Almost cosymplectic statistical manifolds
aa r X i v : . [ m a t h . DG ] J a n ALMOST COSYMPLECTIC STATISTICAL MANIFOLDS
AZIZ YAZLA , ˙IREM K ¨UPELI ERKEN , AND CENGIZHAN MURATHAN Abstract.
This paper is a study of almost contact statistical manifolds. Especiallythis study is focused on almost cosymplectic statistical manifolds. We obtainedbasic properties of such manifolds. It is proved a characterization theorem and acorollary for the almost cosymplectic statistical manifold with Kaehler leaves. We alsostudy curvature properties of an almost cosymplectic statistical manifold. Moreover,examples are constructed. I ntroduction Let (
M, g ) be a Riemannian manifold and ∇ be an affine connection on M . An affineconnection ∇ ∗ is called a conjugate (dual) connection of ∇ if(1.1) Zg ( X, Y ) = g ( ∇ Z X, Y ) + g ( X, ∇ ∗ Z Y )for any X, Y, Z ∈ Γ( M ) . In this situation ∇ g is symmetric. The triple ( g, ∇ , ∇ ∗ ) iscalled a dualistic structure on M and the quadruplet ( M, g , ∇ , ∇ ∗ ) is called statisticalmanifold. A. P. Norden introduced these connections to affine differential geometry andthen U. Simon gave an excellent survey concerning the notion of “conjugate connection”( [18]). The notion of conjugate connection was first initiated into statistics by Amari[1]. In his studies involves statistical problems getting some solutions and developed byLauritzen [12]. If ∇ coincides with ∇ ∗ then statistical manifold simply reduces to usualRiemannian manifold. Clearly, ( ∇ ∗ ) ∗ = ∇ . In a sense, duality is involutive. One canalso show that 2 ∇ = ∇ + ∇ ∗ , where ∇ is Riemannian connection with respect to g . In[11], T.Kurose studied affine immersions of statistical manifolds into the affine space andnoticed that there is a close relationship between the geometry of statistical manifoldsand affine geometry. Otherhand Lagrangian submanifolds of complex space forms are alsonaturally endowed with statistical structures (see [17] pp. 34). So statistical manifoldsplay an important role in differential geometry.Recently, H.Furuhuta [6] defined and studied the holomorphic statistical manifoldwhich can be considered as a Kaehler manifold with a certain connection. Then holomor-phic statistical manifold notion is expanded to the statistical counterparts of a Sasakianmanifold and a Kenmotsu manifold by [7], [8] and [13]. Other hand K. Takano [19],[20]defined Kaehler-like and Sasaki-like statistical manifolds which are considered settingsuitable complex structures and suitable contact structures on statistical manifolds.These studies motivate us to study on almost complex statistical and almost contactstatistical manifolds. Especially our main purpose here is to extend these results toalmost cosymplectic statistical manifolds. Mathematics Subject Classification.
Primary 53B30, 53C15, 53C25; Secondary 53D10.
Key words and phrases.
Almost contact manifold, statistical manifold, conjugate conenction Kaehlerstatistical manifold, Sasakian statistical manifold, Kenmotsu statistical manifold. , ˙IREM K¨UPELI ERKEN , AND CENGIZHAN MURATHAN In the present paper, we are interested in almost Hermitian statistical manifolds andalmost contact statistical manifolds which include Kaehler, Sasakian, Kenmotsu andcosymplectic statistical manifolds. The paper is organized as follows. In Section 2, weprovide a brief the notions of statistical manifolds and almost contact manifolds. InSection 3, we define almost Hermitian statistical manifolds and study almost Kaehlerstatistical manifolds. In Section 4, we introduce almost contact statistical manifolds andprovide a few basic equalities. In Section 5, we study almost cosymplectic statisticalmanifolds and give a characterization, almost Hermitian manifolds and almost contactmanifolds. In Section 6, we give also a characterization of almost cosymplectic statisticalmanifolds with Kaehler statistical leaves and provide examples on almost cosymplecticstatistical manifolds. In the last section, we study curvature properties of an almostcosymplectic statistical manifold. 2.
