(Para-)Holomorphic Connections for Information Geometry

07/11/2017
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On a statistical manifold (M, g, ∇), the Riemannian metric g is coupled to an (torsion-free) affine connection ∇, such that ∇g is totally symmetric; {∇g} is said to form “Codazzi coupling”. This leads ∇*, the g-conjugate of ∇, to have same torsion as that of ∇. In this paper, we investigate how statistical structure interacts with L in an almost Hermitian and almost para-Hermitian manifold (M,g,L) where L denotes, respectively, an almost complex structure J with J2 = - id or an almost para-complex structure K with K2 = - id. Starting with ∇L, the L-conjugate of ∇, we investigate the interaction of (generally
torsion-admitting) ∇ with L, and derive a necessary and sufficient condition (called “Torsion Balancing” condition) for L to be integrable, hence making (M,g,L)  (para-)Hermitian, and for ∇to be (para-)holomorphic.
We further derive that ∇L is (para-)holomorphic if and only if ∇is, and that ∇* is (para-)holomorphic if and only if ∇ is (para-)holomorphic and Codazzi coupled to g. Our investigations provide concise conditions to extend statistical manifolds to (para-)Hermitian manifolds.

(Para-)Holomorphic Connections for Information Geometry

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application/pdf (Para-)Holomorphic Connections for Information Geometry Sergey Grigorian, Jun Zhang
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On a statistical manifold (M, g, ∇), the Riemannian metric g is coupled to an (torsion-free) affine connection ∇, such that ∇g is totally symmetric; {∇g} is said to form “Codazzi coupling”. This leads ∇*, the g-conjugate of ∇, to have same torsion as that of ∇. In this paper, we investigate how statistical structure interacts with L in an almost Hermitian and almost para-Hermitian manifold (M,g,L) where L denotes, respectively, an almost complex structure J with J2 = - id or an almost para-complex structure K with K2 = - id. Starting with ∇L, the L-conjugate of ∇, we investigate the interaction of (generally
torsion-admitting) ∇ with L, and derive a necessary and sufficient condition (called “Torsion Balancing” condition) for L to be integrable, hence making (M,g,L)  (para-)Hermitian, and for ∇to be (para-)holomorphic.
We further derive that ∇L is (para-)holomorphic if and only if ∇is, and that ∇* is (para-)holomorphic if and only if ∇ is (para-)holomorphic and Codazzi coupled to g. Our investigations provide concise conditions to extend statistical manifolds to (para-)Hermitian manifolds.
(Para-)Holomorphic Connections for Information Geometry

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On a statistical manifold (M, g, ∇), the Riemannian metric g is coupled to an (torsion-free) affine connection ∇, such that ∇g is totally symmetric; {∇g} is said to form “Codazzi coupling”. This leads ∇*, the g-conjugate of ∇, to have same torsion as that of ∇. In this paper, we investigate how statistical structure interacts with L in an almost Hermitian and almost para-Hermitian manifold (M,g,L) where L denotes, respectively, an almost complex structure J with J2 = - id or an almost para-complex structure K with K2 = - id. Starting with ∇L, the L-conjugate of ∇, we investigate the interaction of (generally

torsion-admitting) ∇ with L, and derive a necessary and sufficient condition (called “Torsion Balancing” condition) for L to be integrable, hence making (M,g,L)  (para-)Hermitian, and for ∇to be (para-)holomorphic.

We further derive that ∇L is (para-)holomorphic if and only if ∇is, and that ∇* is (para-)holomorphic if and only if ∇ is (para-)holomorphic and Codazzi coupled to g. Our investigations provide concise conditions to extend statistical manifolds to (para-)Hermitian manifolds.

