Product of statistical manifolds with a non-diagonal metric - ntmsci

NTMSCI 5, No. 3, 308-321 (2017)

308

New Trends in Mathematical Sciences http://dx.doi.org/10.20852/ntmsci.2017.206

Product of statistical manifolds with a non-diagonal metric Djelloul Djebbouri and Seddik Ouakkas Department of Mathematics, Tahar Moulay University, Saida E-nasr Saida, Algeria Received: 12 December 2015, Accepted: 8 December 2016 Published online: 17 October 2017.

Abstract: In this paper, we generalize the dualistic structures on warped product manifolds to the dualistic structures on generalized warped product manifolds. We have developed an expression of curvature for the connection of the generalized warped product in relation to the corresponding analogues of its base and fiber and warping functions. We show that the dualistic structures on the base M1 and the fiber M2 induce a dualistic structure on the generalized warped product M1 × M2 and that, conversely, (M1 × M1 , G f f ) or 1 2

(M1 × M1 , g˜ f

1 f2

) is a statistical manifold if and only if (M1 , g1 ) and (M1 , g1 ) are. Finally, Some other interesting consequences are also

given. Keywords: Warped product, dualistic structures, statistical manifold.

1 Introduction The warped product provides a way to construct new pseudo-riemannian manifolds from the given ones, see [8],[4] and [3]. This construction has useful applications in general relativity, in the study of cosmological models and black holes. It generalizes the direct product in the class of pseudo-Riemannian manifolds and it is defined as follows. Let (M1 , g1 ) and (M2 , g2 ) be two pseudo-Riemannian manifolds and let f1 : M1 −→ R∗ be a positive smooth function on M1 , the warped product of (M1 , g1 ) and (M2 , g2 ) is the product manifold M1 × M2 equipped with the metric tensor g f1 := π1∗ g1 + ( f ◦ π1 )2 π2∗ g2 , where π1 and π2 are the projections of M1 × M2 onto M1 and M2 respectively. The manifold M1 is called the base of (M1 × M2 , g f1 ) and M2 is called the fiber. The function f1 is called the warping function. The double warped product is a construction in the class of pseudo-Riemannian manifolds generalizing the warped product and the direct product. It is obtained by homothetically distorting the geometry of each base M1 × {q} and each fiber {p} × M2 to get a new ”doubly warped” metric tensor on the product manifold and it is defined as follows; for i ∈ {1, 2}, let Mi be a pseudo-Riemannian manifold equipped with metric gi , and fi : Mi → R∗ be a positive smooth function on Mi . The well-known notion of doubly warped product manifold M1 × f1 f2 M2 is defined as the product manifold M = M1 × M2 equipped with pseudo-Riemannian metric which is denoted by g f1 f2 , given by g f1 f2 = ( f2 ◦ π2 )2 π1∗ g1 + ( f1 ◦ π1)2 π2∗ g2 . The generalized warped product is defined as follows. let c be an arbitrary real number and let gi , (i = 1, 2) be Riemannian metric tensor on Mi . Given a smooth positive function fi on Mi , the generalized warped product of (M1 , g1 ) and (M2 , g2 ) c 2017 BISKA Bilisim Technology

∗ Corresponding

author e-mail: [email protected]

309

D. Djebbouri and S. Ouakkas: Product of statistical manifolds with a non-diagonal metric

is the product manifold M1 × M2 equipped with the metric tensor G f1 f2 (see [6]), explicitly, given by G f1 , f2 (X,Y ) = ( f2v )2 gπ1 (d π1 (X), d π1 (Y )) + ( f1h )2 gπ2 (d π2 (X), d π2 (Y )) + c f1h f2v X( f1h )Y ( f2v ) + X( f2v )Y ( f1h )) . 1

2

for all X,Y ∈ Γ (T M1 × M2 ). When the warping functions f1 = 1 or f2 = 1 or c = 0, we obtain a warped product or direct product. Dualistic structures are closely related to statistical mathematics. They consist of pairs of affine connections on statistical manifolds, compatible with a pseudo-Riemanniann metric [1]. Their importance in statistical physics is underlined by many authors: [5],[2] etc. ∗

Let M be a pseudo-Riemannian manifold equipped with a pseudo-Riemannian metric g and let ∇, ∇ be the affine ∗ connections on M. We say that a pair of affine connections ∇ and ∇ are compatible (or conjugate ) with respect to g if ∗

X(g(Y, Z)) = g(∇X Y, Z) + g(Y, ∇X Z) for all X,Y, Z ∈ Γ (T M),

(1)

∗

where Γ (T M) is the set of all tangent vector fields on M. Then the triplet (g, ∇, ∇ ) is called the dualistic structure on M. We note that the notion of ”conjugate connection ” has been attributed to A.P. Norden in affine differential geometry literarture (Simon, 2000) and has been independently introduced by (Nagaoka and Amari, 1982) in information geometry, where it was called ” dual connection” (Lauritzen, 1987). The triplet (M, ∇, g) is called a statistical manifold if ∗ ∗ it admits another torsion-free connection ∇ satisfying the equation (1). We call ∇ and ∇ duals of each other with respect to g. In the notions of terms on statistical manifolds, for a torsion-free affine connection ∇ and a pseudo-Riemannian metric g on a manifold M, the triple (M, ∇, g) is called a statistical manifold if ∇g is symmetric. If the curvature tensor R of ∇ vanishes, (M, ∇, g) is said to be flat. This paper extends the study of dualistic structures on warped product manifolds, [9], to dualistic structures on generalized warped products in pseudo-Riemannian manifolds. We develop an expression of curvature for the connection of the generalized warped product in relation to those corresponding analogues of its base and fiber and warping functions. The paper is organized as follows. In section 2, we collect the basic material about Levi-Civita connection, the notion of conjugate, horizontal and vertical lifts and the generalized warped products. In section 3, we show that the projection of a dualistic structure defined on a generalized warped product space (M1 × M2 , G f1 f2 ) induces dualistic structures on the base (M1 , g1 ) and the fiber (M2 , g2 ). Conversely, there exists a dualistic structure on the generalized warped product space induced by its base and fiber. In section 4, we show that the projection of a dualistic structure defined on a generalized warped product space (M1 × M2 , g˜ f1 f2 ) induces dualistic structures on the base (M1 , g1 ) and the fiber (M2 , g2 ). Conversely, there exists a dualistic structure on the generalized warped product space induced by its base and fiber and finally.