Preliminaries
For a statistical manifold (
M, g , ∇ , ∇ ∗ ) the difference (1 ,
2) tensor K of a torsion freeaffine connection ∇ and Levi-Civita connection ∇ is defined as(2.1) K X Y = K ( X, Y ) = ∇ X Y − ∇ X Y. Because of ∇ and ∇ are torsion free, we have(2.2) K X Y = K Y X, g ( K X Y, Z ) = g ( Y, K X Z )for any X, Y, Z ∈ Γ( T M ). By (1.1) and (2.1), one can obtain(2.3) K X Y = ∇ X Y − ∇ ∗ X Y. Using (2.1) and (2.3), we find(2.4) 2 K X Y = ∇ X Y − ∇ ∗ X Y. By (2.1), we have(2.5) g ( ∇ X Y, Z ) = g ( K X Y, Z ) + g ( ∇ X Y, Z ) . An almost Hermitian manifold ( N n , g, J ) is a smooth manifold endowed with an almostcomplex structure J and a Riemannian metric g compatible in the sense J X = − X, g ( JX, Y ) = − g ( X, JY )for any
X, Y ∈ Γ( T N ) . The fundamental 2-form Ω of an almost Hermitian manifold isdefined by Ω(
X, Y ) = g ( JX, Y )for any vector fields
X, Y on N . An almost Hermitian manifold is called an almostKaehler manifold if its fundamental form Ω is closed, that is, d Ω = 0 . If Nijenhuistorsion of J satisfies N J ( X, Y ) = [
X, Y ] − [ JX, JY ] + J [ X, JY ] + J [ JX, Y ] = 0then ( N n , g, J ) is called Kaehler manifold. It is also well known that an almost Hermitianmanifold ( M, J, g ) is Kaehler if and only if its almost complex structure J is parallel withrespect to the Levi-Civita connection ∇ , that is, ∇ J = 0 ([10]).Let M be a (2 n + 1)-dimensional differentiable manifold and φ is a (1 ,
1) tensor field, ξ is a vector field and η is a one-form on M. If φ = − Id + η ⊗ ξ, η ( ξ ) = 1 then ( φ, ξ, η )is called an almost contact structure on M . The manifold M is said to be an almostcontact manifold if it is endowed with an almost contact structure [2]. LMOST COSYMPLECTIC STATISTICAL MANIFOLDS 3
Let M be an almost contact manifold. M will be called an almost contact metricmanifold if it is additionally endowed with a Riemannian metric g , i.e.(2.6) g ( φX, φY ) = g ( X, Y ) − η ( X ) η ( Y ) . For such manifold, we have(2.7) η ( X ) = g ( X, ξ ) , φ ( ξ ) = 0 , η ◦ φ = 0 . Moreover, we can define a skew-symmetric tensor field (a 2-form) Φ by(2.8) Φ(
X, Y ) = g ( φX, Y ) , usually called fundamental form.On an almost contact manifold, the (1 , N (1) is defined by N (1) ( X, Y ) = [ φ, φ ] (
X, Y ) − dη ( X, Y ) ξ, where [ φ, φ ] is the Nijenhuis torsion of φ [ φ, φ ] ( X, Y ) = φ [ X, Y ] + [ φX, φY ] − φ [ φX, Y ] − φ [ X, φY ] . If N (1) vanishes identically, then the almost contact manifold (structure) is said to benormal [2]. The normality condition says that the almost complex structure J definedon M × R J ( X, λ ddt ) = ( φX + λξ, η ( X ) ddt ) , is integrable.An almost contact metric manifold M n +1 , with a structure ( φ, ξ, η, g ) is said to bean almost cosymplectic manifold, if(2.9) dη = 0 , d Φ = 0 . If additionally normality conditon is fulfilled, then manifold is called cosymplectic.On the other hand, Kenmotsu studied in [9] another class of almost contact manifolds,defined by the following conditions on the associated almost contact structure(2.10) dη = 0 , d Φ = 2 η ∧ Φ . A normal almost Kenmotsu manifold is said to be a Kenmotsu manifold.When(2.11) dη = Φan almost contact manifold is called a contact metric manifold [2]. A contact metricmanifold M n +1 is a Sasakian manifold if the structure is normal.3. Dualistic structure on almost Hermitian manifolds
Definition 1.
Let ( N n , g, ∇ , ∇ ∗ ) be a statistical manifold. If ( N n , g, J ) is an almostHermitian manifold then ( N n , g, J, ∇ , ∇ ∗ ) is called almost Hermitian statistical man-ifold. If ( N n , g, J ) is an (almost) Kaehler manifold then ( N n , g, J, ∇ , ∇ ∗ ) is called(almost) Kaehler statistical manifold. After some calculations one can easily get following.
Lemma 1 ( [14]) . Let ( N n , g, ∇ , ∇ ∗ ) be an almost Hermitian statistical manifold. Thenthe following equation (3.1) g (( ∇ X J ) Y, Z ) = − g ( Y, ( ∇ ∗ X J ) Z ) holds for any X, Y, Z ∈ Γ( T M ) . AZIZ YAZLA , ˙IREM K¨UPELI ERKEN , AND CENGIZHAN MURATHAN Using (2.1) and (2.2), we easily find the next result.
Lemma 2.
Let ( N n , g, ∇ , ∇ ∗ ) be an almost Hermitian statistical manifold. Then ( ∇ X J ) Y = ( ∇ X J ) Y + ( K X J ) Y, (3.2) ( ∇ ∗ X J ) Y = ( ∇ X J ) Y − ( K X J ) Y. (3.3) for any X, Y ∈ Γ( T M ) . Lemma 3.
For an almost Hermitian statistical manifold we have (3.4) ( ∇ X Ω)(
Y, Z ) = g (( ∇ X J ) Y, Z ) − g ( K X JY, Z ) , and (3.5) ( ∇ ∗ X Ω)(
Y, Z ) = g (( ∇ ∗ X J ) Y, Z ) + 2 g ( K X JY, Z ) for any X, Y, Z ∈ Γ( T M ) . Proof.
According to a vector field, the derivative of 2-form Φ can be written as( ∇ X Ω)(
Y, Z ) = X Ω( Y, Z ) − Ω( ∇ X Y, Z ) − Ω( Y, ∇ X Z ) . By (1.1), (2.1) and (2.2), we obtain( ∇ X Ω)(
Y, Z ) = Xg ( JY, Z ) − g ( J ∇ X Y, Z ) − g ( JY, ∇ X Z )= g ( ∇ X JY, Z ) + g ( JY, ∇ ∗ X Z ) − g ( J ∇ X Y, Z ) − g ( JY, ∇ X Z )= g (( ∇ X J ) Y, Z ) − g ( K X JY, Z ) . This leads to (3.4). If we similarly calculate the derivative of 2-form Ω respect to conju-gate connection ∇ ∗ , we get (3.5) . (cid:3) By Lemma 2 and the relations (2.1) and (2.2) we easily prove following corollary.
