On the α-connections and the α-conformal equivalence on statistical manifolds

Abstract

Purpose

In this paper, we give some properties of the α-connections on statistical manifolds and we study the α-conformal equivalence where we develop an expression of curvature R¯ for ∇¯ in relation to those for ∇ and ∇^.

Design/methodology/approach

In the first section of this paper, we prove some results about the α-connections of a statistical manifold where we give some properties of the difference tensor K and we determine a relation between the curvature tensors; this relation is a generalization of the results obtained in [1]. In the second section, we introduce the notion of α-conformal equivalence of statistical manifolds treated in [1, 3], and we construct some examples.

Findings

We give some properties of the difference tensor K and we determine a relation between the curvature tensors; this relation is a generalization of the results obtained in [1]. In the second section, we introduce the notion of α-conformal equivalence of statistical manifolds, we give the relations between curvature tensors and we construct some examples.

Originality/value

We give some properties of the difference tensor K and we determine a relation between the curvature tensors; this relation is a generalization of the results obtained in [1]. In the second section, we introduce the notion of α-conformal equivalence of statistical manifolds, we give the relations between curvature tensors and we construct some examples.

Citation

Addad, K. and Ouakkas, S. (2024), "On the α-connections and the α-conformal equivalence on statistical manifolds", Arab Journal of Mathematical Sciences, Vol. 30 No. 1, pp. 2-15. https://doi.org/10.1108/AJMS-12-2020-0126

1. Introduction

Let Mm,g be a Riemannian manifold and ∇ a torsion free linear connection on M. The triple Mm,∇,g is called a statistical manifold if ∇g is symmetric and the pair ∇,g is called a statistical structure. For a statistical manifold Mm,∇,g, let ∇* be an affine connection on M such that,

XgY,Z=g∇XY,Z+gY,∇X*Z,

for all X, Y and Z in ΓTM. The affine connection ∇* is torsion free, and ∇*g symmetric. Then ∇* is called the dual connection of ∇, the triple Mm,∇*,g is called the dual statistical manifold of Mm,∇,g and ∇,∇*,g is the dualistic structure. Denoted by ∇^ the Levi-Civita connection associated with g, the difference tensor K is the 1,2-tensor defined by (see [1]).

KX,Y=∇X*Y−∇XY.

The difference tensor K satisfies for any vector fields X, Y, Z and any smooth function f on M the following properties:

∇XY=∇^XY−12KX,Y,∇X*Y=∇^XY+12KX,Y,

KX,Y=KY,X,KX,Y+Z=KX,Y+KX,Z

and

KfX,Y=KX,fY=fKX,Y.

Moreover, we have,

gKX,Y,Z=gKX,Z,Y.

A statistical structure is called trace-free if ∇v_g = 0 where v_g is the volume form determined by g. This condition is equivalent to the condition Tr_gK = 0. A statistical structure ∇,g is said to be of constant curvature k∈R if the curvature tensor field R of ∇ satisfies,

RX,YZ=kgY,ZX−gX,ZY.

If k = 0, ∇,g is called a Hessian structure. The concept of α-conformally equivalence was treated in [1] where the author develops an expression of the curvature Rα. In [2], the authors studied a 1-conformally flat statistical submanifold of a flat statistical manifold; they proved that a 1-conformally flat statistical manifold with a Riemannian metric can be locally realized as a submanifold of a flat statistical manifold. The author in [3] gives a procedure to realize a statistical manifold, which is α-conformally equivalent to a manifold with an α-transitively flat connection, as a statistical submanifold and in [4], he describe a method to obtain α-conformally equivalent connections from the relation between tensors and the symmetric cubic form. In [5], the authors studied the statistical hypersurfaces of some types of the statistical manifolds, which enable to construct a structure of a constant curvature. The divergence of 1-conformally flat statistical manifolds is studied in [6] where the authors prove that the generalized Pythagorean theorem holds if the statistical manifold has a constant curvature. In the first section of this paper, we prove some results about the α-connections of a statistical manifolds where we give some properties of the difference tensor K and we determine a relation between the curvature tensors Rα and Rβ; this relation is a generalization of the results obtained in [1]. In the second section, we introduce the notion of α-conformal equivalence of statistical manifolds treated in [1, 3], and we give the relations between R¯, R and R^ and we construct some examples.

