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

Khadidja Addad (Laboratory of Geometry, Analysis, Control and Applications, University of Saida, Dr Moulay Tahar Algeria, Saida, Algeria)
Seddik Ouakkas (Laboratory of Geometry, Analysis, Control and Applications, University of Saida, Dr Moulay Tahar Algeria, Saida, Algeria)

Arab Journal of Mathematical Sciences

ISSN: 1319-5166

Article publication date: 22 June 2021

281

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.

Keywords

Citation

Addad, K. and Ouakkas, S. (2021), "On the α-connections and the α-conformal equivalence on statistical manifolds", Arab Journal of Mathematical Sciences, Vol. ahead-of-print No. ahead-of-print. https://doi.org/10.1108/AJMS-12-2020-0126

Publisher

:

Emerald Publishing Limited

Copyright © 2021, Khadidja Addad and Seddik Ouakkas

License

Published in the Arab Journal of Mathematical Sciences. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/4.0/legalcode


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=gXY,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*YXY.

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

XY=^XY12KX,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 ∇vg = 0 where vg is the volume form determined by g. This condition is equivalent to the condition TrgK = 0. A statistical structure ,g is said to be of constant curvature kR if the curvature tensor field R of ∇ satisfies,

RX,YZ=kgY,ZXgX,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 αR, we define a family of torsion-free connections α by,

α=1+α2+1α2*.
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=^XYα2KX,Y.

In general, for any α,βR, it is easy to see that,

(1)XαY=XβYαβ2KX,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,Zαβ2KX,KY,Z+αβ2KKX,Y,Z+αβ2KY,KX,Z.

Proof of Proposition 1. Let X,Y,ZΓTM, by definition, we have,

XαKY,Z=XαKY,ZKXαY,ZKY,XαZ.

The properties of the difference tensor K gives us,

XαKY,Z=XβKY,Zαβ2KX,KY,Z,
KXαY,Z=KXβY,Zαβ2KKX,Y,Z
and
KY,XαZ=KY,XβZαβ2KY,KX,Z,
then
XαKY,Z=XβKY,Zαβ2KX,KY,ZKXβY,Z+αβ2KKX,Y,ZKY,XβZ+KY,KX,Z
Finally, using the fact that,
XβKY,Z=XβKY,ZKXβY,ZKY,XβZ,
we deduce that,
XαKY,Z=XβKY,Zαβ2KX,KY,Z+αβ2KKX,Y,Z+αβ2KY,KX,Z.
Remark 1.

As particular cases of Eqn (2), we have

XαKY,Z=^XKY,Zα2KX,KY,Z+α2KKX,Y,Z+α2KY,KX,Z=XKY,Zα12KX,KY,Z+α12KKX,Y,Z+α12KY,KX,Z=X*KY,Zα+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,βR.

Theorem 1.

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

(3) RαX,YZ=RβX,YZ+βα2XβKY,Zβα2YβKX,Z+βα24KX,KY,Zβα24KY,KX,Z,

for all X,Y,ZΓTM.

Proof of Theorem 1. Let X,Y,ZΓTM, By definition we have,

(4)RαX,YZ=XαYαZXαYαZX,YαZ.

For the first term XαYαZ, we have,

YαZ=YβZ+βα2KY,Z,
then
XαYαZ=Xα^YZα2XαKY,Z.

It is simple to see that,

XαYβZ=XβYβZ+βα2KX,YβZ
and
XαKY,Z=XβKY,Z+βα2KX,KY,Z,
which gives us
(5)XαYαZ=XβYβZ+βα2KX,YβZ+βα2XβKY,Z+βα24KX,KY,Z.

A similar calculation gives,

(6)YαXαZ=YβXβZ+βα2KY,XβZ+βα2YβKX,Z+βα24KY,KX,Z.