Corollary 1.
For an almost Hermitian statistical manifold we have (3.6) ( ∇ X Ω)(
Y, Z ) = ( ∇ X Ω)(
Y, Z ) − g ( K X JY + J K X Y, Z ) and (3.7) ( ∇ ∗ X Ω)(
Y, Z ) = ( ∇ X Ω)(
Y, Z ) + g ( K X JY + J K X Y, Z ) for any X, Y, Z ∈ Γ( T M ) . Theorem 1.
Let ( N n , g, , J, ∇ , ∇ ∗ ) be an almost Hermitian statistical manifold. Thecovariant derivatives ∇ J, ∇ ∗ J of J with respect to the torsion free connections ∇ and ∇ ∗ are given by g (( ∇ X J ) Y, Z ) = 2 g (( K X J ) Y, Z )(3.8) +3 d Ω( X, Y, Z ) − d Ω( X, JY, JZ ) + g ( N J ( Y, Z ) , JX ) , g (( ∇ ∗ X J ) Y, Z ) = − g (( K X J ) Y, Z )(3.9) +3 d Ω( X, Y, Z ) − d Ω( X, JY, JZ ) + g ( N J ( Y, Z ) , JX ) for any X, Y, Z ∈ Γ( T M ) . Proof.
It is well known that the covariant derivative ∇ J satisfies2 g (( ∇ X J ) Y, Z ) = 3 d Ω( X, Y, Z ) − d Ω( X, JY, JZ ) + g ( N J ( Y, Z ) , JX ) , for any X, Y, Z ∈ Γ( T M ) . If we notice the relations (3.2) and (3.3) we reach to ourequations. (cid:3)
LMOST COSYMPLECTIC STATISTICAL MANIFOLDS 5
Corollary 2.
Let ( N n , g, J, ∇ , ∇ ∗ ) be an almost Kaehler statistical manifold. Then g (( ∇ X J ) Y, Z ) = 2 g (( K X J ) Y, Z ) + g ( N J ( Y, Z ) , JX )(3.10) 2 g (( ∇ ∗ X J ) Y, Z ) = − g (( K X J ) Y, Z ) + g ( N J ( Y, Z ) , JX )(3.11) for any X, Y, Z ∈ Γ( T M ) . By (2.2) we can give following.
Proposition 1.
Let ( M n , g, ∇ , ∇ ∗ ) be a statistical manifold and ψ be a skew symmetric (1 , tensor field on M . Then we have (3.12) g ( K X ψY + ψ K X Y, Z ) + g ( K Z ψX + ψ K Z X, Y ) + g ( K Y ψZ + ψ K Y Z, X ) = 0 for any
X, Y, Z ∈ Γ( T M ) . Corollary 3.
Let ( N n , g, J, ∇ , ∇ ∗ ) be an almost Kaehler statistical manifold. Then ( ∇ X Ω)(
Y, Z ) + ( ∇ Z Ω)(
X, Y ) + ( ∇ Y Ω)(
Z, X ) = 0 , (3.13) ( ∇ ∗ X Ω)(
Y, Z ) + ( ∇ ∗ Z Ω)(
X, Y ) + ( ∇ ∗ Y Ω)(
Z, X ) = 0(3.14) for any
X, Y, Z ∈ Γ( T M ) .Proof. Since d Ω = 0 we have(3.15) ( ∇ X Ω)(
Y, Z ) + ( ∇ Z Ω)(
X, Y ) + ( ∇ Y Ω)(
Z, X ) = 0 . Using (3.2), Corollary 1 and Proposition 1 we have requested equations. (cid:3)
Corollary 4.
Let ( N n , g, J, ∇ , ∇ ∗ ) be a Kaehler statistical manifold. Then g (( ∇ X J ) Y, Z ) = g (( K X J ) Y, Z )(3.16) g (( ∇ ∗ X J ) Y, Z ) = − g (( K X J ) Y, Z )(3.17) for any
X, Y, Z ∈ Γ( T M ) . Definition 2 ([6],[7]) . Let ( M n , g, ∇ , ∇ ∗ ) be statistical manifold. A -form ω on M n isdefined by (3.18) ω ( X, Y ) = g ( ψX, Y ) where ψ is skew symmetric (1 , tensor field on M. If K X ψY + ψ K X Y = 0 for any X, Y ∈ Γ( T M ) . then ( M n , g, ∇ , ∇ ∗ ) is called holomorphic statistical manifold. From Lemma 3 and Corollary 4 we have following.
Corollary 5 ([5]) . Let ( N n , g, J, ∇ , ∇ ∗ ) be a Kaehler statistical manifold. Then thefollowing three equations are equivalent:1) ∇ Ω = 0 , N n holomorphic statistical manifold,3) ∇ ∗ Ω = 0 . Statistical Almost Contact Metric Manifolds
Definition 3.
Let ( M n +1 , g, ∇ , ∇ ∗ ) be a statistical manifold. If M n +1 is an almostcontact metric manifold then M n +1 is called almost contact metric statistical manifold. Using anti-symmetry property of φ and the equation (1.1) we have AZIZ YAZLA , ˙IREM K¨UPELI ERKEN , AND CENGIZHAN MURATHAN Lemma 4.