2. Some results on the α-connections of a statistical manifolds

Let Mm,∇,g a statistical manifold with a dualistic structure ∇,∇*,g. For α∈R, we define a family of torsion-free connections ∇α by,

∇α=1+α2∇+1−α2∇*.

∇α is called an α-connection of Mm,∇,g. The triple Mm,∇α,g is also a statistical manifold, and ∇−α is the dual connection of ∇α. In particular,

∇0=∇^,∇1=∇,∇−1=∇*.

Moreover, we have the following equality,

∇XαY=∇^XY−α2KX,Y.

In general, for any α,β∈R, it is easy to see that,

(1)∇XαY=∇XβY−α−β2KX,Y

Proposition 1.

Let Mm,∇,g a statistical manifold with a dualistic structure ∇,∇*,g. For all vector fields X, Y, Z on M, we have,

(2) ∇XαKY,Z=∇XβKY,Z−α−β2KX,KY,Z+ α−β2KKX,Y,Z+α−β2KY,KX,Z.

Proof of Proposition 1. Let X,Y,Z∈ΓTM, by definition, we have,

∇XαKY,Z=∇XαKY,Z−K∇XαY,Z−KY,∇XαZ.

The properties of the difference tensor K gives us,

∇XαKY,Z=∇XβKY,Z−α−β2KX,KY,Z,

K∇XαY,Z=K∇XβY,Z−α−β2KKX,Y,Z

and

KY,∇XαZ=KY,∇XβZ−α−β2KY,KX,Z,

then

∇XαKY,Z=∇XβKY,Z− α−β2KX,KY,Z− K∇XβY,Z+ α−β2KKX,Y,Z− KY,∇XβZ+KY,KX,Z

Finally, using the fact that,

∇XβKY,Z=∇XβKY,Z− K∇XβY,Z−KY,∇XβZ,

we deduce that,

∇XαKY,Z=∇XβKY,Z− α−β2KX,KY,Z+ α−β2KKX,Y,Z+ α−β2KY,KX,Z.

Remark 1.

As particular cases of Eqn (2), we have

∇XαKY,Z=∇^XKY,Z−α2KX,KY,Z+ α2KKX,Y,Z+α2KY,KX,Z=∇XKY,Z−α−12KX,KY,Z+ α−12KKX,Y,Z+α−12KY,KX,Z=∇X*KY,Z−α+12KX,KY,Z+ α+12KKX,Y,Z+α+12KY,KX,Z

For a statistical structure ∇,∇*,g, we denote R, R*, R^ the curvature tensors for ∇, ∇*, ∇^, respectively, and Rα the curvature tensor for ∇α. In the first results, we give the relation between Rα and Rβ for any α,β∈R.

Theorem 1.

Let Mm,∇,g a statistical manifold. The relation between Rα and Rβ is given by,

(3) RαX,YZ=RβX,YZ+β−α2∇XβKY,Z−β−α2∇YβKX,Z+ β−α24KX,KY,Z−β−α24KY,KX,Z,

for all X,Y,Z∈ΓTM.

Proof of Theorem 1. Let X,Y,Z∈ΓTM, By definition we have,

(4)RαX,YZ=∇Xα∇YαZ−∇Xα∇YαZ−∇X,YαZ.

For the first term ∇Xα∇YαZ, we have,

∇YαZ=∇YβZ+β−α2KY,Z,

then

∇Xα∇YαZ=∇Xα∇^YZ−α2∇XαKY,Z.

It is simple to see that,

∇Xα∇YβZ=∇Xβ∇YβZ+β−α2KX,∇YβZ

and

∇XαKY,Z=∇XβKY,Z+β−α2KX,KY,Z,

which gives us

(5)∇Xα∇YαZ=∇Xβ∇YβZ+β−α2KX,∇YβZ+ β−α2∇XβKY,Z+β−α24KX,KY,Z.

A similar calculation gives,

(6)∇Yα∇XαZ=∇Yβ∇XβZ+β−α2KY,∇XβZ+ β−α2∇YβKX,Z+β−α24KY,KX,Z.

