3 Affine Connections with Torsion in (Para-)complexified Structures
53
Let us now writing out explicitly the (2,0)-part of torsion-coupling
L
T
∇
L (X + α
−1 L X, Y + α
−1 LY )
= αT
∇
(X + α
−1 L X, Y + α
−1 LY )
or explicitly,
L
T
∇
L (X, Y ) + α
−2 T
∇
L (L X, LY ) + α
−1
(T
∇
L (L X, Y ) + T
∇
L (X, LY ))
= α
T
∇
(X, Y ) + α
−2 T
∇
(L X, LY )
+ T
∇
(L X, Y ) + T
∇
(X, LY ),
from which we obtain two equations (separating real and imaginary components):
L
2 T
∇
L (X, Y ) + T
∇
L (L X, LY ) = L
T
∇
(X, LY ) + T
∇
(L X, Y )
,
L
T
∇
L (L X, Y ) + T
∇
L (X, LY )
= L
2 T
∇
(X, Y ) + T
∇
(L X, LY ).
The above two equations are equivalent to each other, since substituting L X for
X in the first yields the second. Furthermore they exactly (!) express the condition
N L = 0. This shows that the (2,0)-part of the torsion-coupling condition is exactly
N L = 0. Since N
(1,1)
≡ 0, N L = 0 is equivalent to N
(2,0)
= 0.
3.3.5.2 (1, 1)-Part of the Torsion-Coupling: (Para-)holomorphicity
Let us now only impose torsion-coupling condition to the (1,1)-part, that is, when
inputs are taken from T
+
p M × T
−
p M or (+, −):
LT
∇
(X + α
−1 L X, Y − α
−1 LY ) = αT
∇
L (X + α
−1 L X, Y − α
−1 LY ).
Writing out explicitly,
L
T
∇
(X, Y ) − α
−2 T
∇
(L X, LY ) + α
−1 T
∇
(L X, Y ) − α
−1 T
∇
(X, LY )
= α
T
∇
L (X, Y ) − α
−2 T
∇
L (L X, LY ) + α
−1 T
∇
L (L X, Y ) − α
−1 T
∇
L (X, LY )
from which we obtain two equations (separating real and imaginary components):
L
T
∇
(X, Y ) − L
2 T
∇
(L X, LY )
= T
∇
L (L X, Y ) − T
∇
L (X, LY ),
L
T
∇
(L X, Y ) − T
∇
(X, LY )
= L
2 T
∇
L (X, Y ) − T
∇
L (L X, LY ).
The above two equations are equivalent to each other—we can obtain the second
from subsituting X → L X in the first. Rearranging the second:
T
∇
L (L X, LY ) − LT
∇
(X, LY ) = L
2 T
∇
L (X, Y ) − LT
∇
(L X, Y ).
(3.18)