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(Para-)Holomorphic Connections for Information Geometry Sergey Grigorian1 and Jun Zhang2 1 University of Texas Rio Grande Valley, Edinburg, TX 78539, U.S.A., sergey.grigorian@utrgv.edu, 2 University of Michigan, Ann Arbor, MI 48109, U.S.A., junz@umich.edu, Abstract. On a statistical manifold pM, g, ∇q, the Riemannian metric g is coupled to an (torsion-free) affine connection ∇, such that ∇g is totally symmetric; t∇, gu is said to form “Codazzi coupling”. This leads ∇˚ , the g-conjugate of ∇, to have same torsion as that of ∇. In this paper, we investigate how statistical structure interacts with L in an almost Hermitian and almost para-Hermitian manifold pM, g, Lq, where L denotes, respectively, an almost complex structure J with J2 “ ´ id or an almost para-complex structure K with K2 “ id. Starting with ∇L , the L-conjugate of ∇, we investigate the interaction of (generally torsion-admitting) ∇ with L, and derive a necessary and sufficient condi- tion (called “Torsion Balancing” condition) for L to be integrable, hence making pM, g, Lq (para-)Hermitian, and for ∇ to be (para-)holomorphic. We further derive that ∇L is (para-)holomorphic if and only if ∇ is, and that ∇˚ is (para-)holomorphic if and only if ∇ is (para-)holomorphic and Codazzi coupled to g. Our investigations provide concise conditions to extend statistical manifolds to (para-)Hermitian manifolds. 1 Introduction On the tangent bundle TM of a differentiable manifold M, one can introduce two separate structures: affine connection ∇ and pseudo-Riemannian metric g. A manifold M equipped with a g and a torsion-free connection ∇ is called a statisti- cal manifold if pg, ∇q is Codazzi-coupled [Lau87]. This is the setting of “classical” information geometry, where the pg, ∇q pair arises from a general construction of divergence (“contrast”) functions. To accomodate for torsions in affine con- nections, the concept of pre-contrast functions was introduced [HM11]. Codazzi coupling has been traditionally studied by affine geometers [NS94,Sim00]. The robustness of Codazzi coupling was investigated by perturbing both the metric and the affine connection [SSS09] and by its interaction with other transforma- tions of connection [TZ16]. Below, we provide a succinct overview. 1.1 g-conjugate connection, cubic form, and Codazzi coupling Given the pair pg, ∇q, we construct the (0,3)-tensor C by CpX, Y, Zq :“ p∇ZgqpX, Y q “ ZgpX, Y q ´ gp∇ZX, Y q ´ gpX, ∇ZY q. (1) The tensor C is sometimes referred to as the cubic form associated to the pair p∇, gq. When C “ 0, we say g is parallel under ∇. Given the pair pg, ∇q, we can also construct ∇˚ , called g-conjugate connec- tion, by ZgpX, Y q “ gp∇ZX, Y q ` gpX, ∇˚ ZY q. (2) It can be checked easily that (i) ∇˚ is indeed a connection and (ii) g-conjugation of a connection is involutive, i.e., p∇˚ q˚ “ ∇. These two constructions from an arbitrary pg, ∇q pair are related via CpX, Y, Zq “ gpX, p∇˚ ´ ∇qZY q, (3) so that C˚ pX, Y, Zq :“ p∇˚ ZgqpX, Y q “ ´CpX, Y, Zq. Therefore CpX, Y, Zq “ C˚ pX, Y, Zq “ 0 if and only if ∇˚ “ ∇, that is, ∇ is g-self-conjugate. A connection is both g-self-conjugate and torsion-free defines what is called the Levi-Civita connection ∇LC associated to g. Simple calculation reveals that CpX, Y, Zq ´ CpZ, Y, Xq “ p∇ZgqpX, Y q ´ p∇XgqpZ, Y q, CpX, Y, Zq ´ CpX, Z, Y q “ gpX, T∇˚ pZ, Y q ´ T∇ pZ, Y qq, (4) where T∇ denotes the torsion of ∇ T∇ pX, Y q “ ∇XY ´ ∇Y X ´ rX, Y s. Note that CpX, Y, Zq “ CpY, X, Zq always holds, due to gpX, Y q “ gpY, Xq. Therefore, imposing either of the following is equivalent: 1. CpX, Y, Zq “ CpZ, Y, Xq, 2. CpX, Y, Zq “ CpX, Z, Y q; this is because either (i) or (ii) will make C totally symmetric in all of its indices. In the case of (i), we say that g and ∇ are Codazzi-coupled: p∇ZgqpX, Y q “ p∇XgqpZ, Y