Let ( M n +1 , g, ∇ , ∇ ∗ ) be an almost contact metric statistical manifold. Thenthe following equation (4.1) g (( ∇ X φ ) Y, Z ) = − g ( Y, ( ∇ ∗ X φ ) Z ) holds for any X, Y, Z ∈ Γ( T M ) . Using (2.1) and (2.2), we easily find the next result.
Lemma 5.
Let ( M n +1 , g, ∇ , ∇ ∗ ) be an almost contact statistical manifold. Then ( ∇ X φ ) Y = ( ∇ X φ ) Y + ( K X φ ) Y, (4.2) ( ∇ ∗ X φ ) Y = ( ∇ X φ ) Y − ( K X φ ) Y. (4.3) for any X, Y ∈ Γ( T M ) . Lemma 6.
For an almost contact statistical manifold we have (4.4) ( ∇ X Φ)(
Y, Z ) = g (( ∇ X φ ) Y, Z ) − g ( K X φY, Z ) , and (4.5) ( ∇ ∗ X Φ)(
Y, Z ) = g (( ∇ ∗ X φ ) Y, Z ) + 2 g ( K X φY, Z ) for any X, Y, Z ∈ Γ( T M ) . By Lemma 5 and the relations (2.1) and (2.2) we easily prove the following corollary.
Corollary 6.
For an almost contact metric statistical manifold we have (4.6) ( ∇ X Φ)(
Y, Z ) = ( ∇ X Φ)(
Y, Z ) − g ( K X φY + φ K X Y, Z ) and (4.7) ( ∇ ∗ X Φ)(
Y, Z ) = ( ∇ X Φ)(
Y, Z ) + g ( K X φY + φ K X Y, Z ) for any X, Y, Z ∈ Γ( T M ) . By (4.6) and (4.7) we have
Corollary 7 ([7]) . For an almost contact metric statistical manifold we have (4.8) ( ∇ X Φ)(
Y, Z ) − ( ∇ ∗ X Φ)(
Y, Z ) = − g ( K X φY + φ K X Y, Z ) for any X, Y, Z ∈ Γ( T M ) . For an almost contact metric manifold, the covariant derivative with respect to Rie-mannian connection ∇ is given by2 g (( ∇ φ ) Y, Z ) = 3 d Φ( X, Y, Z ) − d Φ( X, φY, φZ ) + g ( N (1) ( X, Y ) , φZ )+(( L φX η )( Y ) − ( L φY η )( X )) η ( X )(4.9) +2 dη ( φY, X ) η ( Z ) − dη ( φZ, X ) η ( Y ) . (see [2]).Using (4.2), (4.3) and (4.9), we have Theorem 2.
Let ( M n +1 , g, φ, ∇ , ∇ ∗ ) be an almost contact metric statistical manifold.The covariant derivatives ∇ φ, ∇ ∗ φ of J with respect to the torsion free connections ∇ and ∇ ∗ are given by g (( ∇ X φ ) Y, Z ) = 2 g (( K X φ ) Y, Z )+3 d Φ( X, Y, Z ) − d Φ( X, φY, φZ ) + g ( N (1) ( X, Y ) , φZ )+(( L φX η )( Y ) − ( L φY η )( X )) η ( X )(4.10) +2 dη ( φY, X ) η ( Z ) − dη ( φZ, X ) η ( Y ) , LMOST COSYMPLECTIC STATISTICAL MANIFOLDS 7 g (( ∇ ∗ X φ ) Y, Z ) = − g ( K X φ ) Y, Z )3 d Φ( X, Y, Z ) − d Φ( X, φY, φZ ) + g ( N (1) ( X, Y ) , φZ )+(( L φX η )( Y ) − ( L φY η )( X )) η ( X )(4.11) +2 dη ( φY, X ) η ( Z ) − dη ( φZ, X ) η ( Y ) for any X, Y, Z ∈ Γ( T M ) . Almost Cosymplectic Statistical Manifolds
For an almost cosymplectic statistical manifold we define the (1 , A , A ∗ and A by(5.1) A X = −∇ X ξ, A ∗ X = −∇ ∗ X ξ and A X = −∇ X ξ, ∀ X ∈ Γ( T M ) . Since 2 ∇ = ∇ + ∇ ∗ , by (5.1), we obtain(5.2) 2 A = A + A ∗ . Proposition 2.
For an almost cosymplectic statistical manifold we have i ) L ξ η = 0 , ii ) g ( A X, Y ) = g ( X, A Y ) ,iii ) g ( A ∗ X, Y ) = g ( X, A ∗ Y ) , iv ) A ξ = −A ∗ ξ = K ξ ξ,v ) ( ∇ ξ φ ) X = φ A X + A ∗ φX, vi )( ∇ ∗ ξ φ ) X = φ A ∗ X + A φX,vii ) A φ + φ A = − ( A ∗ φ + φ A ∗ ) where L indicates the operator of the Lie differentiation, X, Y are arbitrary vector fieldson M. Proof.