Finally, we have,

(7)∇X,YαZ=∇X,YβZ+β−α2KX,Y,Z

If we replace (5), (6) and (7) in (4), we deduce that,

RαX,YZ=RβX,YZ+β−α2∇XβKY,Z−β−α2∇YβKX,Z+ β−α2KX,∇YβZ−β−α2KY,∇XβZ−β−α2KX,Y,Z+ β−α24KX,KY,Z−β−α24KY,KX,Z

Using the fact that,

∇XβKY,Z=∇XβKY,Z+K∇XβY,Z+KY,∇XβZ,

∇YβKX,Z=∇YβKX,Z+K∇YβX,Z+KX,∇YβZ

and

KX,Y,Z=K∇XβY,Z−K∇YβX,Z,

we get

RαX,YZ=RβX,YZ+β−α2∇XβKY,Z−β−α2∇YβKX,Z+ β−α24KX,KY,Z−β−α24KY,KX,Z.

As particular cases of Theorem 1, we get the following Corollary:

Corollary 1.

Let Mm,∇,g a statistical manifold with a dualistic structure ∇,∇*,g. The relations between Rα, R^, R and R* are given by

(8) RαX,YZ=R^X,YZ−α2∇^XKY,Z+α2∇^YKX,Z+ α24KX,KY,Z−α24KY,KX,Z,

(9) RαX,YZ=RX,YZ+1−α2∇XKY,Z−1−α2∇YKX,Z+ 1−α24KX,KY,Z−1−α24KY,KX,Z

and

(10) RαX,YZ=R*X,YZ−1+α2∇X*KY,Z+1+α2∇Y*KX,Z+ 1+α24KX,KY,Z−1+α24KY,KX,Z,

for all X,Y,Z∈ΓTM.

Remark 2.

From Theorem 1, we can give other relations:

1. The relation between Rα and R−α is given by (see [1]).

   RαX,YZ=R−αX,YZ+αRX,YZ−R*X,YZ.

2. The relation between Rα, R and R* is given by (see [1]).

   RαX,YZ=1+α2RX,YZ+1−α2R*X,YZ− 1−α24KX,KY,Z+1−α24KY,KX,Z

Corollary 2.

Let eii=1m be a local orthonormal frame field on Mm,g, for a statistical structure ∇,∇*,g, if we denote

RicciαX=TrgRαX,⋅⋅=RαX,eiei

and

RicciβX=TrgRβX,⋅⋅=RβX,eiei,

for any X∈ΓTM, then the relation between RicciαX and RicciβX is given by the following formula :

RicciαX=RicciβX+β−α24KX,E+β−α2Trg∇XβK⋅,⋅− β−α2Trg∇⋅βKX,⋅−β−α24TrgKKX,⋅,⋅,

where

Trg∇XβK⋅,⋅=∇XβKei,ei,

Trg∇⋅βKX,⋅=∇eiβKX,ei,

TrgKKX,⋅,⋅=KKX,ei,ei,

and

E=TrgK=Kei,ei.

In particular for β∈−1,0,1, we obtain,

RicciαX=Ricci^X+α24KX,E−α2Trg∇^XK⋅,⋅+ α2Trg∇^⋅KX,⋅−α24TrgKKX,⋅,⋅,

RicciαX=RicciX+1−α24KX,E+1−α2Trg∇XK⋅,⋅− 1−α2Trg∇⋅KX,⋅−1−α24TrgKKX,⋅,⋅

and

RicciαX=Ricci*X+1+α24KX,E−1+α2Trg∇X*K⋅,⋅+ 1+α2Trg∇*⋅KX,⋅−1+α24TrgKKX,⋅,⋅

Example 1.

Let R2,g be a statistical manifold with Riemannian metric g = dx² + dy² and ∇ an affine connection defined by

∇e1e1=e2,∇e2e2=0,∇e1e2=∇e2e1=e1

where e1=∂∂x,e2=∂∂y is an orthonormal frame field. A simple calculation gives,

∇*e1e1=−e2,∇*e2e2=0,∇*e1e2=∇*e2e1=−e1.

We deduce that,

Ke1,e1=−2e2,Ke2,e2=0,Ke1,e2=Ke2,e1=−2e1,

then,

E=TrgK=Ke1,e1+Ke2,e2=−2e2.

In this case, we have,

∇e1αe1=αe2,∇e2αe2=0,∇e1αe2=∇e2αe1=αe1,

Rαe1,e2e2=−α2e1,Rαe2,e1e1=−α2e2.

and

RicciαX=−α2X,RicαX,Y=−α2gX,Y,Sgα=−2α2.