q. (5) In the case of (ii), ∇ and ∇˚ have same torsion. These well-known facts are summarized in the following Lemma. Lemma 1. Let g be a pseudo-Riemannian metric, ∇ an arbitrary affine connec- tion, and ∇˚ be the g-conjugate connection of ∇. Then the following statements are equivalent: 1. p∇, gq is Codazzi-coupled; 2. p∇˚ , gq is Codazzi-coupled; 3. C is totally symmetric; 4. C˚ is totally symmetric; 5. T∇ “ T∇˚ . In the above case, pg, ∇, ∇˚ q is called a Codazzi triple. Codazzi-coupling between g and ∇ or, equivalently, the existence of Codazzi triple pg, ∇, ∇˚ q is the key feature of a statistical manifold. In “quantum” information geometry, ∇ is allowed to carry torsion, and [Mat13] introduced Statistical Manifold Admitting Torsion (SMAT) as a manifold pM, g, ∇q satisfying p∇Y gqpX, Zq ´ p∇XgqpY, Zq “ gpT∇ pX, Y q, Zq. Note that ∇˚ is torsion-free if and only if pM, g, ∇q is a SMAT. However, in a SMAT, neither ∇ nor ∇˚ is Codazzi coupled to g; the deviation from Codazzi coupling is measured by the torsion T∇ of ∇. 2 Structure of T M Arising From L A tangent bundle isomorphism L may induce a splitting of TM, corresponding to the eigenbundles associated with the eigenvalues of L. How the action of an arbitrary connection ∇ respects such splitting is the focus of our current paper. 2.1 Splitting of T M by L For a smooth manifold M, an isomorphism L of the tangent bundle TM is a smooth section of the bundle EndpTMq such that it is invertible everywhere. By definition, L is called an almost complex structure if L2 “ ´ id, or an almost para-complex structure if L2 “ id and the multiplicities of the eigenvalues ˘1 are equal. We will use 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, respectively. Following the standard procedure, we (para-)complexify TM by tensoring with C or para-complex (also known as split-complex) field D, and use TL M to denote the resulting TM b C or TM b D, depending on the type of L. In analogy with standard notation in the complex case, let Tp1,0q M and Tp0,1q M be the eigenbundles of L corresponding to the eigenvalues ˘α, i.e., at each point p P M, the fiber is defined by Tp1,0q ppq :“ tX P TL p M : LppXq “ αXu , Tp0,1q ppq :“ tX P TL p M : LppXq “ ´αXu . As sub-bundles of the (para-)complexified tangent bundle TL M, Tp1,0q M and Tp0,1q M are distributions. A distribution is called a foliation if it is closed under the bracket r¨, ¨s . We will refer to vectors to be of type p1, 0q and p0, 1q if they take values in Tp1,0q M and Tp0,1q M respectively. Moreover, define πp1,0q and πp0,1q to be the projections of a vector field to Tp1,0q M and Tp0,1q M respectively. The Nijenhuis tensor NL associated with L is defined as NLpX, Y q “ ´L2 rX, Y s ` LrX, LY s ` LrLX, Y s ´ rLX, LY s. (6) When NL “ 0, the operator L is said to be integrable. It is well-known that both Tp1,0q M and Tp0,1q M are foliations if and only if L is integrable, i.e., the integrability condition NL “ 0 is satisfied. 2.2 L-conjugate of ∇ Starting from a (not necessarily torsion-free) connection ∇ operating on sections of TM, we can apply an L-conjugate transformation to obtain a new connection ∇L :“ L´1 ∇L, or ∇L XY “ L´1 p∇XpLY qq (7) for any vector fields X and Y ; here L´1 denotes the inverse isomorphism of L. It can be verified that indeed ∇L is an affine connection. Define a (1,2)-tensor (vector-valued bilinear form) S via the expression SpX, Y q “ p∇XLqY ´ p∇Y LqX, (8) where p∇XLqY “ ∇XpLY q ´ Lp∇XY q. We say that L and ∇ are Codazzi-coupled if S “ 0. The following is known. Lemma 2. (e.g., [SSS09]) Let ∇ be an affine connection, and let L be an arbi- trary tangent bundle isomorphism. Then the following statements are equivalent: (i) p∇, Lq is Codazzi-coupled. (ii) T∇ pX, Y q “ T∇L pX, Y q. (iii) p∇L , L´1 q is Codazzi-coupled. Lemma 3. For the special case of (para-)complex operators L2 “ ˘ id, 1. ∇L “ ∇L´1 , i.e., L-conjugate transformation is involutive, p∇L qL “ ∇. 