Using Cartan magic formula L ξ η = di ξ ( η ) + i ξ d ( η ) , and dη = 0 , i ξ η = 1 we obtain i ). Again noting η is closed and using (1.1) we have0 = 2 dη ( X, Y ) = Xη ( Y ) − Y η ( X ) − η ([ X, Y ))= − g ( Y, A ∗ X ) + g ( X, A ∗ Y )and by help of similar calculations respect to torsion free affine connection ∇ , we obtain g ( A X, Y ) = g ( X, A Y ) . So we get ii ) and iii ) . From (2.1) and (2.2) we get iv ) . Cartan magic formula(5.3) L V Ω = di V (Ω) + i V d (Ω)is valid for any form Ω ∈ ∧ ( M ) and V ∈ Γ( T M ) . Let us apply (5.3) to the (2.8). Since i ξ (Φ) = 0 and d Φ = 0, we have(5.4) L ξ Φ = 0 . AZIZ YAZLA , ˙IREM K¨UPELI ERKEN , AND CENGIZHAN MURATHAN On the other hand, the Lie derivative of 2-form Φ with respect to characteristic vectorfield ξ can be expressed as( L ξ Φ)(
X, Y ) = L ξ Φ( X, Y ) − Φ( L ξ X, Y ) − Φ( X, L ξ Y )= ξg ( φX, Y ) − g ( φ ∇ ξ X, Y ) + g ( φ ∇ X ξ, Y ) − g ( φX, ∇ ξ Y ) + g ( φX, ∇ Y ξ )= g ( ∇ ξ φX, Y ) + g ( φX, ∇ ∗ ξ Y ) − g ( φ ∇ ξ X, Y ) + g ( φ ∇ X ξ, Y ) − g ( φX, ∇ ξ Y ) + g ( φX, ∇ Y ξ )= g (( ∇ ξ φ ) X, Y ) − g ( φ A X, Y ) − g ( φX, A Y ) + g ( φX, ( A − A ∗ ) Y )= g (( ∇ ξ φ ) X, Y ) − g ( φ A X, Y ) − g ( A ∗ φX, Y ) . We thus obtain ( ∇ ξ φ ) X = φ A X + A ∗ φX, According to conjugate connection ∇ ∗ weconclude that ( ∇ ∗ ξ φ ) X = φ A ∗ X + A φX. So we have v ) and vi ) . We know that A φ + φ A = 0 is valid for almost cosymplectic manifold. Thus, by (5.2) we get vii ) . (cid:3) Remark 1.
By Proposition 2 , we say that A ξ = 0 if and only if A ∗ ξ = 0 for an almostcosymplectic statistical manifold. Proposition 3.
Let ( M n +1 , g, ∇ , ∇ ∗ ) be an almost cosymplectic statistical manifold.Then ( ∇ X Φ)(
Y, Z ) + ( ∇ Z Φ)(
X, Y ) + ( ∇ Y Φ)(
Z, X ) = 0 , (5.5) ( ∇ ∗ X Φ)(
Y, Z ) + ( ∇ ∗ Z Φ)(
X, Y ) + ( ∇ ∗ Y Φ)(
Z, X ) = 0(5.6) for any
X, Y, Z ∈ Γ( T M ) .Proof. Since M n +1 is almost cosymplectic the relation(5.7) ( ∇ X Φ)(
Y, Z ) + ( ∇ Z Φ)(
X, Y ) + ( ∇ Y Φ)(
Z, X ) = 0holds. If we insert the relation (4.6) into above expression, we find0 = ( ∇ X Φ)(
Y, Z ) + ( ∇ Y Φ)(
Z, X ) + ( ∇ Z Φ)(
X, Y )+ g ( K X φY + φ K X Y, Z ) + g ( K Y φZ + φ K X Z, X )(5.8) + g ( K Z φX + φ K Z X, Y ) . If we make use of the relation (3.12), we have(5.9) 0 = ( ∇ X Φ)(
Y, Z ) + ( ∇ Y Φ)(
Z, X ) + ( ∇ Z Φ)(
X, Y ) . Employing (4.7) into (5.7) and using (5.9), we can easily verify that the equality0 = ( ∇ ∗ X Φ)(
Y, Z ) + ( ∇ ∗ Z Φ)(
X, Y ) + ( ∇ ∗ Y Φ)(
Z, X )holds. (cid:3)
Proposition 4.
For an almost cosymplectic statistical manifold we have i )( L ξ g )( X, Y ) = − g ( A X, Y ) = − g (( A + A ∗ ) X, Y ) ,ii ) ( ∇ X η ) Y = ( ∇ X η ) Y, iii ) ( ∇ ∗ X η ) Y = ( ∇ ∗ X η ) Y. LMOST COSYMPLECTIC STATISTICAL MANIFOLDS 9
Proof.
By direct calculations and using (1.1), we find( L ξ g )( X, Y ) = ξg ( X, Y ) − g ([ ξ, X ] , Y ) − g ( X, [ ξ, Y ])= g ( ∇ ξ X, Y ) + g ( X, ∇ ∗ ξ Y ) − g ( ∇ ξ X, Y ) − g ( A X, Y ) − g ( X, ∇ ξ Y ) − g ( X, A Y ) . Due to the symmetry of the operator A , we get( L ξ g )( X, Y ) = g ( X, ∇ ∗ ξ Y − ∇ ξ Y ) − g ( A X, Y ) . In view of (2.2), we obtain( L ξ g )( X, Y ) = − g ( X, K ξ Y ) − g ( A X, Y )= − g ( X, K Y ξ ) − g ( A X, Y )= − g ( X, A Y ) . It is clear that one has( ∇ X η ) Y = − g ( Y, A ∗ X ) = − g ( A ∗ Y, X ) = ( ∇ Y η ) X which completes ii ) . Since dη = 0 , one can easily get(5.10) ( ∇ X η ) Y = ( ∇ X η ) Y. From ii ) and (5.10), we find that ( ∇ ∗ X η ) Y = ( ∇ ∗ X η ) Y. (cid:3) Proposition 5.