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2 Preliminaries 2.1 Statistical manifolds We recall some standard facts about Levi-Civita connections and the dual statistical manifold. Many fundamental definitions and results about dualistic structure can be found in Amari’s monograph ([1],[2]). Let (M, g) be a pseudo-Riemannian manifold. The metric g defines the musical isomorphisms ♯g : T ∗ M → T M α 7→ ♯g (α ) such that g(♯g (α ),Y ) = α (Y ), and its inverse ♭g . We can thus define the cometric ge of the metric g by : ge(α , β ) = g(♯g (α ), ♯g (β )).

(2)

ˆ X Y, Z) + g(Y, ∇ ˆ X Z) f or X,Y, Z ∈ Γ (T M) X(g(Y, Z)) = g(∇

(3)

A fundamental theorem of pseudo-Riemannian geometry states that given a pseudo-Riemannian metric g on the tangent bundle T M, there is a unique connection (among the class of torsion-free connection) that ”preserves” the metric; as long as the following condition is satisfied:

ˆ is known as the Levi-Civita connection. Its component forms, called Christoffel symbols, Such a connection, denoted as ∇, are determined by the components of pseudo-metric tensor as (”Christoffel symbols of the second Kink ”)

∂ g jl ∂ gi j ∂g 1 Γî kj = ∑ gkl ( ilj + − l) 2 ∂ x ∂ xi ∂x l and (”Christoffel symbols of the first Kink”) 1 ∂ gik ∂ g jk ∂ gi j Γî j,k = ( j + − k ). 2 ∂x ∂ xi ∂x The Levi-Civita connection is compatible with the pseudo metric, in the sense that it treats tangent vectors of the shortest curves on a manifold as being parallel. It turns out that one can define a kind of ”Compatibility” relation more generally than expressed by equation (3), by introducing the notion of ”Conjugate” (denoted by *) between two affine connections. ∗

∗

Let (M, g) be a pseudo-Riemannian manifold and let ∇, ∇ be an affine connections on M. A connection ∇ is said to be ”conjugate” to ∇ with respect to g if ∗

X(g(Y, Z)) = g(∇X Y, Z) + g(Y, ∇X Z) f or X,Y, Z ∈ Γ (T M) Clearly, ∗ ∗

(∇ ) = ∇. ˆ which satisfies equation (3), is special in the sense that it is self-conjugate Otherwise, ∇, ˆ ∗ = ∇. ˆ (∇) c 2017 BISKA Bilisim Technology

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Because pseudo-metric tensor g provides a one-to-one mapping between vectors in the tangent space and co-vectors in the cotangent space, the equation (1) can also be seen as characterizing how co-vector fields are to be parallel-transported in order to preserve their dual pairing < ., . > with vector fields. Writing out the equation 1 explicitly,

∂ gi j = Γki, j + Γk∗j,i , ∂ xk where

(5)

∇∂i ∂ j = ∑ Γi ∗l j ∂l ∗

l

so that

Γk∗j,i = g(∇∂ j ∂k , ∂i ) = ∑ gil Γk∗lj . ∗

l

∗

In the following part, a manifold M with a pseudo-metric g and a pair of conjugate connections ∇, ∇ with respect to g is ∗ ∗ called a ” pseudo-Riemannian manifold with dualistic structure ” and denoted by (M, g, ∇, ∇ ). Obviously, ∇ and ∇ (or equivalently, Γ and Γ ∗ ) satisfy the relation ˆ = 1 (∇ + ∇∗ ) (or equivalently, Γˆ = 1 (Γ + Γ ∗ )). ∇ 2 2 ∗

Thus an affine connection ∇ on (M, g) is metric if and only if ∇ = ∇ ( that it is self-conjugate). For a torsion-free affine connection ∇ and a pseudo-Riemannian metric g on a manifold M, the triplet (M, ∇, g) is called a statistical manifold if ∇g is symmetric. If the curvature tensor R of ∇ vanishes, (M, ∇, g) is said to be flat. ∗

∗

For a statistical manifold (M, ∇, g), the conjugate connection ∇ with respect to g is torsion-free and ∇ g symmetric. ∗ ∗ Then the triplet (M, ∇ , g) is called the dual statistical manifold of (M, ∇, g) and (∇, ∇ , g) is the dualistic structure on M. ∗ ∗ The curvature tensor of ∇ vanishes if and only if that of ∇ does and in such a case, (∇, ∇ , g) is called the dually flat structure [2]. More generally, in information geometry, a one-parameter family of affine connections ∇(λ ) indexed by λ (λ ∈ R), called λ − connections, is introduced by Amari and Nagaoka in ([1],[2]). ∇(λ ) =

1−λ ∗ 1+λ 1+λ 1−λ ∗ Γ+ Γ ). ∇+ ∇ (or equivalently, Γ (λ ) = 2 2 2 2

(6)

ˆ Obviously, ∇(0) = ∇. ∗

∗

It can be shown that for a pair of conjugate connections ∇, ∇ , their curvature tensors R, R satisfy ∗

g(R(X,Y )Z,W ) + g(Z, R (X,Y )W ) = 0,

(7)

g(R (λ )(X,Y )Z,W ) + g(Z, R ∗(λ )(X,Y )W ) = 0.

