CONFORMAL VECTOR FIELDS ON STATISTICAL MANIFOLDS

. Introducing the conformal vector ﬁelds on a statistical manifold, we present necessary and suﬃcient conditions for a vector ﬁeld on a statistical manifold to be conformal. After presenting some examples, we classify the conformal vector ﬁelds on two famous statistical manifolds. Considering three statistical structures on the tangent bundle of a statistical manifold, we study the conditions under which the complete and horizontal lifts of a vector ﬁeld can be conformal on these structures.


Introduction
Nowadays, information geometry as a combination of statistics and differential geometry has an effective role in science. Some of its vast applications can be found in image processing, physics, computer science and machine learning [4,9,13,12,31]. It is a realm that makes it possible to illustrate statistical objects as geometric ones by the way of capturing their geometric properties. Rigid objects in the sense of coordinate transformation are a favourite among differential geometers. So, observing the statistical spaces from the doorway of differential geometry makes it convenient to study the statistical behaviors profoundly. A fundamental detailed survey on information geometry can be found in the monograph [5].
For an open subset Θ of R n and a sample space Ω with parameter θ = (θ 1 , . . . , θ n ), we call the set of probability density functions S = {p(x; θ) : Ω p(x; θ) = 1, p(x; θ) > 0, θ ∈ Θ ⊆ R n } a statistical model. For a statistical model S, the semi-definite Fisher information matrix g(θ) = [g ij (θ)] is defined as where θ = (x; θ) := log p(x; θ), ∂ i := ∂ ∂θ i , and E p [f ] is the expectation of f (x) with respect to p(x; θ). The space S equipped with such information matrices is called a statistical manifold in the literature. the solutions to the Laplace equation on complicated planar domains have extensive applications. An effective approach to the construction of such solutions is based on the use of conformal mappings [24]. Conformal vector fields are vector fields with flows preserving a given conformal class of metrics. These vector fields as the generalizations of conformal functions between Euclidean spaces and conformal maps between (semi-) Riemannian manifolds, are important matters of fact in Riemannian geometry (see for instance [8,30]). Indeed, a smooth vector field X on a Riemannian manifold (M, g) is said to be a conformal vector field if there exists a smooth function ρ on M such that L X g = 2ρg. The function ρ is called the potential function of the conformal vector field X. If f is a constant function, X is called a homothetic vector field. A special category of homothetic vector fields is the set of Killing vector fields, where ρ = 0 (in the presence of a Riemannian metric g, their flow preserves g). These kinds of preserving the metric help to categorize the spaces in the sense of diffeomorphism and symmetry. The study of conformal vector fields on Riemannian manifolds and their tangent bundles is of interest to many researchers (see for instance [1,18,25,32]).
Recently, the study of geometric concepts of statistical manifolds has been considered by many researchers (see e.g. [7,10,17,20,21,35,36,37]). For instance, Sasakian geometry and symplectic geometry on statistical manifolds are introduced by [17] and [37], respectively. Also, Hasegawa and Yamauchi [20,21] introduced the concepts of λ-conformally flat and conformally-projectively flat on statistical manifolds. The importance of conformal vector fields in Riemannian geometry and the concepts introduced in [20,21] led to the idea of introducing conformal vector fields on statistical manifolds. Since the concept of conformal vector field is independent of the choice of linear connection, its introduction on statistical manifolds is similar to that on Riemannian ones and will not be valuable. So, we need to introduce a new concept that uses the statistical connection structure and gives us the definition of conformal vector field in the Riemannian case (for this reason we call it the conformal vector field on statistical manifolds). Therefore, the aim of this paper is to study the conformal geometry on statistical manifolds.
The organization of the paper is as follows. In Section 2 we recall some concepts on statistical manifolds and lift geometry on the tangent bundle of a manifold. Section 3 is devoted to the study of Lie derivatives of tensor fields on statistical manifolds. In Section 4 we introduce the conformal vector fields on statistical manifolds and we present some examples of them. Then we focus on two famous statistical manifolds (the general Gaussian distribution manifold and the 2-dimensional statistical manifold) and we determine the conformal vector fields on these manifolds. In the last section we consider three statistical structures on the tangent bundle of a Riemannian manifold and we find necessary and sufficient conditions under which the horizontal and complete lifts of a vector field can be conformal on these structures. Then we implement these conditions on an example, and also on the tangent bundle of the generalized Gaussian distribution manifold and 2-dimensional statistical manifold.