For an almost cosymplectic statistical manifold we have ( ∇ X Φ)(
Y, φZ ) + ( ∇ ∗ X Φ)(
Z, φY ) = η ( Y ) g ( A X, Z ) + η ( Z ) g ( A ∗ X, Y )(5.11) ( ∇ ∗ X Φ)( φZ, φY ) − ( ∇ X Φ)(
Y, Z ) = η ( Y ) g ( A X, φZ ) − η ( Z ) g ( A X, φY )(5.12) for any
X, Y, Z ∈ Γ( T M ) . Proof.
Differentiating the identity φ = − I + η ⊗ ξ covariantly, we obtain( ∇ X φ ) φY + φ ( ∇ X φ ) Y = − g ( Y, A ∗ X ) ξ − η ( Y ) A X. Projecting this equality onto Z and then using antisymmetry of φ, we get g (( ∇ X φ ) φY, Z ) − g (( ∇ X φ ) Y, φZ ) = − η ( Z ) g ( A ∗ X, Y ) − η ( Y ) g ( A X, Z ) . Recalling g (( ∇ X φ ) Y, Z ) = − g ( Y, ( ∇ ∗ X φ ) Z ) , we obtain g ( φY, ( ∇ ∗ X φ ) Z ) + g (( ∇ X φ ) Y, φZ ) = η ( Z ) g ( A ∗ X, Y ) + η ( Y ) g ( A X, Z ) . Finally, by Lemma 6, we find (5.11).It is easily verified that ( ∇ ∗ X φ ) ξ = φ A ∗ X for any X ∈ Γ( T M ) . So we get( ∇ X Φ)(
Y, ξ ) = g (( ∇ X φ ) Y, ξ ) − g ( K X φY, ξ )= − g ( Y, ( ∇ ∗ X φ ) ξ ) − g ( φY, K X ξ )= − g ( φ A ∗ X, Y ) + 2 g ( A X, φY ) − g ( A X, φY )= − g ( φ A ∗ X, Y ) + 2 g ( A X, φY ) − g ( A X, φY ) − g ( A ∗ X, φY )= g ( A X, φY ) . Replacing Z by φZ in (5.11) and using foregoing equality, we finally arrive at the (5.12). (cid:3) , ˙IREM K¨UPELI ERKEN , AND CENGIZHAN MURATHAN Since an almost cosymplectic manifold is characterized by ∇ φ = 0, by Corollary 5we have Theorem 3.
Let ( M n +1 , g, φ, ∇ , ∇ ∗ ) be a cosymplectic statistical manifold. Then (5.13) ( ∇ X φ ) Y = ( K X φ ) Y (5.14) ( ∇ ∗ X φ ) Y = − ( K X φ ) Y for any X, Y ∈ Γ( T M ) . Theorem 4.
Let ( M n +1 , g, φ, ∇ , ∇ ∗ ) be an almost contact statistical manifold. Then( M n +1 , g, φ, ∇ , ∇ ∗ ) be a cosymplectic statistical manifold if and only if (5.15) ∇ X φY − φ ∇ ∗ X Y = K X φY + φ K X Y for any X, Y ∈ Γ( T M ) . Corollary 8. ( M n +1 , g, φ, ∇ , ∇ ∗ ) is a cosymplectic holomorphic statistical manifoldif and only if ∇ X φY = φ ∇ ∗ X Y for any X, Y ∈ Γ( T M ) . Let t be the coordinate on R . We denote by ∂ t = ∂∂ t the unit vector fileld on R . Defineaffine connections ∇ and ∇ ∗ on R by(5.16) R ∇ ∂ t ∂ t = λ ( t ) ∂ t and R ∇ ∗ ∂ t ∂ t = λ ∗ ( t ) ∂ t = − λ ( t ) ∂ t ,where λ : R → R is a smooth function. It is clear that ( g R = dt , R ∇ , R ∇ ∗ ) is a dualisticstructure on R . By [21], the following proposition can be given.
Proposition 6 ( [21]) . Let ( g N , N ∇ , N ∇ ∗ ) be dualistic structures on N . Let us consider ( M = R × N, <, > = dt + g N ) Riemannian product manifold . If
U, V are vector fieldson N and ¯ ∇ , ¯ ∇ ∗ satisfy following relations on R × N : (a) ¯ ∇ ∂ t ∂ t = λ ( t ) ∂ t (b) ¯ ∇ ∂t U = ¯ ∇ U ∂ t = 0 (c) ¯ ∇ U V = N ∇ U V and(i) ¯ ∇ ∗ ∂ t ∂ t = − λ ( t ) ∂ t (ii) ¯ ∇ ∗ ∂ t U = ¯ ∇ ∗ U ∂ t = 0 (iii) ¯ ∇ ∗ U V = N ∇ ∗ U V then ( <, >, ¯ ∇ , ¯ ∇ ∗ ) is a dualistic structure on M = R × N. It is well known that a cosymplectic manifold is a locally product of an open intervaland a Kahlerian manifold [4]. So we can give
Theorem 5.