(8)

and more generally ∗

If the curvature tensor R of ∇ vanishes, ∇ is said to be flat. So, ∇ is flat if and only if ∇∗ is flat. In this case, (M, g, ∇, ∇ ) is said to be dually falt. When ∇, ∇ is dually flat, then ∇(λ ) is called λ -transitively flat. In such case, (M, g, ∇(λ ) , ∇∗(λ ) ) is called an ”λ -Hessian manifold”, or a manifold with λ -Hessian structure. ∗



312

2.2 Horizontal and vertical lifts Throughout this paper M1 and M2 will be respectively m1 and m2 dimensional manifolds, M1 × M2 the product manifold with the natural product coordinate system and π1 : M1 × M2 → M1 and π2 : M1 × M2 → M2 the usual projection maps. We recall briefly how the calculus on the product manifold M1 × M2 derives from that of M1 and M2 separately. For details see [8]. Let ϕ1 in C∞ (M1 ). The horizontal lift of ϕ1 to M1 × M2 is ϕ1h = ϕ1 ◦ π1 . One can define the horizontal lifts of tangent vectors as follows. Let p1 ∈ M1 and let X p1 ∈ Tp1 M1 . For any p2 ∈ M2 the horizontal lift of X p1 to (p1 , p2 ) is the unique h h h tangent vector X(p ,p ) in T(p1 ,p2 ) (M1 × M2 ) such that d(p1 ,p2 ) π1 (X(p ,p ) ) = X p1 and d(p1 ,p2 ) π2 (X(p ,p ) ) = 0. 1

2

1

2

1

2

We can also define the horizontal lifts of vector fields as follows. Let X1 ∈ Γ (T M1 ). The horizontal lift of X1 to M1 × M2 is the vector field X1h ∈ Γ (T (M1 × M2 )) whose value at each (p1 , p2 ) is the horizontal lift of the tangent vector (X1 )p1 to (p1 , p2 ). For (p1 , p2 ) ∈ M1 × M2 , we will denote the set of the horizontal lifts to (p1 , p2 ) of all the tangent vectors of M1 at p1 by L(p1 , p2 )(M1 ). We will denote the set of the horizontal lifts of all vector fields on M1 by L(M1 ). The vertical lift ϕ2v of a function ϕ2 ∈ C∞ (M2 ) to M1 × M2 and the vertical lift X2v of a vector field X2 ∈ Γ (T M2 ) to M1 × M2 are defined in the same way using the projection π2 . Note that the spaces L(M1 ) of the horizontal lifts and L(M2 ) of the vertical lifts are vector subspaces of Γ (T (M1 × M2 )) but neither is invariant under multiplication by arbitrary functions ϕ ∈ C∞ (M1 × M2 ). Observe that if { ∂∂x , . . . , ∂ x∂m } is the local basis of the vector fields (resp. {dx1 , . . . , dxm1 } is the local basis of 1-forms ) 1

1

relative to a chart (U, Φ ) of M1 and { ∂∂y , . . . , ∂ y∂m } is the local basis of the vector fields (resp. {dy1 , . . . , dym2 } the local 1

2

basis of the 1-forms) relative to a chart (V, Ψ ) of M2 , then {( ∂∂x )h , . . . , ( ∂ x∂m )h , ( ∂∂y )v , . . . , ( ∂ y∂m )v } is the local basis of 1

1

1

2

the vector fields (resp. {(dx1 )h , . . . , (dxm1 )h , (dy1 )v , . . . , (dym2 )v } is the local basis of the 1-forms) relative to the chart (U × V, Φ × Ψ ) of M1 × M2 . The following lemma will be useful later for our computations. Lemma 1. (1) Let ϕi ∈ C∞ (Mi ), Xi ,Yi ∈ Γ (T Mi ) and αi ∈ Γ (T ∗ Mi ), i = 1, 2. Let ϕ = ϕ1h + ϕ2v , X = X1h + X2v and α , β ∈ Γ (T ∗ (M1 × M2 )). Then (i) For all (i, I) ∈ {(1, h), (2, v)}, we have XiI (ϕ ) = Xi (ϕi )I ,

[X,YiI ] = [Xi ,Yi ]I

and

αiI (X) = αi (Xi )I .

(ii) If for all (i, I) ∈ {(1, h), (2, v)} we have α (XiI ) = β (XiI ), then α = β . (2) Let ωi and ηi be r-forms on Mi , i = 1, 2. Let ω = ω1h + ω2v and η = η1h + η2v . We have d ω = (d ω1 )h + (d ω2 )v

and

ω ∧ η = (ω1 ∧ η1 )h + (ω2 ∧ η2 )v .

Proof. See [7]. Remark. Let X be a vector field on M1 ×M2 , such that d π1 (X) = ϕ (X1 ◦ π1 ) and d π2 (X) = φ (X2 ◦ π2 ), then X = ϕ X1h + φ X2v . c 2017 BISKA Bilisim Technology

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2.3 The generalized warped product let ψ : M → N be a smooth map between smooth manifolds and let g be a metric on k-vector bundle (F, PF ) over N. The metric gψ : Γ (ψ −1 F) × Γ (ψ −1 F) → C∞ (M) on the pull-back (ψ −1 F, Pψ −1 F ) over M is defined by gψ (U,V )(p) = gψ (p) (U p ,Vp ), ∀ U,V ∈ Γ (ψ −1 F), p ∈ M. ψ

Given a linear connection ∇N on k-vector bundle (F, PF ) over N, the pull-back connection ∇is the unique linear connection on the pull-back (ψ −1 F, Pψ −1 F ) over M such that N W, ∇X W ◦ ψ = ∇ d ψ (X)

ψ

∀W ∈ Γ (F), ∀X ∈ Γ (T M).

(9)

Further, let U ∈ ψ −1 F and let p ∈ M, X ∈ Γ (T M). Then ψ

e ψ (p)), (∇X U)(p) = (∇N U)(

(10)

dpψ (X p )

e ∈ Γ (F) with U e ◦ ψ = U. where U

i

Now, let πi , i=1,2, be the usual projection of M1 × M2 onto Mi , given a linear connection ∇ on vector bundle T Mi , the πi pull-back connection ∇is the unique linear connection on the pull-back M1 × M2 → πi−1 (T Mi ) such that for each Yi ∈ Γ (T Mi ), X ∈ Γ (T M1 × M2 ) πi i ∇X Yi ◦ πi = ∇ Yi . (11) d πi (X)

Further, let (p1 , p2 ) ∈ M1 × M2 , U ∈ πi−1 (T M) and X ∈ Γ (T M1 × M2 ). Then πi

i

(∇X U)(p1 , p2 ) = ∇dπi (X(p

) (p1 ,p2 ) 1 ,p2 )

e (pi ). U

(12)

