48
J. Zhang and G. Khan
¯
∂
∇
X + Y
+
= ¯
∂
∇
X − Y
−
= 0;
¯
∂
∇
X − Y
+
= ∇ X − Y
+
;
¯
∂
∇
X + Y
−
= ∇ X + Y
−
.
In other words, operating on Y
+
∈ T
+
M, ¯
∂
∇ takes the (0,1)-part of the vector-valued
1-form ∇Y
+ (and conversely the (1,0)-part when operating on Y
−
∈ T
−
M).
Again, we do not assumed integrability of L, so the same caveat applies to ¯
∂
∇
as for the operator ¯
∂ defined above— ¯
∂
∇ is defined on the full (para-)complexified
tangent bundle T
L
M, and therefore a restriction of it to T
−
M would actually be
denoted by ∂
∇ if acting on T
−
M as a (para-)holomorphic bundle.
On a (para-)holomorphic vector bundle, a connection is conventionally called
(para-)holomorphic if the two Dolbeault operators ¯
∂ and ¯
∂
∇ coincide. We extend
this notion to arbitrary connections on T
L
M = T
(1,0)
M ⊕ T
(0,1)
M where ∇ does
not necessarily preserve T
(1,0)
M or T
(0,1)
M.
Definition 2 A connection ∇ is called (para-)holomorphic if ¯
∂
∇
X Y = ¯
∂ X Y for any
vector fields X and Y .
Using (3.14) and (3.16), we can also derive a necessary and sufficient condition for
(para-)holomorphicity in terms of N L , T
∇
, and T
∇
L . In fact the following were
proven in [7]:
Lemma 2 Given an arbitrary pair (∇, L), the following statements are equivalent
(i) ∇ is (para-)holomorphic;
(ii) ∇
L is (para-)holomorphic;
(iii) the following holds
1
2
N L (X, Y ) = L
2 T
∇
L (X, Y ) − LT
∇
(L X, Y ).
(3.17)
Two connections ∇
1 and ∇
2 are said to be ¯
∂-balanced when ¯
∂
∇
1
X Y = ¯
∂
∇
2
X Y holds.
This Lemma implies that a sufficient condition for (∇, ∇
L
) to be ¯
∂-balanced is
that either of them is (para-)holomorphic. (Para-)holomorphicity of ∇ (and ∇
L ) is
stronger than (∇, ∇
L
) being ¯
∂-balanced. These notions were investigated in [7].
3.3.3 (Para-)complexifying N L and T ∇
3.3.3.1 Decomposition of N L
Both T
(1,0)
M and T
(0,1)
M are foliations if and only if L is integrable, i.e., the
integrability condition N L = 0 is satisfied.
We recall that in the (para-)complex case, the (1, 1)-component of the Nijenhuis
tensor vanishes identically. Heuristically, this follows from the fact that the Nijenhuis
J. Zhang and G. Khan
¯
∂
∇
X + Y
+
= ¯
∂
∇
X − Y
−
= 0;
¯
∂
∇
X − Y
+
= ∇ X − Y
+
;
¯
∂
∇
X + Y
−
= ∇ X + Y
−
.
In other words, operating on Y
+
∈ T
+
M, ¯
∂
∇ takes the (0,1)-part of the vector-valued
1-form ∇Y
+ (and conversely the (1,0)-part when operating on Y
−
∈ T
−
M).
Again, we do not assumed integrability of L, so the same caveat applies to ¯
∂
∇
as for the operator ¯
∂ defined above— ¯
∂
∇ is defined on the full (para-)complexified
tangent bundle T
L
M, and therefore a restriction of it to T
−
M would actually be
denoted by ∂
∇ if acting on T
−
M as a (para-)holomorphic bundle.
On a (para-)holomorphic vector bundle, a connection is conventionally called
(para-)holomorphic if the two Dolbeault operators ¯
∂ and ¯
∂
∇ coincide. We extend
this notion to arbitrary connections on T
L
M = T
(1,0)
M ⊕ T
(0,1)
M where ∇ does
not necessarily preserve T
(1,0)
M or T
(0,1)
M.
Definition 2 A connection ∇ is called (para-)holomorphic if ¯
∂
∇
X Y = ¯
∂ X Y for any
vector fields X and Y .
Using (3.14) and (3.16), we can also derive a necessary and sufficient condition for
(para-)holomorphicity in terms of N L , T
∇
, and T
∇
L . In fact the following were
proven in [7]:
Lemma 2 Given an arbitrary pair (∇, L), the following statements are equivalent
(i) ∇ is (para-)holomorphic;
(ii) ∇
L is (para-)holomorphic;
(iii) the following holds
1
2
N L (X, Y ) = L
2 T
∇
L (X, Y ) − LT
∇
(L X, Y ).
(3.17)
Two connections ∇
1 and ∇
2 are said to be ¯
∂-balanced when ¯
∂
∇
1
X Y = ¯
∂
∇
2
X Y holds.
This Lemma implies that a sufficient condition for (∇, ∇
L
) to be ¯
∂-balanced is
that either of them is (para-)holomorphic. (Para-)holomorphicity of ∇ (and ∇
L ) is
stronger than (∇, ∇
L
) being ¯
∂-balanced. These notions were investigated in [7].
3.3.3 (Para-)complexifying N L and T ∇
3.3.3.1 Decomposition of N L
Both T
(1,0)
M and T
(0,1)
M are foliations if and only if L is integrable, i.e., the
integrability condition N L = 0 is satisfied.
We recall that in the (para-)complex case, the (1, 1)-component of the Nijenhuis
tensor vanishes identically. Heuristically, this follows from the fact that the Nijenhuis