3 Affine Connections with Torsion in (Para-)complexified Structures
39
Along with ∇
L
:= L
−1
∇ L, the L-conjugate transformation of ∇, we can define
torsion-coupling of (∇, L) as the following relation (Definition 1) between the torsions T
∇ and T
∇
L of ∇ and ∇
L :
T
∇
(L X, Y ) = L(T
∇
L (X, Y )).
(3.2)
Torsion-coupling is equivalent to Codazzi-coupling when torsion of ∇ vanishes.
So torsion-coupling of ∇ with L provides an appropriate generalization to impose
Codazzi-like coupling requirement for connections with non-vanishing torsion. It is
shown ([7], Theorem 6) that ∇ is torsion-coupled to L if any only L is integrable and
∇ is (para-)holomorphic. Moreover, given a (para-)Hermitian manifold (M, g, L)
with four connections ∇, ∇
L , ∇
∗ , ∇
†
≡ ∇
∗L
= ∇
L∗ , all possibly carrying torsion, if
any of the four is torsion-coupled to L and any of the four is Codazzi-coupled to g,
then all four must be torsion-coupled to L and Codazzi-coupled to g ([7], Theorem
12). This leads to the definition of a Codazzi-(para-)Hermitian structure–which is
a (para-)Hermitian manifold with all four of the aforementioned connections being
(para-)holomorphic and Codazzi-coupled to g. In this case, their torsions satisfy
T = T
∗
, T
L
= T
† , but dω = 0 in general, unless the torsions of the quadruple of
the connections are all zero.
The torsion-coupling condition Eq. (3.2) on an almost (para-)complex manifold
(M, L), as first investigated in [7], is interesting because the same equation (called
the 2nd Matching Condition or MC2 below) encodes two things simulataneously,
both a constraint on L, i.e., L should be integrable, and a constraint on ∇, i.e., ∇
should be (para-)holomorphic. In this paper, we investigate this torsion-coupling
condition further—by decomposing this condition into two parts, one which implies
integrability of L, and another which implies (para-)holomorphicity of ∇. We investigate another coupling of L with ∇ (called the 1st Matching Condition or MC1
below) that would lead to integrability condition of L and certain properties of ∇
in reference to the Gauduchon line [8]. Our goal in this paper is to understand how
torsion of a connection interacts with a (para-)complex structure L on a manifold
when the connection is (in general) not required to be parallel with respect to L.
3.2 Torsion of ∇ and Integrability of L
3.2.1 L Conjugation of ∇
Given a real manifold M of even dimensions dim M = 2n, we study affine connection ∇ (in general carrying torsion and curvature) on M, i.e., ∇ operates on sections
(T M) of its tangent bundle T M, i.e., vector fields with real dimension 2n. Let
L : (T M) → (T M) be an a map induced by tangent bundle endomorphism,
with L
−1 denoting the inverse map. Then, the L-gauge transformed connection is
defined as ∇
L
:= L
−1
◦ ∇ ◦ L, or
39
Along with ∇
L
:= L
−1
∇ L, the L-conjugate transformation of ∇, we can define
torsion-coupling of (∇, L) as the following relation (Definition 1) between the torsions T
∇ and T
∇
L of ∇ and ∇
L :
T
∇
(L X, Y ) = L(T
∇
L (X, Y )).
(3.2)
Torsion-coupling is equivalent to Codazzi-coupling when torsion of ∇ vanishes.
So torsion-coupling of ∇ with L provides an appropriate generalization to impose
Codazzi-like coupling requirement for connections with non-vanishing torsion. It is
shown ([7], Theorem 6) that ∇ is torsion-coupled to L if any only L is integrable and
∇ is (para-)holomorphic. Moreover, given a (para-)Hermitian manifold (M, g, L)
with four connections ∇, ∇
L , ∇
∗ , ∇
†
≡ ∇
∗L
= ∇
L∗ , all possibly carrying torsion, if
any of the four is torsion-coupled to L and any of the four is Codazzi-coupled to g,
then all four must be torsion-coupled to L and Codazzi-coupled to g ([7], Theorem
12). This leads to the definition of a Codazzi-(para-)Hermitian structure–which is
a (para-)Hermitian manifold with all four of the aforementioned connections being
(para-)holomorphic and Codazzi-coupled to g. In this case, their torsions satisfy
T = T
∗
, T
L
= T
† , but dω = 0 in general, unless the torsions of the quadruple of
the connections are all zero.
The torsion-coupling condition Eq. (3.2) on an almost (para-)complex manifold
(M, L), as first investigated in [7], is interesting because the same equation (called
the 2nd Matching Condition or MC2 below) encodes two things simulataneously,
both a constraint on L, i.e., L should be integrable, and a constraint on ∇, i.e., ∇
should be (para-)holomorphic. In this paper, we investigate this torsion-coupling
condition further—by decomposing this condition into two parts, one which implies
integrability of L, and another which implies (para-)holomorphicity of ∇. We investigate another coupling of L with ∇ (called the 1st Matching Condition or MC1
below) that would lead to integrability condition of L and certain properties of ∇
in reference to the Gauduchon line [8]. Our goal in this paper is to understand how
torsion of a connection interacts with a (para-)complex structure L on a manifold
when the connection is (in general) not required to be parallel with respect to L.
3.2 Torsion of ∇ and Integrability of L
3.2.1 L Conjugation of ∇
Given a real manifold M of even dimensions dim M = 2n, we study affine connection ∇ (in general carrying torsion and curvature) on M, i.e., ∇ operates on sections
(T M) of its tangent bundle T M, i.e., vector fields with real dimension 2n. Let
L : (T M) → (T M) be an a map induced by tangent bundle endomorphism,
with L
−1 denoting the inverse map. Then, the L-gauge transformed connection is
defined as ∇
L
:= L
−1
◦ ∇ ◦ L, or