Now, let c be an arbitrary real number and let gi , (i = 1, 2) be a Riemannian metric tensor on Mi . Given a smooth positive function fi on Mi , the generalized warped product of (M1 , g1 ) and (M2 , g2 ) is the product manifold M1 × M2 equipped with the metric tensor (see [6]) G f1 , f2 = ( f2v )2 π1∗ g1 + ( f1h )2 π2∗ g2 + c f1h f2v d f1h ⊙ d f2v , Where πi , (i = 1, 2) is the projection of M1 × M2 onto Mi and d f1h ⊙ d f2v = d f1h ⊗ d f2v + d f2v ⊗ d f1h . For all X,Y ∈ Γ (T M1 × M2 ), we have G f1 , f2 (X,Y ) = ( f2v )2 gπ1 (d π1 (X), d π1 (Y )) + ( f1h )2 gπ2 (d π2 (X), d π2 (Y )) + c f1h f2v X( f1h )Y ( f2v ) + X( f2v )Y ( f1h )) . 1

2

The latter is the unique tensor fields such that for any Xi ,Yi ∈ Γ (T Mi ), (i = 1, 2)

g˜ f1 f2 (XiI ,YkK ) =

 J 2 I   ( f3−i ) gi (Xi ,Yi ) ,

if (i, I) = (k, K)

  c f I f K X ( f )I Y ( f )K , otherwise k k i k i i

(13)



314

If either f1 ≡ 1 or f2 ≡ 1 but not both, then we obtain a singly warped product. If both f1 ≡ 1 and f2 ≡ 1, then we have a product manifold. If neither f1 nor f2 is constant and c = 0, then we have a nontrivial doubly warped product. If neither f1 nor f2 is constant and c 6= 0, then we have a nontrivial generalized warped product. Now, Let us assume that (Mi , gi ), (i = 1, 2) is a smooth connected Riemannian manifold. The following proposition provides a necessary and sufficient condition for a symmetric tensor field G f1 , f2 of type (0, 2) of two Riemannian metrics to be a Riemannian metric. Proposition 1. [6] Let (Mi , gi ), (i = 1, 2) be a Riemannian manifold and let fi be a positive smooth function on Mi and c be an arbitrary real number. Then the symmetric tensor field G f1 f2 is a Riemannian metric on M1 × M2 if and only if 0 ≤ c2 g1 (grad f1 , grad f1 )h g2 (grad f2 , grad f2 )v < 1.

(14)

Corollary 1. [6] If the symmetric tensor field G f1 , f2 of type (0, 2) on M1 × M2 is degenerate, then for any i ∈ {1, 2}, gi (grad fi , grad fi ) is positive constant ki in which ki =

1 . c2 k(3−i)

In all what follows, we suppose that f1 and f2 satisfie the inequality (14). Lemma 2.[6] Let X be an arbitrary vector field of M1 × M2 , if there exist ϕi , ψi ∈ C∞ (Mi ) and Xi ,Yi ∈ Γ (T Mi ), (i = 1, 2) such that  h v h h v h   G f1 f2 (X, Z1 ) = G f1 f2 (ϕ2 X1 + ϕ1 X2 , Z1 ), ∀ Zi ∈ Γ (T Mi ),   G (X, Z v ) = hh G (ψ vY h + ψ hY v , Z v ). f1 f2 f1 f2 1 2 2 2 1 2

Then we have,

n n o o X = ϕ2v X1h + ψ1hY2v + c f1h f2v ψ2vY1 ( f1 )h − ϕ2v X1 ( f1 )h grad( f2v ) − c f1h f2v ψ1hY2 ( f2 )v −ϕ1h X2 ( f2 )v grad( f1h ).

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3 Dualistic structure with respect to G f1 f2 i

∗

i ∗

Proposition 2. Let (G f1 f2 , ∇, ∇ ) be a dualistic structure on M1 × M2 . Then there exists an affine connection ∇, ∇ on Mi , i

i ∗

such that (gi , ∇, ∇ ) is a dualistic structure on Mi (i = 1, 2). Proof. Taking the affine connections on Mi , (i = 1, 2).  1 f 1h h h v   (∇X1 Y1 ) ◦ π1 = d π1 (∇X1h Y1 ) + c f2v (∇X1h Y1 )( f2 )(grad f1 ) ◦ π1, ∀ X1 ,Y1 ∈ Γ (T M1 )   1 ∗  fh (∇  Y ) ◦ π = d π (∇∗ Y h ) + c 1v (∇∗ Y h )( f v )(grad f ) ◦ π , X1 1

2

1

1

 (∇X2 Y2 ) ◦ π2 =      2∗  (∇X2 Y2 ) ◦ π2 =

1 1 2 f2 X1h 1 v f d π2 (∇X2v Y2v ) + c f 2h (∇X2v Y2v )( f1h )(grad f2 ) ◦ π2, 1 f 2v ∗ v d π2 (∇X v Y2 ) + c f h (∇∗X v Y2v )( f1h )(grad f2 ) ◦ π2. 2 2 1 X1h 1

∀ X2 ,Y2 ∈ Γ (T M2 )

Therfore, we have for all Xi ,Yi , Zi ∈ Γ (T Mi ) (i = 1, 2). ∗

XiI (G f1 f2 (YiI , ZiI )) = G f1 f2 (∇X I YiI , ZiI ) + G f1 f2 (YiI , ∇X I ZiI ). i


i

(16)

315


J ) = 0 and for any X ∈ Γ (T M × M ), Since, d π (ZiI ) = 0, XiI ( f3−i 1 2 3−i

J J )Zi ( fi )I , g f1 f2 (X, ZiI ) = ( f3−i )2 giπi (d πi (X), Zi ◦ πi ) + c f1h f2v X( f3−i

then the equation (16) is aquivalent to i i J J ( f3−i )2 (Xi (gi (Yi , Zi )))I = ( f3−i )2 gi (∇Xi Yi , Zi ) + gi (Yi , ∇X∗i Zi )}I

where (i, I), (3 − i, J) ∈ {(1, h), (2, v)}. i

i ∗

Hence, the pair of affine connections ∇ and ∇ are conjugate with respect to gi . i ∗

i

Proposition 3. Let (gi , ∇, ∇ ) be a dualistic structure on Mi (i = 1, 2). Then there exists a dualistic structure on M1 × M2 with respect to G f1 f2 . Proof. Let ∇ and ∇∗ be the connections on M1 × M2 given by   d π1 (∇X Y )              d π2 (∇X Y )      

