a tensor.
Let ⃗ X := d ⃗ Φ · X and ⃗ Y := d ⃗ Φ · Y. Denote also by π T the
orthogonal projection onto ⃗ Φ ∗ T q Σ 2 .
π ⃗n (d 2 ⃗ Φ(X,Y)) = d(d ⃗ Φp·X)·Y −π T (d(d ⃗ Φ p·X)·Y)
= d ⃗ X · ⃗Y −∇ Y X
= ∇ ⃗Y X ⃗ −∇Y X .
(X.10)
where ∇ is the Levi-Civita connection in R m for the canonical
metric and ∇ is the Levi-Civita connection on TΣ 2 induced by
37 The Hodge operator on R m is the linear map from ∧ p R m into ∧ m−p R m which to a
p−vector α assigns the m − p−vector ⋆α on R m such that for any p−vector β in ∧ p R m
the following identity holds
β ∧⋆α =< β,α > ε 1 ∧···∧ε m ,
where (ε 1 ,··· ,ε m ) is the canonical orthonormal basis of R m and < ·,· > is the canonical
scalar product on ∧ p R m .
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