Then R2,∇α,g is a statistical manifold of constant curvature − α² and it is a Hessian structure if and only if α = 0.

Example 2.

Let H2=x,y∈R2,y≻0,g be a statistical manifold with Riemannian metric g=1y2dx2+dy2 and ∇ an affine connection defined by

∇e1e1=∇e2e2=2e2,∇e1e2=0,∇e2e1=e1,

where e1=y∂∂x,e2=y∂∂y is an orthonormal frame field. A simple calculation gives

∇*e1e1=0,∇*e2e2=−2e2,∇*e1e2=−2e1,∇*e2e1=−e1.

Then,

∇e1αe1=1+αe2,∇e2αe2=2αe2,∇e1αe2=−1−αe1,∇e2αe1=αe1.

We deduce that,

Rαe1,e2e2=α2−1e1,Rαe2,e1e1=α2−1e2,

it follows that,

RicciαX=α2−1X,RicαX,Y=α2−1gX,Y,Sgα=2α2−1.

In this case, H2,∇α,g is a statistical manifold of constant curvature α² − 1 and it is a Hessian structure if and only if α = ±1.

3. The α-conformal equivalence

For a real number α, statistical manifolds Mm,∇,g and Mm,∇¯,g¯ are said to be α-conformally equivalent if there exists a function γ on M such that the Riemannian metrics g¯ and g and h are related by the following relation,

(11)g¯X,Y=e2γgX,Y

and the connection ∇¯ is given by,

(12)∇¯XY=∇XY+1−αYγX+1−αXγY−1+αgX,Ygradγ,

for all X,Y,Z∈ΓTM. Using the fact that ∇XY=∇^XY−12KX,Y, we obtain,

(13)∇¯XY=∇^XY+1−αYγX+1−αXγY−1+αgX,Ygradγ−12KX,Y.

Theorem 2.

(14) R¯X,YZ=R^X,YZ−1+αgY,Z∇^Xgradγ+1+αgX,Z∇^Ygradγ+ 1+α2gY,ZXγgradγ−1+α2gX,ZYγgradγ+1−α2YγZγX− 1−α2gY,Zgradγ2X−1−α2KY,ZγX−1−αg∇^Ygradγ,ZX− 1−α2XγZγY+1−αg∇^Xgradγ,ZY+1−α2gX,Zgradγ2Y+ 1−α2KX,ZγY−12∇^XKY,Z+12∇^YKX,Z+ 1+α2gY,ZKX,gradγ−1+α2gX,ZKY,gradγ+ 14KX,KY,Z−14KY,KX,Z.

Proof of Theorem 2. By definition, we have,

(15)R¯X,YZ=∇¯X∇¯YZ−∇¯Y∇¯XZ−∇¯X,YZ.

We will study the right side of this equation term by term. By (13), we obtain,

∇¯YZ=∇^YZ+1−αZγY+1−αYγZ−1+αgY,Zgradγ−12KY,Z,

which gives us,

∇¯X∇¯YZ=∇¯X∇^YZ+1−α∇¯XZγY+1−α∇¯XYγZ− 1+α∇¯XgY,Zgradγ−12∇¯XKY,Z.

Using Eqn (13), we deduce that,

∇¯X∇^YZ=∇^X∇^YZ+1−α∇^YZγX+1−αXγ∇^YZ− 1+αgX,∇^YZgradγ−12KX,∇^YZ,

∇¯XZγY=Zγ∇^XY+1−αYγZγX+1−αXγZγY−1+αgX,YZγgradγ−12ZγKX,Y+g∇^Xgradγ,ZY+ggradγ,∇^XZY,

∇¯XYγZ=Yγ∇^XZ+1−αYγZγX+1−αXγYγZ−1+αgX,ZYγgradγ−12YγKX,Z+g∇^Xgradγ,YZ+ggradγ,∇^XYZ,

∇¯XgY,Zgradγ=gY,Z∇^Xgradγ+1−αgY,Zgradγ2X−2αgY,ZXγgradγ− 12gY,ZKX,gradγ+g∇^XY,Zgradγ+gY,∇^XZgradγ

and

∇¯XKY,Z=∇^XKY,Z+1−αKY,ZγX+1−αXγKY,Z− 1+αgX,KY,Zgradγ−12KX,KY,Z.