π1

1 = ∇X d π1 (Y ) +Y (ln f 2v )d π1 (X) + X(ln f 2v )d π1 (Y )+ f h f v (1−c 2 bh bv ) 1 2 1 2 h v h v v h −c f1 (1 − cb2 ) X( f1 )Y ( f2 ) + X( f2 )Y ( f1 ) (grad f1 ) ◦ π1 , π2

1 = ∇X d π2 (Y ) +Y (ln f 1h )d π2 (X) + X(ln f 1h )d π2 (Y )+ f h f v (1−c 2 bh bv ) 1 2 1 2 v h h v v h −c f2 (1 − cb1 ) X( f1 )Y ( f2 ) + X( f2 )Y ( f1 ) (grad f2 ) ◦ π2 ,

( f h )2

Bf v (X,Y ) − cbv2 f2v Bf h (X,Y )

( f2v )2

B fh (X,Y ) − cbh1 f1h Bfv (X,Y )

1

f2v

f1h

2

1

2

1

π

1 ( f1h )2 ∗  1 v v ∗ ∗ Y ) = ∇∗ d π (Y ) +Y (ln f v )d π (X) + X(ln f v )d π (Y )+  d (∇ π  X 1 1 1 1 2 2 X f2v B f2v (X,Y ) − cb2 f 2 B f1h (X,Y )  f1h f2v (1−c2 bh1 bv2 )      −c f1h (1 − cbv2 ) X( f1h )Y ( f2v ) + X( f2v )Y ( f1h ) (grad f1 ) ◦ π1 ,   π2  ( f v )2 ∗   1 h h ∗ ∗ ∗ h h  d π2 (∇X Y ) = ∇X d π2 (Y ) +Y (ln f 1 )d π2 (X) + X(ln f 1 )d π2 (Y )+ f h f v (1−c2 bh bv ) f2h B f h (X,Y ) − cb1 f1 B f2v (X,Y )  1 2 1 2 1 1    −c f2v (1 − cbh1 ) X( f1h )Y ( f2v ) + X( f2v )Y ( f1h ) (grad f2 ) ◦ π2 ,

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for any X,Y ∈ Γ (T M1 × M2 ), where BfI and BfI ∗(i = 1, 2) the (0, 2) tensors fields of fiI are given respectively by i

i

n πi o 1 BfI (X,Y ) = c fiI X(Y ( fiI )) − gπi i ∇X d πi (Y ), (grad fi ) ◦ πi + cX( fiI )Y ( fiI ) − J giπi d πi (X), d πi (Y ) , i fj and

n πi o 1 BfI ∗(X,Y ) = c fiI X(Y ( fiI )) − gπi i ∇X∗d πi (Y ), (grad fi ) ◦ πi + cX( fiI )Y ( fiI ) − J gπi i d πi (X), d πi (Y ) , i fj

j = i − 3 and (i, I), ( j, J) ∈ {(1, h), (2, v)}. Or, for any Xi ,Yi ∈ Γ (T Mi ) (i = 1, 2)  1  ∇X h Y1h = (∇X1 Y1 )h + f2v B f1 (X1 ,Y1 )h grad( f2v );   1  2    ∇X2v Y2vh = (∇X2 Y2 )v + f1h B f2 (X2 ,Y2 )v grad( f1h );    1   ∇∗ h Y h = (∇X∗1 Y1 )h + f v B∗ (X1 ,Y1 )h grad( f v ); f1 2 2 X 1 1

2   ∇∗X v Y2v = (∇X∗2 Y2 )v + f1h B∗f2 (X2 ,Y2 )v grad( f1h );  2    ∇ h Y v = ∇∗ Y v = −cX ( f )hY ( f )v f v grad( f h ) + f h grad( f v )} + Y (ln f )v X h + X (ln f )hY v  1 1 2 2 2 1 2  X 1 2 2 1 1 2 2 X1h 2 1 2    ∇ v X h = ∇∗ X h = ∇ Y v . Y2 1 Xh 2 Yv 1

(18)

1

2

Where Bf and Bf ∗ (i = 1, 2) are the (0, 2) tensors fields of fi given respectively by i

i

i Bf (Xi ,Yi ) = c fi Xi (Yi ( fi )) − ∇Xi Yi ( fi ) + cXi ( fi )Yi ( fi ) − gi (Xi ,Yi ), i



and

316

i Bf∗ (Xi ,Yi ) = c fi Xi (Yi ( fi )) − ∇X∗i Yi ( fi ) + cXi ( fi )Yi ( fi ) − gi (Xi ,Yi ), i

i

i

Let us assume that (gi , ∇, ∇∗) is a dualistic structures on Mi , i = 1, 2. Let A be a tensor field of type (0, 3) defined for any X,Y, Z ∈ Γ (T M1 × M2 ) by A(X,Y, Z) = X(G f1 f2 (Y, Z)) − G f1 f2 (∇X Y, Z) − G f1 f2 (Y, ∇∗X Z), if Xi ,Yi , Zi ∈ Γ (T Mi ), i = 1, 2, then we have J XiI (G f1 f2 (YiI , ZiI )) = XiI (( f3−i )2 gi (Xi ,Yi )I ). J ) = 0, and hence Since d π3−i (XiI ) = 0, it follows that d π3−i (XiI )( f3−i = XiI ( f3−i J XiI (G f1 f2 (YiI , ZiI )) = ( f3−i )2 (X(gi (Yi , Zi )))I , i

i

as (gi , ∇, ∇)∗ is dualistic structure, we have thus i

i

J XiI (G f1 f2 (YiI , ZiI )) = ( f3−i )2 {gi (∇Xi Yi , Zi )I + gi (Yi , ∇X∗i Zi )I },

From Equations (13), (18), then it’s easily observed that the following equation holds A(XiI ,YiI , ZiI ) = 0 In the different lifts (i 6= j), we have XiI (G f1 f2 (YiI , Z Jj )) = c f Jj (Z j ( f j ))J Xi (( fi (Y ( fi ))))I , G f1 f2 (∇X J YiI , Z Jj ) = f Jj {c fi Xi (Yi ( fi )) + cXi ( fi )Yi ( fi ) − gi (Xi ,Yi )}I Z j ( f j )J , i

and G f1 f2 (∇∗X I Z Jj ,YiI ) = f Jj gi (Xi ,Yi )I Z j ( f j )J . i

We add these equations and obtain A(XiI ,YiI , Z Jj ) = 0 Hence the same applies for A(X Jj ,YiI , ZiI ) = A(XiI ,Y jJ , ZiI ) = 0. This proves that ∇∗ is conjugate to ∇ with respect to G f1 f2 . 1

