38
J. Zhang and G. Khan
for any vector fields X, Y on M, that is, X, Y ∈ (T M), leads to an almost (para-)
Hermitian structure of (M, g, L). Here, g is positive-definite and hence a Riemannian metric when L = J , and is of split signature and hence a Norden metric when
L = K . When L is integrable, (M, g, L) becomes a (para-)Hermitian manifold.
Compatibility of g and ∇ has been investigated in the context of affine differential geometry and information geometry, through the notion of “Codazzi-coupling”
(definition to follow). A manifold M equipped with a metric g and a torsion-free
connection ∇ is called a statistical manifold if (g, ∇) is Codazzi-coupled [1]. Finally,
compatibility of L with (a torsion-free) ∇ has also been studied, in terms of Codazzicoupling between them (e.g. [2, 3]). In [4], it was further shown that when a torsionfree connection ∇ is Codazzi-coupled both to g and to L, then (M, g, L) in fact turns
into a (para-)Kähler manifold. Such a (para-)Kähler manifold is called Codazzi(para-)Kähler manifold [4] because of the additional structure imposed by ∇. A
special case is the class of “special Kähler manifold” of [2], with ∇ assumed to be
curvature-free in addition to being torsion-free.
Given (g, L) on M, one can define g-conjugate transformation (denoted ∇
∗ ) and
L-gauge transformation (denoted ∇
L ), respectively, given any connection ∇ on M.
These transformations are involutive (∇
∗
)
∗
= ∇ = (∇
L
)
L . Details were in [4] and
will be given below. When Eq. (3.1) is satisfied, then these two transformations of ∇
are commutative (∇
∗
)
L
= (∇
L
)
∗
≡ ∇
† , so that {id, ∗, L , †} form a 4-element Klein
group of transformation of affine connections on M (Theorem 2.13 of [4]). Under
this scenario, ∇
† then becomes the conjugate connection of ∇ with respect to the
fundamental form ω defined by ω(X, Y ) = g(L X, Y ); here (g, L , ω) is known as
the “compatible triple.”
How this quadruple of connections (∇, ∇
∗
, ∇
L
, ∇
†
) interact with g and L on an
almost (para-)Hermitian manifold (M, g, L) deserves further investigation. Since
Codazzi-coupling of ∇ with either g or L ensures that ∇
∗ or ∇
L will have same
torsion as that of ∇, enforcing Codazzi couplings of a torsion-free ∇ both with g
and with L will lead to vanishing torsion for the entire quadruple of connections
(∇, ∇
∗
, ∇
L
, ∇
†
). In this case (see [4]) ∇
†
= ∇, so Codazzi-(para-)Kähler manifolds
admit pairs of torsion-free connections. These manifolds are both (para-)Kähler
manifolds (with integrable L and d-closed ω) and statistical manifolds (with a family
of α-connections [5]). Manifolds where ∇
†
= ∇, or equivalently ∇ω = 0, are called
holomorphic statistical manifold [6] (a notion originally due to Takashi Kurose) when
torsion of ∇ is zero; in this case, ∇ is a symplectic connection. So Codazzi-(para)Kähler manifolds are important examples of holomorphic statistical manifolds.
In a recent paper, Grigorian and Zhang [7] relaxed the restriction of torsionfreeness of ∇, and investigated integrable structures on an almost (para-)Hermitian
manifold (M, g, L) that nevertheless admits affine connections with torsion. Connections with torsion on Hermitian manifolds have always been a topic of interest,
e.g. [8, 9]. The work of [4, 7] investigated connections that are not necessarily parallel to L, ∇ L = 0. That is, neither ∇ nor ∇
L is (para-)complex, though
1
2
(∇ + ∇
L
)
is always parallel to L:
1
2
(∇ + ∇
L
)L = 0.
J. Zhang and G. Khan
for any vector fields X, Y on M, that is, X, Y ∈ (T M), leads to an almost (para-)
Hermitian structure of (M, g, L). Here, g is positive-definite and hence a Riemannian metric when L = J , and is of split signature and hence a Norden metric when
L = K . When L is integrable, (M, g, L) becomes a (para-)Hermitian manifold.
Compatibility of g and ∇ has been investigated in the context of affine differential geometry and information geometry, through the notion of “Codazzi-coupling”
(definition to follow). A manifold M equipped with a metric g and a torsion-free
connection ∇ is called a statistical manifold if (g, ∇) is Codazzi-coupled [1]. Finally,
compatibility of L with (a torsion-free) ∇ has also been studied, in terms of Codazzicoupling between them (e.g. [2, 3]). In [4], it was further shown that when a torsionfree connection ∇ is Codazzi-coupled both to g and to L, then (M, g, L) in fact turns
into a (para-)Kähler manifold. Such a (para-)Kähler manifold is called Codazzi(para-)Kähler manifold [4] because of the additional structure imposed by ∇. A
special case is the class of “special Kähler manifold” of [2], with ∇ assumed to be
curvature-free in addition to being torsion-free.
Given (g, L) on M, one can define g-conjugate transformation (denoted ∇
∗ ) and
L-gauge transformation (denoted ∇
L ), respectively, given any connection ∇ on M.
These transformations are involutive (∇
∗
)
∗
= ∇ = (∇
L
)
L . Details were in [4] and
will be given below. When Eq. (3.1) is satisfied, then these two transformations of ∇
are commutative (∇
∗
)
L
= (∇
L
)
∗
≡ ∇
† , so that {id, ∗, L , †} form a 4-element Klein
group of transformation of affine connections on M (Theorem 2.13 of [4]). Under
this scenario, ∇
† then becomes the conjugate connection of ∇ with respect to the
fundamental form ω defined by ω(X, Y ) = g(L X, Y ); here (g, L , ω) is known as
the “compatible triple.”
How this quadruple of connections (∇, ∇
∗
, ∇
L
, ∇
†
) interact with g and L on an
almost (para-)Hermitian manifold (M, g, L) deserves further investigation. Since
Codazzi-coupling of ∇ with either g or L ensures that ∇
∗ or ∇
L will have same
torsion as that of ∇, enforcing Codazzi couplings of a torsion-free ∇ both with g
and with L will lead to vanishing torsion for the entire quadruple of connections
(∇, ∇
∗
, ∇
L
, ∇
†
). In this case (see [4]) ∇
†
= ∇, so Codazzi-(para-)Kähler manifolds
admit pairs of torsion-free connections. These manifolds are both (para-)Kähler
manifolds (with integrable L and d-closed ω) and statistical manifolds (with a family
of α-connections [5]). Manifolds where ∇
†
= ∇, or equivalently ∇ω = 0, are called
holomorphic statistical manifold [6] (a notion originally due to Takashi Kurose) when
torsion of ∇ is zero; in this case, ∇ is a symplectic connection. So Codazzi-(para)Kähler manifolds are important examples of holomorphic statistical manifolds.
In a recent paper, Grigorian and Zhang [7] relaxed the restriction of torsionfreeness of ∇, and investigated integrable structures on an almost (para-)Hermitian
manifold (M, g, L) that nevertheless admits affine connections with torsion. Connections with torsion on Hermitian manifolds have always been a topic of interest,
e.g. [8, 9]. The work of [4, 7] investigated connections that are not necessarily parallel to L, ∇ L = 0. That is, neither ∇ nor ∇
L is (para-)complex, though
1
2
(∇ + ∇
L
)
is always parallel to L:
1
2
(∇ + ∇
L
)L = 0.