2. p∇, Lq is Codazzi-coupled if and only if p∇L , Lq is Codazzi-coupled. As an affine connection, ∇ gives rise to a map ∇ : Ω0 pTMq Ñ Ω1 pTMq, where Ωi pTMq is the space of smooth i-forms with value in TM. We may extend this to a map d∇ : Ωi pTMq Ñ Ωi`1 pTMq by d∇ pα b vq “ dα ˆ v ` p´1qi α ^ ∇v for any i-form α and vector field v. In the case that ∇ is flat, then pd∇ q2 “ 0 and we get a chain complex whose cohomology is the de Rham cohomology twisted by the local system determined by ∇. Regarding L as an element of Ω1 pTMq, it is easy to check using local coordinates that pd∇ LqpX, Y q “ p∇XLqY ´ p∇Y LqX ` LT∇ pX, Y q. (9) Therefore, Codazzi coupling of ∇ and L can also be expressed as pd∇ LqpX, Y q “ T∇ pLX, Y q. (10) 2.3 Integrability of L In [FZ17, Lemma 2.5] an expression for NL pX, Y q in terms of T∇ has been derived assuming S “ 0. Using exactly the same procedure, we can write down NL pX, Y q for an arbitrary S. Lemma 4. Given a connection ∇ with torsion T∇ , the Nijenhuis tensor NL of a (para-)complex operator L is given by NL pX, Y q “ L2 T∇ pX, Y q ´ LT∇ pX, LY q ´ LT∇ pLX, Y q ` T∇ pLX, LY q `LS pX, Y q ´ L´1 S pLY, LXq . Now, define θ to be θpX, Y q “ 1 2 p∇L XY ´ ∇XY q “ 1 2 L´1 p∇XLqY. (11) with Lθ pX, Y q ` θ pX, LY q “ 0. (12) In particular, we see that 1 2 L´1 pS pX, Y qq “ θ pX, Y q ´ θ pY, Xq , and therefore, θ is symmetric if and only if L and ∇ are Codazzi-coupled. Intro- duce ˜ ∇ “ 1 2 p∇ ` ∇L q, which satisfies ˜ ∇L ” 0. A connection with respect to which L is parallel is called (para-)complex connec- tion, and in particular, such a connection preserves the decomposition TL M – Tp1,0q M ‘ Tp0,1q M. So starting from any connection ∇, we can construct its conjugate ∇L , the average of which is the (para-)complex connection ˜ ∇. This situation mirrors the relationship between Levi-Civita connection and the pair of g-conjugate connections ∇, ∇˚ . Note that we can also write ∇ “ ˜ ∇ ´ θ and ∇L “ ˜ ∇ ` θ, so the quantity θ measures the failure of both ∇ and ∇L to be a (para-)complex connection. 3 (Para-)Holomorphicity of ∇ Associated to L 3.1 (Para-)holomorphic connections The (para-)Dolbeault operator B̄ for a given L on TL M is defined as B̄XY “ 1 4 ` rX, Y s ´ L2 rLX, LY s ´ L´1 rLX, Y s ` L´1 rX, LY s ˘ (13) for any vector fields X and Y . It can be checked easily that this expression is tensorial in X, that is B̄fXY “ f ` B̄XY ˘ and is a derivation. In the case when L “ J, this defines the holomorphic structure on TC M and locally defines the differentiation of vector fields of type p1, 0q with respect to the anti-holomorphic coordinates B Bz̄i . Similarly for para-holomorphic structure on TD M when L “ K. From (13) we obtain that if X and Y are of the same type, then B̄XY “ 0. However, if Y P Tp1,0q M and X P Tp0,1q M, then B̄XY “ πp1,0q rX, Y s (14) and similarly B̄XY “ πp0,1q rX, Y s if Y P Tp0,1q M and X P Tp1,0q M. Equivalently, note that if X P Tp1,0q M, then B̄X is a vector-valued 1-form, of type p1, 0q as a vector and type p0, 1q as a 1-form, and conversely if X P Tp0,1q M. Given a connection ∇ operating on TL M, we can ask the question whether ∇ is compatible with B̄. To understand this we may define an alternative operator B̄∇ , which for Y P Tp1,0q M is defined as taking the p0, 1q-part of the vector-valued 1-form ∇Y (and conversely on Tp0,1q M). This can be expressed as B̄∇ XY “ 1 2 ` ∇XY ´ ∇LX ` L´1 Y ˘˘ (15) for any vector fields X and Y in TL M. Clearly, B̄∇ XY “ 0 if X and Y are of the same type and is just ∇XY if X and Y are of opposite type. On a (para-)holomorphic vector bundle, a connection is said to be (para)-holomorphic if these two Dolbeault operators coincide. We extend this notion to arbitrary connections on TL M – Tp1,0q M ‘ Tp0,1q M (that do not