3 Affine Connections with Torsion in (Para-)complexified Structures
55
connections may carry torsion. So our paper characterizes the torsion tensors T
∇
and T
∇
L that will lead to integrable (para-)complex structure. The case of ∇ L = 0
is recovered as the special case where ∇ = ∇
L .
A Hermitian connection ∇ is one that preserves the Hermitian form h, ∇h = 0,
defined by h(X, Y ) = g(X, Y ) +
√ −1ω(X, Y ). Equivalently, a Hermitian connection is one which is metric (∇g = 0) and complex (∇ J = 0). In general, Hermitian
connections carry non-zero torsion. In fact, there is a line of Hermitian connections
called the Gauduchon connections, all with differerent torsions. This line passes
through the first canonical connection and the second canonical connection which
satisfies, respectively, Eq. (3.19) and Eq. (3.20). But there are infinitely many other
Hermitian connections that do not fall on the Gauduchon line.
Our investigations generalize the first and second canonical connections by removing the assumption of their being (para-)complex connections ∇ L = 0. Specifically,
MC2 generalizes Eq. (3.20) to involve both ∇, ∇
L , while MC1 is to be enforced separately on ∇ and ∇
L . The set of torsion-compatible connections (satisfying MC1)
intersects the line of Gauduchon connections at the first canonical connection. This
intersection is unique (and transversal) because for all other Gauduchon connections,
the torsion is not (1, 1). The set of torsion-coupled connections (satisfying MC2)
intersects the line of Gauduchon connections at the Chern connection (the second
canonical connection). This intersection is also unique and transversal because the
Chern connection is the unique holomorphic Hermitian (complex and metric) connection.
MC1 condition expressed a constraint about the torsion of ∇, namely, the (2,0)and (0,2)-component of T
∇ must vanish. However, this constraint can be separately
enforced upon ∇ and ∇
L . Only when both T
∇ and T
∇
∗ satisfy MC1 would an
integrable L result. In the presence of Codazzi coupling, which stipulates T
∇
= T
∇
L ,
(∇, L) satisfies MC1 if and only if (∇
L
, L) satisfies MC1. Thus MC1 articulates a
generalization of the first canonical connection.
MC2 condition, on the other hand, expresses a “balance” of T
∇ and T
∇
L , such
that it involves the torsions of ∇ and ∇
L in an intertwined fashion. (∇, L) satisfies
MC2 if and only if (∇
L
, L) satisfies MC2. It is in this scenario that ∇ and ∇
L both
become (para-)holomorphic connections. MC2 articulates a generalization of the
second canonical connection.
(2, 0)-part of MC2 ⇔ N
(2,0)
= 0 ⇔ N L (X, Y ) = 0, since N
(1,1)
≡ 0
(1, 1)-part of MC2 ⇔ Cond. H ⇔
(para-)holomorphicity of ∇
MC1 and MC2 conditions are independent of each other. When torsion-coupling and
torsion-compatibility are both imposed, then the connections must be torsion-free
T
∇
= T
∇
L = 0. These are torsion-free holomorphic connections yet generally not
(para-)complex.
Acknowledgements The project is supported by DARPA/ARO Grant W911NF-16-1-0383 (“Information Geometry: Geometrization of Science of Information”) and a sub-contract from AFOSR
Grant FA9550-19-1-0213 (“Brain-Inspired Networks for Multifunctional Intelligent Systems and
55
connections may carry torsion. So our paper characterizes the torsion tensors T
∇
and T
∇
L that will lead to integrable (para-)complex structure. The case of ∇ L = 0
is recovered as the special case where ∇ = ∇
L .
A Hermitian connection ∇ is one that preserves the Hermitian form h, ∇h = 0,
defined by h(X, Y ) = g(X, Y ) +
√ −1ω(X, Y ). Equivalently, a Hermitian connection is one which is metric (∇g = 0) and complex (∇ J = 0). In general, Hermitian
connections carry non-zero torsion. In fact, there is a line of Hermitian connections
called the Gauduchon connections, all with differerent torsions. This line passes
through the first canonical connection and the second canonical connection which
satisfies, respectively, Eq. (3.19) and Eq. (3.20). But there are infinitely many other
Hermitian connections that do not fall on the Gauduchon line.
Our investigations generalize the first and second canonical connections by removing the assumption of their being (para-)complex connections ∇ L = 0. Specifically,
MC2 generalizes Eq. (3.20) to involve both ∇, ∇
L , while MC1 is to be enforced separately on ∇ and ∇
L . The set of torsion-compatible connections (satisfying MC1)
intersects the line of Gauduchon connections at the first canonical connection. This
intersection is unique (and transversal) because for all other Gauduchon connections,
the torsion is not (1, 1). The set of torsion-coupled connections (satisfying MC2)
intersects the line of Gauduchon connections at the Chern connection (the second
canonical connection). This intersection is also unique and transversal because the
Chern connection is the unique holomorphic Hermitian (complex and metric) connection.
MC1 condition expressed a constraint about the torsion of ∇, namely, the (2,0)and (0,2)-component of T
∇ must vanish. However, this constraint can be separately
enforced upon ∇ and ∇
L . Only when both T
∇ and T
∇
∗ satisfy MC1 would an
integrable L result. In the presence of Codazzi coupling, which stipulates T
∇
= T
∇
L ,
(∇, L) satisfies MC1 if and only if (∇
L
, L) satisfies MC1. Thus MC1 articulates a
generalization of the first canonical connection.
MC2 condition, on the other hand, expresses a “balance” of T
∇ and T
∇
L , such
that it involves the torsions of ∇ and ∇
L in an intertwined fashion. (∇, L) satisfies
MC2 if and only if (∇
L
, L) satisfies MC2. It is in this scenario that ∇ and ∇
L both
become (para-)holomorphic connections. MC2 articulates a generalization of the
second canonical connection.
(2, 0)-part of MC2 ⇔ N
(2,0)
= 0 ⇔ N L (X, Y ) = 0, since N
(1,1)
≡ 0
(1, 1)-part of MC2 ⇔ Cond. H ⇔
(para-)holomorphicity of ∇
MC1 and MC2 conditions are independent of each other. When torsion-coupling and
torsion-compatibility are both imposed, then the connections must be torsion-free
T
∇
= T
∇
L = 0. These are torsion-free holomorphic connections yet generally not
(para-)complex.
Acknowledgements The project is supported by DARPA/ARO Grant W911NF-16-1-0383 (“Information Geometry: Geometrization of Science of Information”) and a sub-contract from AFOSR
Grant FA9550-19-1-0213 (“Brain-Inspired Networks for Multifunctional Intelligent Systems and