It follows that,

(16)∇¯X∇¯YZ=∇^X∇^YZ−1+αgY,Z∇^Xgradγ+1+α2gX,KY,Zgradγ−1−α2gX,YZγgradγ−1−α2gX,ZYγgradγ+2α1+αgY,ZXγgradγ−1+αgX,∇^YZgradγ−1+αg∇^XY,Zgradγ−1+αgY,∇^XZgradγ−1−α2gY,Zgradγ2X+21−α2YγZγX+1−α∇^YZγX+1−α2XγZγY+1−αg∇^Xgradγ,ZY+1−αggradγ,∇^XZY+1−α2XγYγZ+1−αg∇^Xgradγ,YZ+1−αggradγ,∇^XYZ+1−αXγ∇^YZ−12KX,∇^YZ+1−αZγ∇^XY+1−αYγ∇^XZ−1−α2YγKX,Z−1−α2ZγKX,Y+1+α2gY,ZKX,gradγ−12∇^XKY,Z−1−α2KY,ZγX−1−α2XγKY,Z+14KX,KY,Z.

A similar calculation gives us,

(17)∇¯Y∇¯XZ=∇^Y∇^XZ−1+αgX,Z∇^Ygradγ+1+α2gY,KX,Zgradγ−1−α2gX,YZγgradγ−1−α2gY,ZXγgradγ+2α1+αgX,ZYγgradγ−1+αgY,∇^XZgradγ−1+αg∇^YX,Zgradγ−1+αgX,∇^YZgradγ−1−α2gX,Zgradγ2Y+21−α2XγZγY+1−α∇^XZγY+1−α2YγZγX+1−αg∇^Ygradγ,ZX+1−αggradγ,∇^YZX+1−α2XγYγZ+1−αg∇^Xgradγ,YZ+1−αggradγ,∇^YXZ+1−αYγ∇^XZ−12KY,∇^XZ+1−αZγ∇^YX+1−αXγ∇^YZ−1−α2XγKY,Z−1−α2ZγKX,Y+1+α2gX,ZKY,gradγ−12∇^YKX,Z−1−α2KX,ZγY−1−α2YγKX,Z+14KY,KX,Z.

Finally, it is easy to see that,

(18)∇¯X,YZ=∇^X,YZ−1+αg∇^XY,Zgradγ+1+αg∇^YX,Zgradγ+1−α∇^XYγZ−1−α∇^YXγZ+1−αZγ∇^XY−1−αZγ∇^YX−12K∇^XY,Z+12K∇^YX,Z.

By replacing (16), (17) and (18) in (15), we conclude that,

R¯X,YZ=R^X,YZ−1+αgY,Z∇^Xgradγ+1+αgX,Z∇^Ygradγ+ 1+α2gY,ZXγgradγ−1+α2gX,ZYγgradγ+1−α2YγZγX− 1−α2gY,Zgradγ2X−1−α2KY,ZγX−1−αg∇^Ygradγ,ZX− 1−α2XγZγY+1−αg∇^Xgradγ,ZY+1−α2gX,Zgradγ2Y+ 1−α2KX,ZγY−12∇^XKY,Z+12∇^YKX,Z+ 1+α2gY,ZKX,gradγ−1+α2gX,ZKY,gradγ+14KX,KY,Z− 14KY,KX,Z.

The same method of calculation used in Theorem 2 and the following equations,

∇^XY=∇XY+12KX,Y,

R^X,YZ=RX,YZ+12∇XKY,Z−12∇YKX,Z+ 14KX,KY,Z−14KY,KX,Z

gives us the following theorem

Theorem 3.

(19) R¯X,YZ=RX,YZ−1+αgY,Z∇Xgradγ+1+αgX,Z∇Ygradγ− 1+α2gX,ZYγgradγ+1+α2gY,ZXγgradγ− 1−α2gY,Zgradγ2X+1−α2gX,Zgradγ2Y− 1−αg∇Ygradγ,ZX+1−αg∇Xgradγ,ZY+ 1−α2YγZγX−1−α2XγZγY− 1−αgKY,Z,gradγX+1−αgKX,Z,gradγY

Corollary 3.