necessarily preserve Tp1,0q M and Tp0,1q M) – we say a connection ∇ is (para-)holomorphic if B̄∇ XY “ B̄XY for any vector fields X and Y . It can be readily shown that Theorem 1. ∇L is (para-)holomorphic if and only if ∇ is (para-)holomorphic. Theorem 2. When ∇ is (para-)holomorphic, the quantity θ pX, Y q satisfies: Lθ pX, Y q “ ´θ pX, LY q “ ´θ pLX, Y q “ L´1 θ pLX, LY q . (16) Theorem 2 shows that θ pX, Y q vanishes whenever X and Y are of different types. Moreover, if X and Y are both of type p1, 0q, θ pX, Y q is of type p0, 1q, and vice versa. Using (13) and (15), we can also prove Lemma 5. Given an arbitrary connection ∇ and an L on a manifold, the con- nection ∇ is (para-)holomorphic if and only if SpX, Y q “ T∇ pLX, Y q ´ LT∇ pX, Y q ´ 1 2 L2 NLpLX, Y q. (17) From this, we prove the main theorem of our paper. Theorem 3. Given the an arbitrary pair p∇, Lq on a manifold, the connection ∇ is (para-)holomorphic and L is integrable if and only if SpX, Y q “ T∇ pLX, Y q ´ LT∇ pX, Y q. (18) The significance of Theorem 3 is that this gives us a generalization of the Codazzi coupling condition for L that was used in [FZ17] in the case T∇ “ 0. In fact, it follows immediately that if T∇ “ 0 then Codazzi coupling of ∇ with L makes L integrable and makes ∇ (para-)holomorphic. The condition (18) can be recast in another form to reveal its meaning: Theorem 4. Given ∇ and L on a manifold, then ∇ is (para-)holomorphic and L is integrable if and only if T∇ pLX, Y q “ LpT∇L pX, Y qq. (19) Theorem 4 shows that the (para-)holomorphicity condition on ∇ can be thought of as requiring “Torsion-Balancing” between ∇ and ∇L . 3.2 Almost (para-)Hermitian structure The compatibility condition between g and an almost (para-)complex structure J(K) is well-known. We say that g is compatible with J if J is orthogonal, i.e. gpJX, JY q “ gpX, Y q (20) holds for any vector fields X and Y . Similarly we say that g is compatible with K if gpKX, KY q “ ´gpX, Y q (21) is always satisfied, which implies that g must be of split signature. When ex- pressed using L, (20) and (21) have the same form gpX, LY q ` gpLX, Y q “ 0. (22) When specified in terms of compatible g and L, the manifold pM, g, Lq is said to be almost (para-)Hermitian, and (para-)Hermitian manifold if L is integrable. For any almost (para)-Hermitian manifold, we can define the 2-form ωpX, Y q “ gpLX, Y q, called the fundamental form, which turns out to satisfy ωpX, LY q ` ωpLX, Y q “ 0. The three structures, a pseudo-Riemannian metric g, a nonde- generate 2-form ω, and a tangent bundle isomorphism L : TM Ñ TM forms a “compatible triple” such that given any two, the third one is uniquely specified; the triple is rigidly “interlocked”. It can be shown that for almost (para-)Hermitian manifolds, p∇L XgqpLY, Zq ` p∇XgqpY, LZq “ 0. (23) 3.3 (Para-)holomorphicity of ∇˚ We have seen in Theorem 1 that ∇ is (para-)holomorphic if and only if ∇L is also (para-)holomorphic. We now investigate conditions under which ∇˚ is also (para-)holomorphic whenever ∇ is. Lemma 6. Given arbitrary g and L on a manifold, with a (para-)holomorphic connection ∇. Then ∇˚ is also (para-)holomorphic if and only if C pLX, Y, Zq “ C pX, Y, LZq (24) for any vector fields X, Y, Z. If moreover, g and L are compatible, i.e., (22) holds, then (24) is equivalent to C pX, Y, Zq “ g pθ pZ, Xq , Y q ` g pX, θ pZ, Y qq . (25) The condition that ∇˚ is (para-)holomorphic is a very strong one as the Theorem below shows. Theorem 5. Let ∇ be a (para-)holomorphic connection ∇ on an almost (para- )Hermitian manifold pM, g, Lq. Then, the connection ˜ ∇ “ 1 2 ` ∇ ` ∇L ˘ is metric- compatible if and only if ∇˚ is also (para-)holomorphic. In fact, since we already know that ˜ ∇ is a (para-)complex connection, i.e. it preserves L, the condition of ∇˚ being (para-)holomorphic is then equiva- lent to ˜ ∇ being an