2

We recall that the connection ∇ on M1 × M2 induced by ∇ and ∇ on M1 and M2 respectively, is given by Equation (18). 1

2

Proposition 4. (M1 , ∇, g1 ) and (M2 , ∇, g2 ) are statistical manifolds if and only if (M1 × M2 , G f1 f2 , ∇) is a statistical manifold. i

Proof. Let us assume that (Mi , ∇, gi ) (i = 1, 2) is statistical manifold. Firstly, we show that ∇ is torsion-free. Indeed; by Equation (17), we have for any X,Y ∈ Γ (T M1 × M2 ) πi

πi

d πi (T (X,Y )) = ∇X d πi (Y ) − ∇Y d πi(X) − d πi ([X,Y ]) i

Since for i = 1, 2, ∇ is torsion-free, then πi

πi

∇X d πi (Y ) − ∇Y d πi (X) = d πi ([X,Y ])

Therefore, from Remark 2.2, the connection ∇ is torsion-free.


317


Secondly, we show that ∇G f1 , f2 is symmetric. In fact; for i = 1, 2, (∇G f1 f2 )(XiI ,YiI , ZiJ ) = XiI (G f1 f2 (YiI , ZiI )) − G f1 f2 (∇X I YiI , ZiI ) − G f1 f2 (YiI , ∇X I ZiI ) i

i

i

by Equations (13) and (18) and since (∇gi ), i = 1, 2, is symmetric, we have i

J )2 ((∇g )(X ,Y , Z ))I (∇G f1 f2 )(XiI ,YII , ZiI ) = ( f3−i i i i i i

J )2 ((∇g )(Y , X , Z ))h = ( f3−i i i i i = (∇G f1 f2 )(YiI , XII , ZiI ).

In the different lifts, we have J J I (∇G f1 f2 )(XiI ,YiI , Z3−i ) = (∇G f1 f2 )(X3−i ,YiI , ZiI ) = (∇G f1 f2 )(XiI ,Y3−i , ZiI ) = 0,

Therefore, (∇G f1 f2 ) is symmetric. Thus (M1 × M2 , g f1 f2 , ∇) is a statistical manifold. Conversely, if (M1 × M2 , G f1 f2 , ∇) is a statistical manifold, then (∇G f1 f2 ) is symmetric and ∇ is torsion-free, particularly, when Xi ,Yi , Zi ∈ Γ (T Mi ), we have  (∇G

f1 f2

)(XiI ,YII , ZiI ) = (∇G f1 f2 )(YiI , XII , ZiI ), ∀ i = 1, 2,

T (X I ,Y I ) = 0, ∀ i = 1, 2, . i i

i

i

i

Then, by Equations (13) and (18), we obtained, for i = 1, 2, ∇gi , is symmetric and ∇, is torsion-free. Therefore, (Mi , ∇, gi ), i = 1, 2, is a statistical manifold.

4 Dualistic structure with respect to g˜ f1 f2 Let c be an arbitrary real number and let gi , (i = 1, 2) be a Riemannian metric tensors on Mi . Given a smooth positive function fi on Mi , we define a metric tensor field on M1 × M2 by g˜ f1 , f2 = π1∗ g1 + ( f1h )2 π2∗ g2 +

c2 v 2 h ( f ) d f1 ⊙ d f1h , 2 2

(19)

where πi , (i = 1, 2) is the projection of M1 × M2 onto Mi (see [6]). For all X,Y ∈ Γ (T M1 × M2 ), we have g˜ f1 , f2 (X,Y ) = gπ1 (d π1 (X), d π1 (Y )) + ( f1h )2 gπ2 (d π2 (X), d π2 (Y )) + (c f2v )2 X( f1h )Y ( f1h )). 1

2

The latter is the unique tensor fields such that for any Xi ,Yi ∈ Γ (T Mi ), (i = 1, 2)  h h h v 2 h   g˜ f1 f2 (X1 ,Y1 ) = g1 (X1 ,Y1 ) + (c f2 ) X1 ( f1 )Y1 ( f1 ) , g˜ f f (X1h ,Y2v ) = g˜ f1 f2 (Y2v , X1h ) = 0,   12 v v g˜ f1 f2 (X2 ,Y2 ) = ( f1h )2 g2 (X2 ,Y2 )v .

(20)

i

i

Proposition 5. Let (g˜ f1 f2 , ∇, ∇∗ ) be a dualistic structure on M1 × M2 . Then there exists an affine connections ∇, ∇∗on Mi , i

i

such that (gi , ∇, ∇∗) is a dualistic structure on Mi (i = 1, 2). c 2017 BISKA Bilisim Technology


318

Proof. Taking the affine connections on Mi , (i = 1, 2).  1  (∇X Y1 ) ◦ π1 = d π1 (∇ h Y h ) − (c f v )2 H f1h (X h ,Y h )(grad f1 ) ◦ π1, 1 2 1 1 X 1 1

1

(21)

 (∇X∗ Y1 ) ◦ π1 = d π1 (∇∗ h Y h ) − (c f v )2 H ∗ f1h (X h ,Y h )(grad f1 ) ◦ π1, X1 1

1

 2  (∇X2 Y2 ) ◦ π2 = 2 ∗  (∇X2 Y2 ) ◦ π2 =

1

2

1

1 d π2 (∇X2v Y2v ) ( f 1h )2 1

( f 1h )2

(22)

d π2 (∇∗X v Y2v ). 2

Therefore, we have for all Xi ,Yi , Zi ∈ Γ (T Mi ) (i = 1, 2).

