Statistical Manifolds with almost Quaternionic Structures and Quaternionic Kähler-like Statistical Submersions

In this paper we investigate statistical manifolds with almost quaternionic structures. We define the concept of quaternionic Kähler-like statistical manifold and derive the main properties of quaternionic Kähler-like statistical submersions, extending in a new setting some previous results obtained by K. Takano concerning statistical manifolds endowed with almost complex and almost contact structures. Finally, we give a nontrivial example and propose some open problems in the field for further research.


Introduction
It is well known that the concept of statistical manifold arises naturally from divergencies-like Kullback-Leibler relative entropy-in statistics, information theory and related fields [1,2].On the other hand, the notion of statistical submersion between statistical manifolds was introduced in 2001 by N. Abe and K. Hasegawa [3], the authors generalizing some basic results of B. O'Neill [4,5] concerning Riemannian submersions and geodesics.Later, K. Takano defined the concepts of Kähler-like statistical manifold and Kähler-like statistical submersion [6], Sasaki-like statistical manifold and Sasaki-like statistical submersion [7], and obtained several geometric properties.Particularly relevant examples of statistical manifolds are the exponential families, whose points are probability densities of exponential form depending on a finite number of parameters.For some important exponential families, like the multinomial distribution, the multivariate normal distribution, and the Dirichlet and von Mises-Fisher distributions, it is proved in [8] that they admit almost complex structures.Also, in [9] H. Matsuzoe and J. Inoguchi investigate the extensions of statistical structures on manifolds to their tangent bundles, proving that the tangent bundle of a flat statistical manifold has a natural almost complex statistical structure with Norden metric.Moreover, in [10] the author considers the statistical model of the multivariate normal distribution as the Riemannian manifold and constructs an interesting example of statistical submersion.
We remark that a complex version of the notion of statistical structure was also considered in [11], where the author derived a condition for the curvature of a statistical manifold to admit a kind of standard hypersurface.On the other hand, the existence of symplectic structures on statistical manifolds was investigated in [12], where the author obtained a duality relation between the Fubini-Study metric on a projective space and the Fisher metric on a statistical model on a finite set.Other interesting results concerning the geometry of statistical manifolds were recently obtained in [13][14][15][16][17][18][19][20][21].In this paper, we investigate very natural kind of statistical manifold, namely those endowed with almost quaternionic structures, extending the results of K. Takano in a new setting and obtaining new curvature properties of statistical submersions.In particular, we generalize some previous results of S. Ianuş et al. [22] concerning Riemannian submersions between quaternionic manifolds.Recall that an almost quaternionic structure on a smooth manifold M is a 3-dimensional subbundle of End(T M ) which is locally spanned by an almost hypercomplex structure, i.e., three almost complex structures satisfying the quaternionic identities [23].We also note that the quaternionic structures generalize many relevant properties of 4-dimensional semi-Riemannian manifolds to higher 4n-dimensional manifolds, some of them being relevant for mathematical physics, with important applications in string theory, solitons, theory of liquid crystals, gravity and general relativity (see [24,25] and references therein).
The present work is organized as follows.Section 2 contains definitions and basic properties of statistical manifolds and statistical submersions.In Section 3 we investigate statistical manifolds with almost quaternionic structures and introduce the concept of quaternionic Kähler-like statistical manifold.Section 4 is devoted to the study of the quaternionic Kähler-like statistical submersions.This paper ends with conclusions and several open problems in the field for further research.

Preliminaries
Let (M, g) be a semi-Riemannian manifold and ∇ a torsion free linear connection on M .Then ∇ is said to be compatible to g if the covariant derivative ∇g is symmetric.Moreover, the pair (∇, g) is called a statistical structure on M and the triple (M, ∇, g) is said to be a statistical manifold.
For a statistical manifold (M, ∇, g), let ∇ * be an affine connection on M such that for all E, F, G ∈ Γ(T M ), where Γ(T M ) denotes the set of smooth tangent vector fields on M .Then it is easy to see that the affine connection ∇ * is torsion free and ∇ * g is symmetric.This connection, ∇ * , is called the dual connection of ∇; the triple (M, ∇ * , g) is said to be the dual statistical manifold of (M, ∇, g); and the triple (∇, ∇ * , g) is called the dualistic structure on M [26].We note that the concept of dual connections, whose name is motivated by the fact that (∇ * ) * = ∇, was originally introduced by S. Amari in his seminal work [1] and later applied in various fields, like statistical physics, neural networks and information theory.