Preliminaries
Let M be a smooth manifold with a Riemannian metric g and let ∇ be a linear symmetric connection. The triple (M, g, ∇) is called a statistical manifold if ∇g is symmetric for all X, Y, Z ∈ χ(M ). In other words, ∇g = C, where C is a symmetric tensor of degree (0,3), namely, ∇g satisfies the Codazzi equations (2.1) In this case, ∇ is called a statistical connection. When C = 0, we have the unique Levi-Civita connection ∇ (0) . Now we define the skewness operator K of degree (1,2) on M as follows: It is easy to see that K satisfies the following relations: The dual connection of a linear connection ∇ is defined by It is known that if (M, g, ∇) is a statistical manifold, then (M, g, * ∇) is a statistical manifold as well. Moreover, we have ∇ (0) = 1 2 (∇ + * ∇). Using ∇ and * ∇, we have the family of α-connections as follows [35]: Let M be an n-dimensional manifold, let T M be its tangent bundle and let π : T M → M be the projection map. The space T T M can be split into two subspaces at every point (p, v) as follows: and Γ k ij are the Christoffel symbols of a linear connection ∇. From now on to simplify we use ∂ i , ∂ī and δ i instead of ∂ ∂x i , where R k ijr are the components of the curvature tensor of M given by It is known that the components of the curvature tensor of a statistical connection satisfy the following relations: The above lift operations are extended to the tensor algebra J (M ) by the following rules [33]: (2.7) In particular, for any tensor fields P, Q ∈ J r s (M ) with r = 0, 1, we have P c (X c 1 , . . . , X c s ) = (P (X 1 , . . . , X s )) c , P c (X v 1 , . . . , X v s ) = 0, (2.8) and (2.9)

Lie derivation and statistical connection
First, we recall some concepts and notations on the Lie derivative and the covariant derivative of tensors.

LEILA SAMEREH AND ESMAEIL PEYGHAN
In the local expression we have and which gives us (3.3). Since * ∇ = ∇ − 2K, we get
Here we study some properties of the Lie derivative of tensors on statistical manifolds in the local format. Note that the local expression of the tensors C and K introduced by (2.1) and (2.2) From the above equations we conclude that C ijk is totally symmetric, i.e., . Then we have (3.9) But using (2.1) we get Similarly, we have Putting (3.10) and (3.11) in (3.9) and considering that C jrk is totally symmetric we obtain Using the third equation of (3.8) in the above equation implies

Proof. Equation (3.5) has the local expression
Since ∇ is a statistical connection, we get Using ∇ = * ∇ + 2K, the above equation reduces to The expression (3.12) and the above equations imply Using the local expression of * R (see (2.6)) we obtain

Conformal vector fields on statistical manifolds
In this section we present the definition of a conformal vector field on a statistical manifold and we study some examples.
It is known that the concept of conformal vector field is independent of the choice of linear connection. So, its introduction on statistical manifolds is similar to that on Riemannian ones. This motivates us to introduce a new concept that uses the statistical connection structure and gives us the definition of conformal vector field in the Riemannian case. Since the skewness operator K has a basic role in statistical manifolds, we need to consider a certain condition on it. It is known that the geometrical symmetries of spacetime (which have many applications in general relativity) are often defined through the vanishing of the Lie derivative of certain tensors with respect to a vector (see [26] for more details). So, we are interested in skewness operators whose Lie derivative is zero.
where ρ is a function on M .
According to (3.2), the above equations have local expressions Using the relations we can rewrite (4.2) as follows From Definition 4.1 and equations (4.1) and (4.2) we can conclude the following:

Example 4.3. Consider the Fisher metric
The equations (3.8) imply Considering i, j = 1, 2, the above equation implies