XiI (g˜ f1 f2 (YiI , ZiI )) = g˜ f1 f2 (∇X I YiI , ZiI ) + g˜ f1 f2 (YiI , ∇∗X I ZiI ). i

(23)

i

J ) = 0 and for any X ∈ Γ (T M × M ), Since, d π (ZiI ) = 0, XiI ( f3−i 1 2 3−i

 gπ1 (d π (X), Z ◦ π ) + (c f v )2 X( f h )Z ( f )h , if (i, I) = (1, h) i 1 1 1 1 2 1 1 I g˜ f1 f2 (X, Zi ) = ( f h )2 gπ2 (d π2 (X), Z2 ◦ π2), (i, I) = (2, v) 1

2

Substituting from Equations (21) and (22) into Formula (23) we get

 1 1 ∗   (X1 (g1 (Y1 , Z1 )))h = gπ1 1 (∇X1 Y1 , Z1 ◦ π1 ) + g1π1 (∇X1 Z1 ,Y1 ◦ π1 ), π2 2 π2 2 ∗ v h 2 h 2  Z ,Y ◦ (∇ Y , Z ◦ (∇ g ( f ) (X (g (Y , Z ))) = ( f ) ) + g ) , π π  1 X2 2 2 X2 2 2 2 2 2 2 2 2 2 2 1 i ∗

i

Hence, the pair of affine connections ∇ and ∇ are conjugate with respect to gi . i

i ∗

Proposition 6. Let (gi , ∇, ∇) be a dualistic structure on Mi (i = 1, 2). Then there exists a dualistic structure on M1 × M2 with respect to g˜ f1 f2 . Proof. Let ∇ and ∇∗ be the connections on M1 × M2 given by  1 (c f v )2 H f1 (X ,Y )h  ∇X h Y1h = (∇X1 Y1 )h + 21+(c f v )21bh 1 (grad f1 )h − c2 f2v (X1 (ln f1 )Y1 (ln f1 ))h (grad f2 )v ,   1 2 1   2  f h g (X ,Y )v  ∇X2v Y2vh = (∇X2 Y2 )v − 1 2 v2 22 h (grad f1 )h ,  1+(c f 2 ) b1    1 v 2 ∗f h  ∇∗ h Y h = (∇X∗ Y1 )h + (c f2 ) H 1 (X1 ,Y1 ) (grad f1 )h − c2 f v (X1 (ln f1 )Y1 (ln f1 ))h (grad f2 )v , 1 2 X 1 1+(c f v )2 bh 1

2

1

2 f 1h g2 (X2 ,Y2 )v   ∇∗X v Y2v = (∇X∗2 Y2 )v − 1+(c (grad f1 )h ,   f 2v )2 bh1 2   h  c2 f vY2 ( f 2 )v X1 ( f 1 )h   ∇X h Y2v = ∇∗X h Y2v = 21+(c (grad f1 )h + X1 (ln f1 ) Y2v ,  v )2 b h f  1 1 2 1    v h ∇Y2 X1 = ∇Y∗ v X1h = ∇X h Y2v . 2

(24)

1

1

1 ∗

for any Xi ,Yi ∈ Γ (T Mi ) (i = 1, 2) and where H f1 and H ∗ f1 are the Hessian of f1 with respect to ∇ and ∇ respectively. i

i ∗

Let us assume that (gi , ∇, ∇ ) is a dualistic structure on Mi , i = 1, 2. Let A be a tensor field of type (0, 3) defined for any X,Y, Z ∈ Γ (T M1 × M2 ) by ∗

A(X,Y, Z) = X(g˜ f1 f2 (Y, Z)) − g˜ f1 f2 (∇X Y, Z) − g˜ f1 f2 (Y, ∇X Z), c 2017 BISKA Bilisim Technology

319


Since d π3−i (XiI ) = 0, it follows that J XiI ( f3−i ) = d π3−i (XiI )( f3−i ) = 0,

∀(i, I), ( j, J) ∈ {(i, h), (2, v)},

and hence, for all Xi ,Yi , Zi ∈ Γ (T Mi ) (i = 1, 2), we have ( i

i

X1h g˜ f1 f2 (Y1h , Z1h ) = (X1 (g1 (Y1 , Z1 )))h + (c f2v )2 {Y1 ( f1 )X1 (Z1 ( f1 )) + Z1 ( f1 )X1 (Y1 ( f1 ))}h , X2v g˜ f1 f2 (Y2v , Z2v ) = (c f2v )2 (X2 (g2 (Y2 , Z2 )))h .

as (gi , ∇, ∇∗) is a dualistic structure and from Equations (20), (24), then it’s easily seen that the following equation holds A(XiI ,YiI , ZiI ) = 0,

∀(i, I), ( j, J) ∈ {(i, h), (2, v)}.

In the different lifts (i 6= j), we have XiI (g˜ f1 f2 (YiI , Z Jj )) = 0, ( and

g˜ f1 f2 (∇X h Y1h , Z2v ) = −c2 f2v X1 ( f1 )hY1 ( f1 )h Z2 ( f2 )v , 1

g˜ f1 f2 (∇X2v Y2v , Z1h ) = − f1h g2 (X2 ,Y2 )v Z1 ( f1 )h ,

(

∗

g˜ f1 f2 (Y1h , ∇X h Z2v ) = c2 f2v X1 ( f1 )hY1 ( f1 )h Z2 ( f2 )v , ∗

1

g˜ f1 f2 (Y2v , ∇X v Z1h ) = f1h g2 (X2 ,Y2 )v Z1 ( f1 )h , 2

We add these equations and obtain A(XiI ,YiI , Z Jj ) = 0,

∀(i, I), ( j, J) ∈ {(i, h), (2, v)}. ∗

Hence the same applies for A(X jJ ,YiI , ZiI ) = A(XiI ,Y jJ , ZiI ) = 0. This proves that ∇ is conjugate to ∇ with respect to g˜ f1 f2 . 1

2

We recall that the connection ∇ on M1 × M2 induced by ∇ and ∇ on M1 and M2 respectively, is given by Equation (24). 1

2

Proposition 7. (M1 , ∇, g1 ) and (M2 , ∇, g2 ) are statistical manifolds if and only if (M1 × M2 , g˜ f1 f2 , ∇) is a statistical manifold. i

Proof. Let us assume that (Mi , ∇, gi ) (i = 1, 2) is a statistical manifold. Firstly, we show that ∇ is torsion-free. Indeed; by Equation (24), we have for any X,Y ∈ Γ (T M1 × M2 ) πi