It is also easy to check that the curvature tensor R * of ∇ * vanishes if and only if the curvature tensor R of ∇ does, and then the triple (∇, ∇ * , g) is called the dually flat structure [2].In fact, the two curvature tensors R and R * on M , defined with the sign convention for all E, F, G, H ∈ Γ(T M ).
We remark that the geometry of statistical manifolds simply reduces to the usual semi-Riemannian geometry when ∇ and ∇ * coincide [27].Moreover, we note that on a statistical manifold one can define a parametric family of torsion free connections {∇ α } α∈R , called α-connections, by We remark that ∇ (1) = ∇, ∇ (−1) = ∇ * and ∇ (0) is the Levi-Civita connection of the metric g.This family of α-connections has been investigated in [28], where the author obtains that ∇ α is equiaffine for any real number α, provided that (∇, ∇ * , g) is a dually flat structure, as previously noted in [29].
Let (M, g) and (M , g ) be two connected semi-Riemannian manifolds of index s and s respectively, with 0 ≤ s ≤ dimM , 0 ≤ s ≤ dimM and s ≤ s.A semi-Riemannian submersion is a smooth map π : M → M which is onto and satisfies the following conditions [30]: (ii) The fibers π −1 (p ), p ∈ M , are semi-Riemannian submanifolds of M ; (iii) π * preserves scalar products of vectors normal to fibers.
It is well known that the vectors tangent to fibers are called vertical and those normal to fibers are called horizontal.We denote by V the vertical distribution, by H the horizontal distribution and by v and h the vertical and horizontal projection.An horizontal vector field X on M is said to be basic if X is π-related to a vector field X on M .It is clear that every vector field X on M has a unique horizontal lift X to M and X is basic.Moreover, if X and Y are basic vector fields on M , π-related to X and Y on M , then we have the following properties (see [5,31]): Next we consider (M, ∇, g) a statistical manifold, (M , g ) a semi-Riemannian manifold and let π : M → M be a semi-Riemannian submersion.We denote by ∇ and ∇ * the affine connections induced on fibers by the dual connections ∇ and ∇ * from M .We remark that ∇ and ∇ * are well-defined, namely for all U, V ∈ Γ(V).Moreover, we can easily see that ∇ and ∇ * are torsion free and conjugate to each other with respect to the induced metric on fibers.On the other hand, if we define S := ∇ − ∇ * , then S is symmetric, i.e., S E F = S F E, for all vector fields E, F on M , and we also find [6]: for all X, Y, Z ∈ Γ(H).Similarly, if ∇ and ∇ * are affine connections on M , then we can define S = ∇ − ∇ * and we have that hS X Y is basic and π for all basic vector fields X, Y on M π-related to X and Y on M , and p ∈ M .
If π : M → M is a statistical submersion, then we can define as well as in the semi-Riemannian case [32], two (1,2) tensor fields T and A on M , by the formulas: and similarly: for any E, F ∈ Γ(T M ).
We can also define, in a similar way, the tensor fields T * and A * on M by replacing ∇ by ∇ * in Equations ( 6) and (7).It is easy to check now that T * * = T and A * * = A.Moreover, using the above Definitions one can easily prove the following result.Lemma 1. [3,6] T , A, T * and A * have the following properties: for all X, Y ∈ Γ(H) and U, V ∈ Γ(V).
Therefore, we deduce that T (or A) vanishes identically if and only if T * (or A * ) vanishes identically.Moreover, from (9) we deduce that if A = 0 then H is integrable.We note that if T U V = 0, for all U, V ∈ Γ(V) then π is called a statistical submersion with isometric fibers [6].
We also recall that N. Abe and K. Hasegawa [3] provided necessary and sufficient conditions for the total space of a semi-Riemannian submersion to be a statistical manifold.In particular, we note that if π : M → M is a statistical submersion then any fiber is a statistical manifold (see also [6,7]).

Statistical Manifolds with almost Quaternionic Structures
Let M be a differentiable manifold and assume that there is a rank 3-subbundle σ of End(T M ) such that a local basis {J 1 , J 2 , J 3 } exists on sections of σ satisfying for all α ∈ {1, 2, 3}: where Id denotes the identity tensor field of type (1, 1) on M and the indices are taken from {1, 2, 3} modulo 3. Then the bundle σ is called an almost quaternionic structure on M and {J 1 , J 2 , J 3 } is called a canonical local basis of σ.Moreover, (M, σ) is said to be an almost quaternionic manifold [33].It is easy to see that any almost quaternionic manifold is of dimension 4m, m ≥ 1.