almost (para-)Hermitian connection. Moreover, if we assume L to be integrable, since ˜ ∇ is also (para-)holomorphic, we can conclude that when restricted to bundle Tp1,0q M, it must be equal to the (para-)Chern con- nection. In the theory of holomorphic vector bundles, Chern connection is the unique Hermitian holomorphic connection on a holomorphic vector bundle, and in particular on Tp1,0q M on complex manifolds [Mor07]. In general, the Chern connection has torsion, however it is torsion-free on Tp1,0q M if and only if pg, Jq define a Kähler structure. It is significant that if g is Codazzi-coupled to a (para-)holomorphic connec- tion ∇, then ∇˚ is (para-)holomorphic, and hence ˜ ∇ is (para-)Hermitian. Theorem 6. Let pM, g, Lq be a (para-)Hermitian manifold and let p∇, ∇˚ , gq be a Codazzi triple. Then p∇˚ , gq is (para-)holomorphic if and only if p∇, gq is (para-)holomorphic. This generalizes the results on a Codazzi-(para-)Kähler manifold [FZ17] which admit a pair of torsion-free connections to a (para-)Hermitan manifold which admits holomorphic connections with torsion. The Torsion-Balancing condition, while breaking the requirements of (para-)Kähler structure by possibly violating dω “ 0, still preserves the integrability of L. 4 Summary and Discussions (Para-)holomorphic connections have hardly been systematically studied in in- formation geometry except in restricted setting of flat connections (see [Fur09]). Connections investigated in this paper are neither curvature-free nor torsion-free. We gave a necessary and sufficient condition(“Torsion Balance”) of a ∇ to be (para-)holomorhic in the presence of a (para-)complex structure L on the man- ifold. Given a (para-)holomorphic connection ∇, we then showed that (i) ∇L , its L-conjugate, is also (para-)holomorphic; (ii) ∇˚ , its g-conjugate, is (para- )holomorphic if and only if g and ∇ are Codazzi coupled. These concise char- acterizations allow us to enhance a statistical structure to a (para-)Hermitian structure, as well as understand the properties of L-conjugaty and g-conjugacy of a connection of a (para-)Hermitian manifold. Acknowledgement This research is supported by DARPA/ARO Grant W911NF- 16-1-0383 to the University of Michigan (PI: Jun Zhang). References [Fur09] H. Furuhata. Hypersurfaces in statistical manifolds. Differential Geometry and Its Applications, 27(3):420–429, 2009. [HM11] M. Henmi and H. Matsuzoe. Geometry of Preâecontrast Functions and Nonâeconservative Estimating Functions. International Workshop on Complex Structures, Integrability and Vector Fields. Vol. 1340. No. 1. AIP Publishing, 2011. [Lau87] S.L. Lauritzen. Statistical manifolds. In Differential Geometry in Statistical Inference, volume 10 of IMS Lecture Notes Monograph Series, pages 163–216. Institute of Mathematical Statistics, 1987. [Mat13] H. Matsuzoe. Statistical manifolds and geometry of estimating functions, Recent Progress in Differential Geometry and Its Related Field, World Sci. Publ., (2013), 187-202. [Mor07] A. Moroianu. Lectures on Kähler Geometry, volume 69 of London Mathemat- ical Society Student Texts. Cambridge University Press, 2007. [NS94] K. Nomizu and T. Sasaki. Affine Differential Geometry: Geometry of Affine Immersions, volume 111 of Cambridge Tracts in Mathematics. Cambridge Uni- versity Press, 1994. [Sim00] U. Simon. Affine differential geometry. In Handbook of Differential Geometry, volume 1, pages 905–961. North-Holland, 2000. [SSS09] A. Schwenk-Schellschmidt and U. Simon. Codazzi-equivalent affine connec- tions. Results in Mathematics, 56(1-4):211–229, 2009. [TZ16] J. Tao and J. Zhang. Transformations and coupling relations for affine connec- tions. Differential Geometry and Its Applications, 49:111–130, 2016 [FZ17] T. Fei and J. Zhang. Interaction of Codazzi couplings with (para-)Kahler geometry. Results in Mathematics, 2017.