LEILA SAMEREH AND ESMAEIL PEYGHAN
Note that the first two equations of (4.7) imply ∂ 1 (X 1 ) = ∂ 2 (X 2 ). Considering i, j, r = 1, 2 in (4.5) we obtain the following equations: where a is non-zero constant, then we obtain a conformal vector field on (R 2 , g, ∇). then using (2.2) and (4.16) we deduce that the coefficients of the statistical connection ∇ are as follows: Example 4.4. We consider the normal distribution Riemannian manifold (M, g) introduced in Example 4.3. Let η be a 1-form on M . It is easy to see that where η(X) = g(X, η ) satisfies (i) and (ii) of (2.3). So, the linear connection ∇ given by X is a statistical connection on (M, g) (this connection has been introduced by Blaga and Crasmareanu [10]). Now, let X = X 1 ∂ 1 + X 2 ∂ 2 be a vector field on M . Using (4.3), we get where K r ij = g ij η r + η i δ r j + η j δ r i . Considering i, j = 1, 2 and using η i = g ij η j , the above equation induces the following: .7), the six equations above reduce to So, X is conformal if and only if it satisfies these two equations. For instance, if we consider η = λ kµ+c dµ + λ kµ+c dσ, where λ, k and c are constants, then X = (kµ + c)∂ 1 + (kσ + c)∂ 2 is a conformal vector field on (R 2 , g, ∇) with ρ = − c σ . Here we study the conformal geometry on two famous statistical manifolds. One of them is the generalized Gaussian distribution manifold and the other is a p-dimensional manifold (of course for p = 2). The geometric structures of these manifolds were studied in [34].
The generalized Gaussian distribution manifold is defined as where Γ(x) is the gamma function and µ, σ and β are called the location, scale and shape parameters, respectively. When β = 1 or β = 2, this distribution reduces to the Laplace distribution or the Gaussian distribution, respectively. Note that if β is odd, the manifold is not smooth. Hence we only consider the case when β is a known even number. In [34], Yuan proved that the Riemannian metric on the generalized Gaussian statistical manifold M 1 is as follows: where Also, g −1 , given by , is the inverse of g. The Christoffel symbols of the α-connection are as follows: (4.20) Let X = X 1 ∂ 1 + X 2 ∂ 2 be a conformal vector field on M 1 . Considering i, j = 1, 2 in (4.4) we obtain The above equations imply Also, putting i, j, r = 1, 2 in (4.5), we obtain the following equations: If ∂ 1 (X 2 ) = 0, then the last two equations imply c 222 = 0, which is a contradiction. Thus ∂ 1 (X 2 ) = 0. Finally, considering (4.21), we get ∂ 2 (X 1 ) = 0 and ∂ 1 (X 1 ) = 1 σ X 2 . The differential equation system has the solution X 2 = Aσ and X 1 = Aµ + B.
Here we consider a p-dimensional statistical manifold. The importance of this distribution family lies in that its member is a non-Gaussian multivariate distribution, while the marginal distribution is Gaussian, which implies that a set of marginal distributions does not uniquely determine the multivariate normal distribution [14]. A p-dimensional statistical manifold is defined by where . . , p}. The distribution in M 2 can be rewritten as where θ i = − 1 2 λ i . This is one member of the exponential family with the natural coordinates (θ 1 , . . . , θ p ) and the potential function ψ(θ) = − 1 2 p i=1 log(−θ i ). It is known that, for the exponential family, the Fisher information is just the second derivative of the potential function, (4.23) and the α-connection is the third derivative of the potential function, where δ ii = 1 for i = 1, . . . , p, δ ij = 0 for i = j, δ iii = 1 for i = 1, . . . , p, and δ ijk = 0 for unequal i, j, k (see [34] for more details). For p = 2, the matrix expression of the metric g given by (4.23) and its inverse matrix are as follows: (4.25) From (4.24) and (4.23), we get Γ (α)1 11 Proof. From (4.26) we have the Christoffel symbols of the Levi-Civita connection as follows: (4.26) and (4.27) we get Now let X = X 1 ∂ 1 + X 2 ∂ 2 be a conformal vector field on M 2 . By (4.4) we obtain Setting i = j = r = 1 in (4.5) implies ∂ 1 (X 1 ) = X 1 θ1 . Considering this equation and the first equation of (4.28) we conclude that ρ = 0. So X is a Killing vector field. Putting i = j = 1, r = 2 in (4.5) yields K 1 11 ∂ 1 (X 2 ) = 0, which gives us ∂ 1 (X 2 ) = 0. This equation together with the third equation of (4.28) results in ∂ 2 (X 1 ) = 0. From this equation and ∂ 1 (X 1 ) = X 1 θ1 we get X 1 = Aθ 1 . Similarly, the second equation of (4.28) and ∂ 2 (X 1 ) = 0 imply X 2 = Bθ 2 . It is easy to see that X = Aθ 1 ∂ 1 + Bθ 2 ∂ 2 satisfies all the equations of (4.4) and (4.5).

Conformal vector fields on the tangent bundle
In this section, we consider two statistical structures on the tangent bundle of a statistical manifold and we study the conformal vector fields on these structures.
Let (M, g, ∇) be a statistical manifold with the skewness operator K. Using (2.7) and (2.9), the horizontal lift metric g h with respect to ∇ is described by the formulas for all X, Y ∈ χ(M ). Again, using (2.7) and (2.9), the horizontal lift of K is defined by 2) In [23] Matsuzoe and Inoguchi proved that if (M, g, ∇) is a statistical manifold, then (T M, g h , K h ) is a statistical manifold. Here we study the conditions under which X h can be conformal on (T M, g h , K h ). According to Definition 4.1, a vector field X ∈ χ(T M ) is called a conformal vector field on a statistical manifold (T M, g, ∇) if there exists a function ρ(x, y) on T M such that L X g = 2 ρ g and L X K = 0, where K is the skewness operator associated to ∇. If ρ is a function that depends only on x h , then X is called an inessential vector field. Theorem 5.1. Let (M, g, ∇) be a statistical manifold with the skewness operator K and let X h be the horizontal lift of a vector field on M . If X h is conformal with respect to (T M, g h , K h ), then X h is an inessential vector field. Moreover, X h is an inessential vector field if and only if where R rikj := R h rik g hj . Proof. We can rewrite the metric g h and K h defined by (5.1) and (5.2) as follows: Using (2.5) and (5.8) we get (L X h g h )(∂ī, ∂j) = 0. In a similar way, we obtain