Let ( N, g, ∇ , ∇ ∗ , J ) be a Kaehler statistical manifold and ( R , ∇ R , ∇ ∗ R dt ) be statistical manifold. Under the Proposition 6, R × N is a cosymplectic statisticalmanifold. Example 1 ( [3]) . If a group operation in R is defined as ( t, x, y ) ∗ ( s, u, v ) = ( t + s, e − t u + x, e t v + y ) for any ( t, x, y ) , ( s, u, v ) ∈ R then ( R , ∗ ) is a Lie group which is called the solvable non-nilpotent Lie group. The following set of left-invariant vector fields forms an orthonormalbasis for the corresponding Lie algebra: E = ∂∂t , E = e − t ∂∂x , E = e t ∂∂y . LMOST COSYMPLECTIC STATISTICAL MANIFOLDS 11
According to this base , one can construct almost contact metric structure ( φ, ξ, η, g ) on R as follows: η = dt, ξ = E , φ = e t dx ⊗ ∂∂y − e − t dy ⊗ ∂∂x , (5.17) g = dt ⊗ dt + e t dx ⊗ dx + e − t dy ⊗ dy. We obviously get ( R , φ, ξ, η, g ) is an almost cosymplectic manifold. With respect to Example 1, we provide an example on almost cosymplectic statisticalmanifold on R . Example 2.
Consider Example 1 for almost cosymplectic statistical manifolds . By(5.17) and Koszula formula, we can now proceed to calculate the Levi-Civita connections (5.18) ∇ E E = − E , ∇ E E = 0 , ∇ E E = 0 , ∇ E E = 0 , ∇ E E = E , ∇ E E = 0 , ∇ E E = E , ∇ E E = − E , ∇ E E = 0 . Now we define torsion-free affine connections ∇ , ∇ ∗ as follows (5.19) ∇ E E = − E + E , ∇ E E = E + E , ∇ E E = E , ∇ E E = E + E , ∇ E E = E + E , ∇ E E = E , ∇ E E = E + E , ∇ E E = − E + E , ∇ E E = E . (5.20) ∇ ∗ E E = − E − E , ∇ ∗ E E = − E − E , ∇ ∗ E E = − E , ∇ ∗ E E = − E − E , ∇ ∗ E E = E − E , ∇ ∗ E E = − E , ∇ ∗ E E = E − E , ∇ ∗ E E = − E − E , ∇ ∗ E E = − E . where (5.21) K E E = E , K E E = E + E K E E = E , K E E = E + E , K E E = E , K E E = E , K E E = E , K E E = E , K E E = E . Hence we have Zg ( X, Y ) = g ( ∇ Z X, Y ) + g ( X, ∇ ∗ Z Y ) for any X, Y, Z ∈ Γ( T M ) . It means that ( R , g, ∇ , ∇ ∗ , φ ) is an almost cosymplecticstatistical manifold. We notice that A E and A ∗ E are different from zero.If torsion-free affine connections ∇ , ∇ ∗ satisfy the followings ∇ E E = − E + E , ∇ E E = E , ∇ E E = 0 , ∇ E E = E , ∇ E E = E + E , ∇ E E = 0 , ∇ E E = E , ∇ E E = − E , ∇ E E = 0 . ∇ ∗ E E = − E − E , ∇ ∗ E E = − E , ∇ ∗ E E = 0 , ∇ ∗ E E = − E , ∇ ∗ E E = E − E , ∇ ∗ E E = 0 , ∇ ∗ E E = E , ∇ ∗ E E = − E , ∇ ∗ E E = 0 . then ( R , g, ∇ , ∇ ∗ , φ ) is again an almost cosymplectic statistical manifold. In this case A = A ∗ = A and also the integral curves of E = ξ are geodesics respect to affineconnections ∇ , ∇ ∗ . , ˙IREM K¨UPELI ERKEN , AND CENGIZHAN MURATHAN Now we will give an application for Equations (4.2) and (4.3). Let M n +1 = ( M, φ, ξ, η, g )be an almost cosymplectic statistical manifold. By the definition, the form η is closed,therefore distribution D : η = 0 is completely integrable. Each leaf of the foliation,determined by D , carries an almost Kaehler structure ( J, <, > ) J ¯ X = φ ¯ X, (cid:10) ¯ X, ¯ Y (cid:11) = g ( ¯ X, ¯ Y ) , ¯ X, ¯ Y are vector fields tangent to the leaf. If this structure is Kaehler statistical, leaf iscalled a Kaehler statistical leaf. Other hand Dacko and Olszak [4],[15] announced thatan almost cosymplectic manifold has Kaehler leaves if and only if( ∇ X φ ) Y = g ( A X, φY ) ξ + η ( Y ) φ A X, A = −∇ ξ. Using the last equation in (4.2) and (4.3), we have following.
Theorem 6.
An almost cosymplectic statistical manifold has Kaehler statistical leavesif and only if ( ∇ X φ ) Y = ( K X φ ) Y + g ( A X, φY ) ξ + η ( Y ) φ A X, ( ∇ ∗ X φ ) Y = − ( K X φ ) Y + g ( A X, φY ) ξ + η ( Y ) φ A X. The (1,1) tensor field
K ◦ φ + φ ◦ K is important to define hololmorphic statisticalmanifolds. So above theorem can be given following. Corollary 9.
An almost cosymplectic statistical manifold has Kaehler statistical leavesif and only if (5.22) ∇ X φY − φ ∇ ∗ X Y = g ( A X, φY ) ξ + η ( Y ) φ A X + K X φY + φ K X Y for any X, Y ∈ Γ( T M ) . Curvature properties on almost cosymplectic statistical manifolds
In this section, we study curvature properties of an almost cosymplectic statisticalmanifold . By simple computations, we have the following theorem.
Theorem 7.
