3 Affine Connections with Torsion in (Para-)complexified Structures
51
T
∇
(L X
+
, Y
−
) + T
∇
(X
+
, LY
−
) = αT
∇
(X
+
, Y
−
) − αT
∇
(X
+
, Y
−
) ≡ 0
for any real vector fields X, Y . This is to say, the (1,1)-part of the torsion-compatibility
condition is always satisfied.
3.3.4.2 (2,0)-Part of Torsion-Compatibility
Let us now require torsion-compatibility to hold for the (2,0)-part:
T
∇
(L X
+
, Y
+
) + T
∇
(X
+
, LY
+
) = 0.
Writing this out explicitly,
0 = 2αT
∇
(X
+
, Y
+
) = 2α(T
∇
(X + α
−1 L X, Y + α
−1 LY ))
= 2α(T
∇
(X, Y ) + α
−2 T
∇
(L X, LY ) + α
−1 T
∇
(L X, Y ) + α
−1 T
∇
(X, LY ))
= 2 T
∇
(L X, Y ) + 2 T
∇
(X, LY ) + α
−1
2 L
2 T
∇
(X, Y ) + 2 T
∇
(L X, LY )
.
Therefore, we obtain
T
∇
(L X, Y ) + T
∇
(X, LY ) = 0,
L
2 T
∇
(X, Y ) + T
∇
(L X, LY ) = 0.
The first equation is the torsion-compatibility we started with. The second equation is identical to the first one, after setting L X for X in the first equation. So no
new conditions are obtained after (para-)complexifying torsion-compatibility. Summarizing the situation for both (1,1)- and (2,0)-components of the torsion tensor,
(para-)complexifying torsion-compatibility (MC1) does not yield any new information beyond saying that the MC1 condition amounts to the (2,0)-component of torsion
being zero.
3.3.4.3 Para-Complexifying Codazzi Coupling
Codazzi coupling of ∇ and L leads to, according to Proposition 1,
T
∇
(X, Y ) = T
∇
L (X, Y ).
(Para-)complexifying the above, whether taking (2,0)- or (1,1)-component, leads to
T
∇
(L X, Y ) + T
∇
(X, LY ) = T
∇
L (L X, Y ) + T
∇
L (X, LY ),
L
2 T
∇
(X, Y ) + T
∇
(L X, LY ) = L
2 T
∇
L (X, Y ) + T
∇
L (L X, LY ).