LEILA SAMEREH AND ESMAEIL PEYGHAN
The relations (2.5) and (5.9) imply Applying ∂k to this relation gives Multiplying the last equation with g ij we get 0 = 2n∂k ρ, which implies that ρ is a function with respect to x h . So using (5.10) and ( Now we consider the tensor K with the following components: It is easy to check that the above components satisfy (3.8). So is a statistical connection on (M, g) with the following Christoffel symbols: , , .
Using (2.6), we can show that all of the curvature components are zero except for the following ones: From the above equations we deduce that It is worth remarking that (M, g, ∇) is a non-flat statistical manifold, because R 1222 = 0. Applying i = j = 2 in (5.3) and using the above equations we get where y 1 = dµ and y 2 = dσ. Differentiating the above equation with respect to y 2 implies X 1 = 0. Setting i = 1 and j = 2 in (5.3) and using X 1 = 0 and (5.12) lead to Differentiating the above equation with respect to y 2 implies X 2 = 0. So X = 0 is the only conformal vector field on the statistical manifold (M, g, ∇) such that X h can be a conformal vector field on (T M, g h , K h ) (with ρ = 0). Indeed, X h = 0 is the only Killing vector field on (T M, g h , K h ).
Since β = 0, 1, from the above equation we conclude that R Proof. Let X = X 1 ∂ 1 + X 2 ∂ 2 be a vector field on M 1 such that X h is a conformal vector field with respect to (T M 1 , g h , K h ). We consider two cases: In this case, we have R  (4.20) in the above equation gives us But from (3.1) and (4.18) we obtain the following: Putting (5.15) and (5.16) in (5.14) we get 112 ) = 0. (5.17) Setting i = j = m = 1 in (5.7) and using (4.20) we get Using (3.1), (4.18) and (4.20) we get Setting (5.19) in (5.18) gives us From (4.18), (4.20) and the above equation we deduce that (note that c 121 = 0):  12 we get the contradiction 1 = 0. So the possible case is X 2 = 0. Therefore, we conclude that X 1 = X 2 = 0, i.e., X = 0.
Let (M, g, ∇) be a statistical manifold with the skewness operator K. Using the splitting (2.4), we can define the Riemannian metric g S on T M : (5.27) which is called the diagonal lift or Sasaki lift of g [29]. In [23], Matsuzoe Proof. We can rewrite the metric g S defined by (5.27) as follows: ). There does not exist any non-zero vector field on M 1 such that X h is a conformal vector field with respect to (T M 1 , g S , K h ).
Proof. Setting i = j = 1 in (5.28) we obtain Similarly, applying i = j = 2 in (5.28) implies Since β = 1, the two equations above imply α = 2. Setting i = 1, j = 2 in (5.28) and considering α = 2 we get 121 − 1 2 c 121 = 0, which gives us X 1 = 0 (since β = 1, the coefficient of X 1 in the above equation is non-zero). Putting i = j = 1 in (5.30) gives us X 2 = − ρσ. If X 2 = 0, using this equation and (5.32) we deduce that β = 1 2 , which is a contradiction (because β is an even number). So X 2 = 0, and consequently X = 0. Theorem 5.9. Consider the 2-dimensional statistical manifold (M 2 , g, ∇ (α) ) with the skewness operator K. If X h is a conformal vector field with respect to (T M 2 , g S , K h ), then X h reduces to a Killing vector field. Moreover, X h is a Killing vector field if and only if X = A(θ 1 + θ 2 ), where A is a constant function.
Let (M, g, ∇) be a statistical manifold with the skewness operator K. Using (2.7) and (2.8), the complete lift metric g c with respect to ∇ is described by the formulas for all X, Y ∈ χ(M ). Also, using (2.7) and (2.8), the complete lift of K is defined by and relations (5.40)-(5.45) we conclude that X c is a conformal vector field if and only if (∂ r X k )∂ k g ij + X k ∂ r ∂ k g ij + g ik ∂ r ∂ j X k + ∂ j X k ∂ r g ik + g jk ∂ r ∂ i X k + ∂ i X k ∂ r g jk = 2 ρ(x)∂ r g ij , (5.46) Differentiating (5.47) with respect to x r we get The relations (5.46) and (5.49) imply (∂ r ρ)g ij = 0. So ∂ r ρ = 0, i.e., ρ is a constant function. Indeed, conformal vector fields reduce to homothetic (Killing) vector fields. Also, (5.46)-(5.48) reduce to the following: where c is a constant. But these last two conditions are equivalent to the homothetic (Killing) property of X. Thus we conclude the following: