44
J. Zhang and G. Khan
N L (X, Y ) = L
2 T
∇
L (X, Y ) + T
∇
L (L X, LY ).
Further assuming (i) leads to (iii).
Finally, we show that (iv) will lead to T
∇
(L X, Y ) = T
∇
(X, LY ):
T
∇
(L X, Y ) = LT
∇
L (X, Y ) = −LT
∇
L (Y, X ) = −T
∇
(LY, X ) = T
∇
(X, LY ).
And likewise, T
∇
L (L X, Y ) = T
∇
L (X, LY ). Therefore, further imposing (ii) or (iii)
forces T
∇
= T
∇
L = 0.
Remark. In the special case ∇ = ∇
L , then ∇ is a (para-)complex connection ∇ L =
0. In this case, the last two terms of the righthand side of Eq. (3.12) vanish. So
T
∇
(L X, Y ) = LT
∇
(X, Y ). A complex connection that is metric and satisfies Eq.
(3.12) (torsion-coupling condition) is known as a second canonical connection for
Hermitian manifolds. It is also called the Chern connection. On the other hand,
a complex connection that is metric and satisfies Eq. (3.11) (torsion-compatibility
condition), T
∇
(L X, Y ) + T
∇
(X, LY ) = 0, is known as a first canonical connection
for Hermitian manifolds. So our definitions of MC1 (torsion-compatibility) and MC2
(torsion-coupling) generalize the appropriate definitions of first and second canonical
connections with torsion, to non-(para-)complex connections with torsion. Part (d)
of Proposition 4 further claims that MC1 and MC2 are genuinely different types of
coupling, if ∇ and ∇
L must carry non-zero torsions.
3.3 Torsion of ∇ Under (Para-)complexification
3.3.1 Splitting of T M ⊗ L by L
For a smooth manifold M, an endomorphism of the tangent bundle T M induces
a smooth map L of the tangent bundle. By definition, L is called an almost complex structure if L
2
= −id, or an almost para-complex structure if L
2
= id and the
multiplicities of the eigenvalues ±1 are equal. We have used J and K to denote
almost complex structures and almost para-complex structures, respectively, and use
L when these two structures can be treated in a unified way. It is clear from our
definition that such structures exist only when M is of even dimension.
Denote eigenvalues of L as ±α, where α = 1 for L = K and α = i for L = J ,
depending on the nature of L. We have
L
4
= id,
α
4
= 1,
α
2 L
2
= id.
Following the standard procedure, we (para-)complexify T M by tensoring with
complex field C or the para-complex (a.k.a. split-complex) field D, and use T
L
M to
denote the resulting T M ⊗ C =
p∈M T p M ⊗ C or T M ⊗ D =
p∈M T p M ⊗
D, depending on the type of L. In analogy with standard notation in the complex
J. Zhang and G. Khan
N L (X, Y ) = L
2 T
∇
L (X, Y ) + T
∇
L (L X, LY ).
Further assuming (i) leads to (iii).
Finally, we show that (iv) will lead to T
∇
(L X, Y ) = T
∇
(X, LY ):
T
∇
(L X, Y ) = LT
∇
L (X, Y ) = −LT
∇
L (Y, X ) = −T
∇
(LY, X ) = T
∇
(X, LY ).
And likewise, T
∇
L (L X, Y ) = T
∇
L (X, LY ). Therefore, further imposing (ii) or (iii)
forces T
∇
= T
∇
L = 0.
Remark. In the special case ∇ = ∇
L , then ∇ is a (para-)complex connection ∇ L =
0. In this case, the last two terms of the righthand side of Eq. (3.12) vanish. So
T
∇
(L X, Y ) = LT
∇
(X, Y ). A complex connection that is metric and satisfies Eq.
(3.12) (torsion-coupling condition) is known as a second canonical connection for
Hermitian manifolds. It is also called the Chern connection. On the other hand,
a complex connection that is metric and satisfies Eq. (3.11) (torsion-compatibility
condition), T
∇
(L X, Y ) + T
∇
(X, LY ) = 0, is known as a first canonical connection
for Hermitian manifolds. So our definitions of MC1 (torsion-compatibility) and MC2
(torsion-coupling) generalize the appropriate definitions of first and second canonical
connections with torsion, to non-(para-)complex connections with torsion. Part (d)
of Proposition 4 further claims that MC1 and MC2 are genuinely different types of
coupling, if ∇ and ∇
L must carry non-zero torsions.
3.3 Torsion of ∇ Under (Para-)complexification
3.3.1 Splitting of T M ⊗ L by L
For a smooth manifold M, an endomorphism of the tangent bundle T M induces
a smooth map L of the tangent bundle. By definition, L is called an almost complex structure if L
2
= −id, or an almost para-complex structure if L
2
= id and the
multiplicities of the eigenvalues ±1 are equal. We have used J and K to denote
almost complex structures and almost para-complex structures, respectively, and use
L when these two structures can be treated in a unified way. It is clear from our
definition that such structures exist only when M is of even dimension.
Denote eigenvalues of L as ±α, where α = 1 for L = K and α = i for L = J ,
depending on the nature of L. We have
L
4
= id,
α
4
= 1,
α
2 L
2
= id.
Following the standard procedure, we (para-)complexify T M by tensoring with
complex field C or the para-complex (a.k.a. split-complex) field D, and use T
L
M to
denote the resulting T M ⊗ C =
p∈M T p M ⊗ C or T M ⊗ D =
p∈M T p M ⊗
D, depending on the type of L. In analogy with standard notation in the complex