A semi-Riemannian metric g on M is said to be adapted to the almost quaternionic structure σ if it satisfies: for all vector fields E,F on M and any canonical local basis {J 1 , J 2 , J 3 } of σ.Moreover, (M, σ, g) is said to be an almost Hermite quaternionic manifold [33].
Definition 2. Let (M, g) be a semi-Riemannian manifold endowed with an almost quaternionic structure σ which has for any canonical local basis {J 1 , J 2 , J 3 } of σ three other tensor fields for all vector fields E,F on M .Then (M, σ, g) is said to be an almost Hermite-like quaternionic manifold.Moreover, if (M, σ, g) is equipped with a torsion free linear connection ∇ such that ∇g is symmetric, then (M, ∇, σ, g) is said to be an almost Hermite-like quaternionic statistical manifold.
We note that if (20), then (M, ∇, σ, g) is said to be a locally hyper-Kähler-like statistical manifold.Moreover, if J 1 , J 2 , J 3 are globally defined on M , then (M, ∇, J 1 , J 2 , J 3 , g) is said to be a hyper-Kähler-like statistical manifold.
We remark that, if in the above definition ∇ is the Levi-Civita connection of g, then (M, ∇, σ, g), usually denoted by (M, σ, g), is called a quaternionic Kähler manifold [23,33,34].Definition 4. Let (M, ∇, σ, g) be a quaternionic Kähler-like statistical manifold.If the curvature tensor R with respect to ∇ satisfies for all vector fields E, F, G on M , where c is a real constant, then the statistical manifold (M, ∇, σ, g) is said to be of type quaternionic space form.
We remark that changing J α for J * α in (21), we get the curvature tensor R * with respect to the dual connection ∇ * .If (M, σ, g) is a quaternionic Kähler manifold satisfying (21), then M is said to be a space of constant quaternionic sectional curvature, or quaternionic space form.It is known that quaternionic space forms are locally congruent to either a quaternionic projective space HP n (c) of quaternionic sectional curvature c > 0, a quaternionic Euclidean space H n of null quaternionic sectional curvature or a quaternionic hyperbolic space HH n (c) of quaternionic sectional curvature c < 0 [35].
Proof.First of all, it is obvious that the triple (M, σ, g) is an almost Hermite-like quaternionic manifold if and only if (M, σ * , g) is.Now, we take a canonical local basis {J 1 , J 2 , J 3 } of σ.Then, using (1) and (19) we derive for all α ∈ {1, 2, 3}: for all vector fields E, F, G on M , where the indices are taken from {1, 2, 3} modulo 3.
Proof.The assertion is clear from Theorem 1.
Remark 1.We note that the concepts of almost Hermite-like quaternionic manifold and quaternionic Kähler-like statistical manifold proposed in this section generalize the classical notions of almost quaternionic Hermitian manifold and quaternionic Kähler manifold [23,33].In fact, an almost quaternionic Hermitian manifold is a particular case of almost Hermite-like quaternionic manifold with J * α = J α , α ∈ {1, 2, 3}, and hence with σ = σ * .Similarly, any quaternionic Kähler manifold is a particular case of quaternionic Kähler-like statistical manifold, where ∇ = ∇ * is the Levi-Civita connection of the metric g.
Example 1.Let (M, ∇, φ, g) be an almost Hermite-like statistical manifold (see [6,36] for basic definitions and examples).Next we prove that T M can be endowed with an almost Hermite-like quaternionic statistical structure.First of all, we note that the tangent bundle T M can be equipped with the Sasaki metric G defined by for all vector fields A, B on T M , where π is the natural projection of T M onto M and K is the connection map associated with the Levi-Civita connection of the metric g (see [37]).
