Hyper-Lagrangian Submanifolds of Hyperkähler Manifolds and ...

346 Naichung Conan Leung and Tom Y. H. Wan
Most of the equations used in [11] are still valid for general J ∈ J . In particular, we have
N(e i ) = Je i −
2n∑
j=1
ω j i e j , (2.1)
η ij = g ij + ω k i ω kj . (2.2)
∇ ek e j = Ɣ n kj e n − η mn h mkj N(e n ). (2.3)
Note that only quantities in the second term on the right-hand side of the Equation (2.3) depends
on J . We also have
∇ l h ikj −∇ k h ilj = R ijkl + η mn ωn
s ( )
hmlj h ski − h mkj h sli
+ η mn ωi
s ( )
hmkj h nls − h mlj h nks , (2.4)
where R ijkl = g(R(e k ,e l )e j ,N(e i )), and
R ij kl = R ij kl + η mn ( h mik h nj l − h mil h nj k
)
. (2.5)
However, when ω is no longer parallel or closed, other equations have to be modified.
Proposition 2.1. For any orthogonal set of parallel complex structures {J α } 3 α=1
and any J =
∑ 3α=1
a α J α , the corresponding h = h(J ) and ω = ω(J) satisfy
3∑
h kij = h ikj +∇ j ω ik − (∇ j a α )ω α,ik .
Proof. The proof is straightforward by the corresponding proposition of [11]. As each J α
is parallel,
α=1
h kij = ∑ α
= ∑ α
a α h α,kij
a α (h α,ikj +∇ j ω α,ik )
= h ikj + ∑ α
a α ∇ j ω α,ik
= h ikj +∇ j ω ik − ∑ α
(∇ j a α )ω α,ik .
Proposition 2.2.
For any J ∈ J , the corresponding 2-form ω = ω(J) satisfies
∇ k ∇ j ω li −∇ l ∇ j ω ki =∇ j ∇ i ω lk + R s ilj ω ks + R s ij k ω ls − R s jkl ω si +∇ j (dω) lik .
Proof. The proof is exactly the same as in [11]. The only modification is that ω is not closed,
so the last term does not vanish in general.
Proposition 2.3. For any orthogonal set of parallel complex structures {J α } 3 α=1
and any J =