Finally, we have,

(7)X,YαZ=X,YβZ+βα2KX,Y,Z

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

RαX,YZ=RβX,YZ+βα2XβKY,Zβα2YβKX,Z+βα2KX,YβZβα2KY,XβZβα2KX,Y,Z+βα24KX,KY,Zβα24KY,KX,Z

Using the fact that,

XβKY,Z=XβKY,Z+KXβY,Z+KY,XβZ,
YβKX,Z=YβKX,Z+KYβX,Z+KX,YβZ
and
KX,Y,Z=KXβY,ZKYβX,Z,
we get
RαX,YZ=RβX,YZ+βα2XβKY,Zβα2YβKX,Z+βα24KX,KY,Zβα24KY,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,YZα2^XKY,Z+α2^YKX,Z+α24KX,KY,Zα24KY,KX,Z,
(9) RαX,YZ=RX,YZ+1α2XKY,Z1α2YKX,Z+1α24KX,KY,Z1α24KY,KX,Z
and
(10) RαX,YZ=R*X,YZ1+α2X*KY,Z+1+α2Y*KX,Z+1+α24KX,KY,Z1+α24KY,KX,Z,
for all X,Y,ZΓTM.

Remark 2.

From Theorem 1, we can give other relations:

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

    RαX,YZ=RαX,YZ+αRX,YZR*X,YZ.

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

    RαX,YZ=1+α2RX,YZ+1α2R*X,YZ1α24KX,KY,Z+1α24KY,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 XΓTM, then the relation between RicciαX and RicciβX is given by the following formula :
RicciαX=RicciβX+βα24KX,E+βα2TrgXβK,βα2TrgβKX,βα24TrgKKX,,,
where
TrgXβK,=XβKei,ei,
TrgβKX,=eiβKX,ei,
TrgKKX,,=KKX,ei,ei,
and
E=TrgK=Kei,ei.

In particular for β1,0,1, we obtain,

RicciαX=Ricci^X+α24KX,Eα2Trg^XK,+α2Trg^KX,α24TrgKKX,,,
RicciαX=RicciX+1α24KX,E+1α2TrgXK,1α2TrgKX,1α24TrgKKX,,
and
RicciαX=Ricci*X+1+α24KX,E1+α2TrgX*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,yR2,y0,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=yx,e2=yy is an orthonormal frame field. A simple calculation gives
*e1e1=0,*e2e2=2e2,*e1e2=2e1,*e2e1=e1.
Then,
e1αe1=1+αe2,e2αe2=2αe2,e1αe2=1αe1,e2αe1=αe1.
We deduce that,
Rαe1,e2e2=α21e1,Rαe2,e1e1=α21e2,
it follows that,
RicciαX=α21X,RicαX,Y=α21gX,Y,Sgα=2α21.

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+1αYγX+1αXγY1+αgX,Ygradγ,
for all X,Y,ZΓTM. Using the fact that XY=^XY12KX,Y, we obtain,
(13)¯XY=^XY+1αYγX+1αXγY1+αgX,Ygradγ12KX,Y.
Theorem 2.

(14) R¯X,YZ=R^X,YZ1+αgY,Z^Xgradγ+1+αgX,Z^Ygradγ+1+α2gY,ZXγgradγ1+α2gX,ZYγgradγ+1α2YγZγX1α2gY,Zgradγ2X1α2KY,ZγX1αg^Ygradγ,ZX1α2XγZγY+1αg^Xgradγ,ZY+1α2gX,Zgradγ2Y+1α2KX,ZγY12^XKY,Z+12^YKX,Z+1+α2gY,ZKX,gradγ1+α2gX,ZKY,gradγ+14KX,KY,Z14KY,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+1αZγY+1αYγZ1+αgY,Zgradγ12KY,Z,
which gives us,
¯X¯YZ=¯X^YZ+1α¯XZγY+1α¯XYγZ1+α¯XgY,Zgradγ12¯XKY,Z.