We note that if X ∈ Γ(T M ), then there exists exactly one vector field on T M , denoted by X h and called the horizontal lift, and denoted X v and called the vertical lift of X, such that we have for all . We recall now that, according to Theorem 3 in [38], one can define a torsion free linear connection ∇ on T M compatible to the Sasaki metric G. Hence (T M, ∇ , G) is a statistical manifold.Moreover, using the almost complex structure φ on M , we can also define three tensor fields J 1 , J 2 , J 3 on T M by the equalities: It is easy to see that J 1 , J 2 , J 3 satisfy the quaternionic identities (17) and, defining σ to be the 3-subbundle of End(T M ) generated by (J α ) α=1,2,3 , we derive immediately that (T M, ∇ , σ, G) is an almost Hermite-like quaternionic statistical manifold.Moreover it can be proved that (T M, ∇ , σ, G) is a hyper-Kähler-like statistical manifold if and only if (M, ∇, φ, g) is a flat Kähler-like statistical manifold.
Remark 2. We can easily check that: i.A map f : M → M between two almost Hermite-like quaternionic manifolds is a (σ, σ )-holomorphic map at a point p ∈ M if and only if for any canonical local basis {J 1 , J 2 , J 3 } of σ p there exists a canonical local basis ii.A semi-Riemannian submersion π : M → M between two almost Hermite-like quaternionic statistical manifolds (M, ∇, σ, g) and (M , ∇ , σ , g ) is a (σ, σ )-holomorphic map if and only if it is a (σ * , σ * )-holomorphic map.
Property 1.Let π : M → M be an almost Hermite-like quaternionic statistical submersion.Then: i. V and H are invariant under each J ∈ σ p and J * ∈ σ p , ∀p ∈ M .Moreover, J and J * commute with the horizontal and vertical projectors.
ii.If X is a basic vector field, then from i. J α X and J * α X are horizontal vector fields.On the other hand, since π is a (σ, σ )-holomorphic map and X is π-related to X on M we derive that Proof.Let F = π −1 (p ) be a fiber of the submersion, where p ∈ M .Then it is known from [3,6,7] that (F, ∇, g = g| F ) is a statistical manifold.Moreover, for any canonical local basis {J 1 , J 2 , J 3 } of σ, we can define and we can consider the subbundle σ locally spanned by { J 1 , J 2 , J 3 }.Now it follows immediately that (F, ∇, σ, g) is an almost Hermite-like quaternionic statistical manifold.
Proof.If we take two basic vector fields X, Y on M π-related to X , Y on M , then using Proposition 1 we derive: Since (M, ∇, σ, g) is a quaternionic Kähler-like statistical manifold, we have (20) and we can define 1-forms ω 1 , ω 2 , ω 3 on M by: for any local vector field X on M and X a basic vector field on M such that π * X = X .Next, making use of ( 20), ( 24) and ( 25), we obtain: where the indices are taken from {1, 2, 3} modulo 3. Therefore (M , ∇ , σ , g ) is a quaternionic Kähler-like statistical manifold.Next, we consider F = π −1 (p ), p ∈ M , a fiber of the submersion.Then, from Theorem 2, it follows that (F, ∇, σ, g) is an almost Hermite-like quaternionic statistical manifold.Using (12) we derive for all U, V ∈ Γ(V): On the other hand, from (20) we have From ( 27) and ( 28) we deduce and Finally, from (29) it follows that (F, ∇, σ, g) is a quaternionic Kähler-like statistical manifold and the proof is now complete.
Proof.Since T has the symmetry property for vertical vector fields (cf.( 8)), using (17) and (30) we derive for all U, V ∈ Γ(V) and α = 1, 2, 3: Therefore, we deduce In particular, from (30) it follows that On the other hand, replacing in (31) U by J 1 U and V by J 1 V , we derive Now, from (32) and (33) we deduce that and finally, from (31) and (34) we conclude that T U V = 0, for all U, V ∈ Γ(V).Assertion ii.follows in a similar way.
Proof.The assertion is an obvious consequence of Theorem 4.
Proof.The conclusion follows immediately from Theorem 4 and (10).
Corollary 5.If π : M → M is a quaternionic Kähler-like statistical submersion, then the horizontal distribution is completely integrable.
Proof.This assertion is clear from Theorem 4 and (9).Theorem 5. Let π : M → M be a quaternionic Kähler-like statistical submersion.If the total space of the submersion is of type quaternionic space form, then the base space of the submersion is of type quaternionic space form and each fiber is a totally geodesic submanifold of M of type quaternionic space form.
Proof.The conclusions follow easily using the analogues of the O'Neill equations for a statistical submersion (Theorem 2.1 in [6]) and taking account of Theorem 4.