Let us choose ei1≤i≤m to be an orthonormal frame on Mm,∇,g, an orthonormal frame on Mm,∇¯,g¯=e2γg is given by e¯i=e−γei1≤i≤m. For any X,Y∈ΓTM, we define

RicciX=TrgRX,⋅⋅=RX,eiei,Ricci¯X=Trg¯R¯X,⋅⋅=R¯X,ei¯ei¯,

RicX,Y=gRicciX,Y,Ric¯X,Y=g¯Ricci¯X,Y

and

Sg=TrgRic=Ricei,ei,Sg¯=Trg¯Ric¯=Ric¯ei¯,ei¯.

Using Theorem 3, we obtain the following relations,

Ricci¯X=e−2γRicciX+m−2α2+2mα+m−2e−2γXγgradγ− mα+m−2e−2γ∇Xgradγ+mα2−2α−m+2e−2γgradγ2X− 1−αe−2γΔ^γX−1−α2e−2γEγX+1−αe−2γKX,gradγ,

Ric¯X,Y=RicX,Y+m−2α2+2mα+m−2XγYγ+ mα2−2α−m+2gradγ2gX,Y−1−αΔ^γgX,Y− mα+m−2g∇Xgradγ,Y−1−α2EγgX,Y+ 1−αgKX,Y,gradγ

and

Sg¯=e−2γSg+m−1m+2α2−m+2e−2γgradγ2− 2m−1e−2γΔ^γ+m−1αe−2γEγ

Corollary 4.

Theorem 3 and Corollary 3 gives us two particular cases:

1. If α = 1, we obtain,

   R¯X,YZ=RX,YZ−2gY,Z∇Xgradγ+2gX,Z∇Ygradγ− 4gX,ZYγgradγ+4gY,ZXγgradγ,

   Ricci¯X=e−2γRicciX+4m−1Xγgradγ−2m−1∇Xgradγ,

   Ric¯X,Y=RicX,Y+4m−1XγYγ−2m−1g∇Xgradγ,Y

   and

   Sg¯=e−2γSg+4m−1gradγ2−2m−1Δ^γ+m−1Eγ.

2. If α = −1, we obtain,

   R¯X,YZ=RX,YZ−2g∇Ygradγ,ZX+2g∇Xgradγ,ZY+ 4YγZγX−4XγZγY−2gKY,Z,gradγX+ 2gKX,Z,gradγY,

   Ricci¯X=e−2γRicciX−4e−2γXγgradγ+2e−2γ∇Xgradγ+ 4e−2γgradγ2X−2e−2γΔ^γX−e−2γEγX+ 2e−2γKX,gradγ,

   Ric¯X,Y=RicX,Y−4XγYγ+2g∇Xgradγ,Y+ 4gradγ2gX,Y−2Δ^γgX,Y− EγgX,Y+2gKX,Y,gradγ

   and

   Sg¯=e−2γSg+m−1e−2γ4gradγ2−2Δ^γ+Eγ,

   where

   Δ^γ=g∇^eigradγ,ei=eieiγ−∇^eieiγ.

Example 3.

Let R2,g be a statistical manifold with Riemannian metric g = dx² + dy² and ∇ an affine connection defined by

∇e1e1=e2,∇e2e2=0,∇e1e2=∇e2e1=e1

where e1=∂∂x,e2=∂∂y is an orthonormal frame field. Then R2,∇,g is a statistical manifold of constant curvature − 1 and S_g = −2. We want to determine γ such that Sg¯=0. By Corollary\enleadertwodots , we deduce that Sg¯ vanish if and only if

2Δ^γ−αEγ−4α2gradγ2+2=0.

To solve this equation, we will present two cases :

• (1)

  If we assume that γ depends only on the variable x, then Sg¯ vanish if and only if.

  γ″−2α2γ′2+1=0.

Note that if α = 0, the solution of this last equation is,

γx=−12x2+ax+b.

In the case where α ≠ 0, a particular solution is given by γx=1α2x+b.

• (2)

  If the function γ depends only on the variable y, we conclude that Sg¯=0 if and only if,

  γ″+αγ′−2α2γ′2+1=0.

Using the same method, if α = 0, the solution obtained is,

γy=−12y2+ay+b

and if we take α ≠ 0, a particular solution is γy=1αy+b.