Using Eqn (13), we deduce that,

¯X^YZ=^X^YZ+1α^YZγX+1αXγ^YZ1+αgX,^YZgradγ12KX,^YZ,
¯XZγY=Zγ^XY+1αYγZγX+1αXγZγY1+αgX,YZγgradγ12ZγKX,Y+g^Xgradγ,ZY+ggradγ,^XZY,
¯XYγZ=Yγ^XZ+1αYγZγX+1αXγYγZ1+αgX,ZYγgradγ12YγKX,Z+g^Xgradγ,YZ+ggradγ,^XYZ,
¯XgY,Zgradγ=gY,Z^Xgradγ+1αgY,Zgradγ2X2αgY,ZXγgradγ12gY,ZKX,gradγ+g^XY,Zgradγ+gY,^XZgradγ
and
¯XKY,Z=^XKY,Z+1αKY,ZγX+1αXγKY,Z1+αgX,KY,Zgradγ12KX,KY,Z.
It follows that,
(16)¯X¯YZ=^X^YZ1+αgY,Z^Xgradγ+1+α2gX,KY,Zgradγ1α2gX,YZγgradγ1α2gX,ZYγgradγ+2α1+αgY,ZXγgradγ1+αgX,^YZgradγ1+αg^XY,Zgradγ1+αgY,^XZgradγ1α2gY,Zgradγ2X+21α2YγZγX+1α^YZγX+1α2XγZγY+1αg^Xgradγ,ZY+1αggradγ,^XZY+1α2XγYγZ+1αg^Xgradγ,YZ+1αggradγ,^XYZ+1αXγ^YZ12KX,^YZ+1αZγ^XY+1αYγ^XZ1α2YγKX,Z1α2ZγKX,Y+1+α2gY,ZKX,gradγ12^XKY,Z1α2KY,ZγX1α2XγKY,Z+14KX,KY,Z.

A similar calculation gives us,

(17)¯Y¯XZ=^Y^XZ1+αgX,Z^Ygradγ+1+α2gY,KX,Zgradγ1α2gX,YZγgradγ1α2gY,ZXγgradγ+2α1+αgX,ZYγgradγ1+αgY,^XZgradγ1+αg^YX,Zgradγ1+αgX,^YZgradγ1α2gX,Zgradγ2Y+21α2XγZγY+1α^XZγY+1α2YγZγX+1αg^Ygradγ,ZX+1αggradγ,^YZX+1α2XγYγZ+1αg^Xgradγ,YZ+1αggradγ,^YXZ+1αYγ^XZ12KY,^XZ+1αZγ^YX+1αXγ^YZ1α2XγKY,Z1α2ZγKX,Y+1+α2gX,ZKY,gradγ12^YKX,Z1α2KX,ZγY1α2YγKX,Z+14KY,KX,Z.

Finally, it is easy to see that,

(18)¯X,YZ=^X,YZ1+αg^XY,Zgradγ+1+αg^YX,Zgradγ+1α^XYγZ1α^YXγZ+1αZγ^XY1αZγ^YX12K^XY,Z+12K^YX,Z.

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

R¯X,YZ=R^X,YZ1+αgY,Z^Xgradγ+1+αgX,Z^Ygradγ+1+α2gY,ZXγgradγ1+α2gX,ZYγgradγ+1α2YγZγX1α2gY,Zgradγ2X1α2KY,ZγX1αg^Ygradγ,ZX1α2XγZγY+1αg^Xgradγ,ZY+1α2gX,Zgradγ2Y+1α2KX,ZγY12^XKY,Z+12^YKX,Z+1+α2gY,ZKX,gradγ1+α2gX,ZKY,gradγ+14KX,KY,Z14KY,KX,Z.

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

^XY=XY+12KX,Y,
R^X,YZ=RX,YZ+12XKY,Z12YKX,Z+14KX,KY,Z14KY,KX,Z

gives us the following theorem

Theorem 3.