Example 2. Let (M, ∇, σ, g) be an almost Hermite-like quaternionic statistical manifold.Then we can define a torsion free linear connection ∇ on T M such that (T M, ∇ , G) is a statistical manifold [38], where G is the Sasaki metric.Next, we consider for any canonical local basis {J 1 , J 2 , J 3 } of σ the following tensor fields on T M , denoted by J 1 , J 2 , J 3 : Defining now the vector bundle σ over T M generated by {J 1 , J 2 , J 3 } (see [22,39]), one can easily conclude that (T M, ∇ , σ , g ) is an almost Hermite-like quaternionic statistical manifold.Moreover, we remark that Hence π * J α = J α π * , α = 1, 2, 3, and we conclude that the canonical projection π : T M → M is a (σ, σ )-holomorphic map.Therefore π is an almost Hermite-like quaternionic statistical submersion.Moreover, it follows that π is a locally hyper-Kähler-like statistical submersion if and only if (M, ∇, σ, g) is a flat locally hyper-Kähler-like statistical manifold.

Conclusions and Future Research
It is well known there is a deep relationship between statistics and differential geometry.A first step in this connection was given by C.R. Rao [40], who introduced a Riemannian metric in the space of probability distributions, providing a general framework for discussing problems of statistical inference, information loss and estimation, and giving an impulse to construct a geometrical theory of statistics (see, e.g., [41][42][43][44][45][46][47][48]).The most natural frame in this context is the concept of a statistical manifold [49].As it was pointed out in [50], the statistical manifolds are naturally associated to a family of affine-metric geometries and one can obtain interesting properties relating self-parallel curves to the relative entropy.Recently, H.V. Lê [51] proved that any smooth statistical manifold can be embedded into the space of probability measures on a finite set, giving a positive answer to an open problem of S. Amari and S.L. Lauritzen (see [2]).Therefore, any smooth statistical manifold is a finite-dimensional statistical model.
In the present paper, we introduced the notions of almost Hermite-like quaternionic statistical manifold and quaternionic Kähler-like statistical submersion, obtaining several properties.We also proved that the tangent bundle of an almost Hermite-like quaternionic statistical manifold has a natural almost Hermite-like quaternionic statistical structure and showed that the canonical projection provides us a very natural example of an almost Hermite-like quaternionic statistical submersion.We believe that the concepts investigated in this work can be also studied in some new settings, namely for statistical manifolds endowed with quaternionic structures of second kind [52] (also called paraquaternionic structures [39]), Kenmotsu structures [53], 3-Sasakian structures [54], almost para-Hermitian structures [55,56] and almost para-contact structure [57,58].We note that all these structure are of great interest not only in differential geometry, but also in various fields of science and engineering, such as string theory, integrable systems, quantum systems, statistical mechanics, motion planning, robot control and sensing, sensor networks, and digital signal processing.We look forward to studying some of these problems in detail later.Finally, we would like to note another five open problems in the field for further research.
Problem 1.To investigate if it is possible to construct an infinite family of quaternionic Kähler-like statistical structures on the tangent bundle of an almost Hermite-like quaternionic statistical manifold.A possible answer could be obtained by deforming the almost Hermite-like quaternionic statistical structure defined in Example 2 in a similar way to [59].Problem 4. To investigate the behavior of submanifolds in statistical manifolds of type quaternionic space form, as a quaternionic version of some recent results from [13] concerning submanifolds in statistical manifolds of constant curvature.Problem 5. To define affine connections compatible with a hypercosymplectic structure [60] and to obtain necessary and sufficient conditions for two compatible connections to form a dualistic structure, as an extension of the results from [61].Moreover, to define and investigate statistical submersions from almost Hermite-like quaternionic statistical submersions onto statistical manifolds equipped with hypercosymplectic structures.

3 Theorem 2 .
and the conclusion is now clear.If π : M → M is an almost Hermite-like quaternionic statistical submersion, then the fibers are almost Hermite-like quaternionic statistical manifolds.

Problem 2 .
To construct examples of locally hyper-Kähler-like statistical manifolds which are not hyper-Kähler ones.A possible solution could be to deform the almost Hermite-like quaternionic statistical structure from Example 1. Problem 3. To investigate under what conditions the exponential families, including the well-known multinomial distribution, negative multinomial distribution, and multivariate normal distribution, admit hyper-Kähler or quaternionic Kähler structure.