πi

d πi (T (X,Y )) = ∇X d πi (Y ) − ∇Y d πi (X) − d πi([X,Y ]) i

Since for i = 1, 2, ∇ is torsion-free, then πi

πi

∇X d πi (Y ) − ∇Y d πi (X) = d πi ([X,Y ])

Therefore, from Remark 2.2, the connection ∇ is torsion-free. Secondly, we show that ∇G f1 , f2 is symmetric. In fact; for (i, I) ∈ {(i, h), (2, v)}, (∇g˜ f1 f2 )(XiI ,YiI , ZiI ) = XiI (g˜ f1 f2 (YiI , ZiI )) − g˜ f1 f2 (∇X I YiI , ZiI ) − g˜ f1 f2 (YiI , ∇X I ZiI ) i

i



320

i

by Equations (20), (24) and since (∇gi ), i = 1, 2, is symmetric, we have (∇g˜ f1 f2 )(XiI ,YiI , ZiI ) = (∇g˜ f1 f2 )(YiI , XiI , ZiI ). In the different lifts, for all (i, I), ( j, J) ∈ {(i, h), (2, v)}, we have J J J (∇g˜ f1 f2 )(XiI ,YiI , Z3−i ) = (∇g˜ f1 f2 )(X3−i ,YiI , ZiI ) = (∇g˜ f1 f2 )(XiI ,Y3−i , ZiI ) = 0.

Therefore, (∇g˜ f1 f2 ) is symmetric. Thus (M1 × M2 , g˜ f1 f2 , ∇) is a statistical manifold. Conversely, if (M1 × M2 , g˜ f1 f2 , ∇) is a statistical manifold, then (∇g˜ f1 f2 ) is symmetric and ∇ is torsion-free, particularly, when Xi ,Yi , Zi ∈ Γ (T Mi ), we have (

(∇g˜ f1 f2 )(XiI ,YII , ZiI ) = (∇g˜ f1 f2 )(YiI , XII , ZiI ), ∀ i = 1, 2, T (XiI ,YiI ) = 0. i

i

i

Then, by Equations (20) and (24), we obtain, for i = 1, 2, ∇gi , is symmetric and ∇, is torsion-free. Therefore, (Mi , ∇, gi ), i = 1, 2, is statistical manifold. 1

2

At first, note that (M1 × M2 , g˜ f1 f2 , ∇) is the statistical manifold induced from (M1 , g1 , ∇) and (M2 , g2 , ∇). 1

2

1

2

Now, let (M1 , ∇, g1 ) and (M2 , ∇, g2 ) be two statistical manifolds and let R, R and R be the curvature tensors with respect 1

2

to ∇, ∇ and ∇ respectively. i i ∗

Proposition 8. Let (Mi , ∇, ∇, gi ), (i = 1, 2) be a connected statistical manifold. Assume that the gradient of fi is parallel 1 ∗

i

with respect to ∇and ∇(i = 1, 2). Then for any Xi ,Yi , Zi ∈ Γ (T Mi ) (i = 1, 2) we have 1 (1) R(X1 h ,Y1 h )Z1 h= (R (X1 ,Y1 )Z1 )h , v v 2 c2 f 1h f 2v b1 (2) R(X2 v ,Y2 v )Z2 v= (R (X2 ,Y2 )Z2 )v − 1+(cbf1v )2 b (X2 ∧g2 Y2 )Z2 + (grad f1 )h , 2 ((X2 ∧g2 Y2 )Z2 ) ( f 2 ) v 2 1 2 (1+(c f2 ) b1 ) (3) R(X1 h ,Y1 h )Z2 v= 0, (4) R(X1 h ,Y2 v )Z1 h=

c2 X1 (ln f 1 )h Z1 (ln f 1 )hY2 ( f 2 )v (grad f2 )v , 1+(c f 2v )2 b1

where the wedge product (X2 ∧g2 Y2 )Z2 = g2 (Y2 , Z2 )X2 − g2 (X2 , Z2 )Y2 . Proof. After long and straightforward calculations, as in proof of proposal (2), and where it uses the fact that connections are compatible with the metric, we obtain the same results as in (2), knowing we use only the connections are symmetrical. i 1 ∗

Corollary 2. Let (Mi , ∇, ∇, gi ), (i = 1, 2) be a connected statistical manifold. Assume that f1 is a non-constant positive function and c 6= 0. 1 1 ∗

∗

2 2 ∗

If (∇, ∇ , g˜ f1 f2 ) is a dually flat structure then (∇, ∇, g1 ) is also dually flat and (∇, ∇, g2 ) has a constant sectional curvature. ∗

Proof. Let (∇, ∇ , g˜ f1 f2 ) be a dually flat structure. By Proposition 8, for any X1 ,Y1 , Z1 ∈ Γ (T M1 ), we have 1

R(X1 ,Y1 )Z1 = 0, 1

From Equation (7), Since (M1 , ∇, g1 ) (i = 1, 2) is a statistical manifold, we have 1∗

R (X1 ,Y1 )Z1 = 0. c 2017 BISKA Bilisim Technology

321


1 1 ∗

Hence (M1 , ∇, ∇, g1 ) is dually flat. By 4. of Proposition 8, for any X1 , Z1 ∈ Γ (T M1 ) and Y2 ∈ Γ (T M2 ) , we have c2 X1 (ln f1 )h Z1 (ln f1 )hY2 ( f2 )v (grad f2 )v = 0. 1 + (c f2v )2 b1 So f2 is a constant function since f1 is a non-constant function and M2 is assumed to be connected. Moreover, by 2. of Proposition 8, for any X2 ,Y2 , Z2 ∈ Γ (T M2 ), we have 2

R(X2 ,Y2 )Z2 =

v b1 (X2 ∧g2 Y2 )Z2 , 1 + (c f2v )2 b1 2 2 ∗

Since b1 and f2 are constants, it follows from the previous equality that (∇, ∇, g2 ) has a constant sectional curvature b1 . 1+(c f v )2 b 2

1

Competing interests The authors declare that they have no competing interests.

Authors’ contributions All authors have contributed to all parts of the article. All authors read and approved the final manuscript.

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