(19) R¯X,YZ=RX,YZ1+αgY,ZXgradγ+1+αgX,ZYgradγ1+α2gX,ZYγgradγ+1+α2gY,ZXγgradγ1α2gY,Zgradγ2X+1α2gX,Zgradγ2Y1αgYgradγ,ZX+1αgXgradγ,ZY+1α2YγZγX1α2XγZγY1αgKY,Z,gradγX+1αgKX,Z,gradγY

Corollary 3.

Let us choose ei1im to be an orthonormal frame on Mm,,g, an orthonormal frame on Mm,¯,g¯=e2γg is given by e¯i=eγei1im. For any X,YΓTM, 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=e2γRicciX+m2α2+2mα+m2e2γXγgradγmα+m2e2γXgradγ+mα22αm+2e2γgradγ2X1αe2γΔ^γX1α2e2γEγX+1αe2γKX,gradγ,
Ric¯X,Y=RicX,Y+m2α2+2mα+m2XγYγ+mα22αm+2gradγ2gX,Y1αΔ^γgX,Ymα+m2gXgradγ,Y1α2EγgX,Y+1αgKX,Y,gradγ
and
Sg¯=e2γSg+m1m+2α2m+2e2γgradγ22m1e2γΔ^γ+m1αe2γEγ

Corollary 4.

Theorem 3 and Corollary 3 gives us two particular cases:

  1. If α = 1, we obtain,

    R¯X,YZ=RX,YZ2gY,ZXgradγ+2gX,ZYgradγ4gX,ZYγgradγ+4gY,ZXγgradγ,
    Ricci¯X=e2γRicciX+4m1Xγgradγ2m1Xgradγ,
    Ric¯X,Y=RicX,Y+4m1XγYγ2m1gXgradγ,Y
    and
    Sg¯=e2γSg+4m1gradγ22m1Δ^γ+m1Eγ.

  2. If α = −1, we obtain,

    R¯X,YZ=RX,YZ2gYgradγ,ZX+2gXgradγ,ZY+4YγZγX4XγZγY2gKY,Z,gradγX+2gKX,Z,gradγY,
    Ricci¯X=e2γRicciX4e2γXγgradγ+2e2γXgradγ+4e2γgradγ2X2e2γΔ^γXe2γEγX+2e2γKX,gradγ,
    Ric¯X,Y=RicX,Y4XγYγ+2gXgradγ,Y+4gradγ2gX,Y2Δ^γgX,YEγgX,Y+2gKX,Y,gradγ
    and
    Sg¯=e2γSg+m1e2γ4gradγ22Δ^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 Sg = −2. We want to determine γ such that Sg¯=0. By Corollary\enleadertwodots , we deduce that Sg¯ vanish if and only if
2Δ^γαEγ4α2gradγ2+2=0.

To solve this equation, we will present two cases :

  1. (1)

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

    γ2α2γ2+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.

  1. (2)

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

    γ+αγ2α2γ2+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.

References

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[2]Uohashi K, Ohara A, Fujii T. 1-Conformally flat statistical submanifolds. Osaka J Math. 2000; 37: 501-7.

[3]Uohashi K. On α-conformal equivalence of statistical submanifolds. J Geom. 2002; 75: 179-84.

[4]Uohashi K. α-Connections and a symmetric cubic form on a riemannian manifold. Entropy (MDPI). 2017; 19: 344.

[5]Min CR, Choe SO, An YH. Statistical immersions between statistical manifolds of constant curvature. Glob J Adv Res Class Mod Geom. 2014; 3: 66.

[6]Okamoto I, Amari S, Takeuchi K. Asymptotic theory of sequential estimation: differential geometrical approach. Ann Statist. 1991; 19: 961-81.

Acknowledgements

The authors would like to thank the referee for some useful comments and their helpful suggestions that have improved the quality of this paper.

Corresponding author

Seddik Ouakkas can be contacted at: seddik.ouakkas@gmail.com and seddik.ouakkas@univ-saida.dz

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