Let ( M n +1 , φ, ξ, η, g ) be an almost cosymplectic statistical manifold. Then,for any X, Y ∈ Γ( T M n +1 ) ,R ( X, Y ) ξ = ( ∇ X A ) Y − ( ∇ Y A ) X, (6.1) R ∗ ( X, Y ) ξ = ( ∇ ∗ X A ∗ ) Y − ( ∇ ∗ Y A ∗ ) X. (6.2)We define (1 ,
1) tensor fields h , h and h ∗ on M n +1 by h = 12 ( L ξ φ )(6.3) h = 12 ( A φ − φ A ) , h ∗ = 12 ( A ∗ φ − φ A ∗ ) , (6.4)respevtively. It is proved that h is symmetric and h = A φ in [4]. In the followingproposition we establish some properties of the tensor fields h and h ∗ and provide arelation between h , h and h ∗ . Proposition 7.
Let ( M n +1 , φ, ξ, η, g ) be an almost cosymplectic statistical manifold.Then (6.5) g ( hX, Y ) = g ( X, hY ) and g ( h ∗ X, Y ) = g ( X, h ∗ Y ) , (6.6) h X = hX + 12 ( K ζ φ ) X and h X = h ∗ X −
12 ( K ζ φ ) X. LMOST COSYMPLECTIC STATISTICAL MANIFOLDS 13
Proof.
Using the antisymmetry of φ and the symmetry of A, A ∗ , we have the equation(6.5). By direct computations we obtain( L ζ φ ) X = ( ∇ ζ φ ) X + A φX − φ A X = ( ∇ ζ φ ) X + 2 hX ( . ) = ( K ξ φ ) X + 2 hX and ( L ζ ϕ ) X = ( ∇ ∗ ζ ϕ ) X + 2 h ∗ X . − ( K ξ φ ) X + 2 h ∗ X. So from above the last equations we get(6.7) h ∗ X − hX = ( K ξ φ ) X. On the other hand one can easily obtain that(6.8) h ∗ X + hX = 2 h X. By (6.7) and (6.8) we are led to (6.6). (cid:3)
Corollary 10.
Let ( M n +1 , φ, ξ, η, g ) be an almost cosymplectic statistical manifold.Then K ξ ϕ = 0 if and only if A φ = − φ A ∗ and A ∗ φ = − φ A if and only if ∇ ξ φ = 0 = ∇ ∗ ξ φ By (5.2), (6.1) and (6.2) we find following.
Theorem 8.
Let ( M n +1 , φ, ξ, η, g ) be an be an almost cosymplectic statistical manifold.Then, for any X, Y ∈ Γ( T M ) , R ( X, Y ) ξ = R ( X, Y ) ξ + R ∗ ( X, Y ) ξ (6.9) +( ∇ ∗ Y A ) X − ( ∇ ∗ X A ) Y + ( ∇ Y A ∗ ) X − ( ∇ X A ∗ ) Y. For an almost cosymplectic manifold, we can give well known equality (see [16]):(6.10) R ( X, ξ ) ξ − φR ( φX, ξ ) ξ = − A ) . Theorem 9.
Let ( M n +1 , φ, ξ, η, g ) be an almost cosymplectic statistical manifold. Weassume that K ξ ϕ = 0 and A ξ = 0 . Then, for any X ∈ Γ( T M ) , we have (6.11) R ( X, ξ ) ξ − φR ( φX, ξ ) ξ + R ∗ ( X, ξ ) ξ − φR ∗ ( φX, ξ ) ξ ) = − A + ( A ∗ ) ) X, (6.12) S ( ξ, ξ ) + S ∗ ( ξ, ξ ) = − tr ( A + ( A ∗ ) ) . Proof.
If we replace Y by ξ in (6.9) and recall Remark 1 we have4 R ( X, ξ ) ξ = R ( X, ξ ) ξ + R ∗ ( X, ξ ) ξ +( ∇ ∗ ξ A ) X + ( ∇ ξ A ∗ ) X + A∇ ∗ X ξ + A ∗ ∇ X ξ = R ( X, ξ ) ξ + R ∗ ( X, ξ ) ξ (6.13) +( ∇ ∗ ξ A ) X + ( ∇ ξ A ∗ ) X − ( AA ∗ + A ∗ A ) X. , ˙IREM K¨UPELI ERKEN , AND CENGIZHAN MURATHAN Replacing X by φX in (6.13) and then applying the tensor field φ both sides of theobtained equation and recalling that A ξ = 0 Remark A ∗ ξ we readily find4 φR ( φX, ξ ) ξ = φR ( φX, ξ ) ξ + φR ∗ ( φX, ξ ) ξ + φ ( ∇ ∗ ξ A ) φX + φ ( ∇ ξ A ∗ ) φX (6.14) +( AA ∗ + A ∗ A ) X. Substracting (6.13) from (6.14) we get4( R ( X, ξ ) ξ − φR ( φX, ξ ) ξ ) = R ( X, ξ ) ξ − φR ( φX, ξ ) ξ + R ∗ ( X, ξ ) ξ − φR ∗ ( φX, ξ ) ξ +( ∇ ∗ ξ A ) X + ( ∇ ξ A ∗ ) X − φ ( ∇ ∗ ξ A ) φX − φ ( ∇ ξ A ∗ ) φX (6.15) − AA ∗ + A ∗ A ) X. Other hand, using (5.2) and (6.10) we conclude that(6.16) − AA ∗ + A ∗ A ) = 4(( R ( X, ξ ) ξ − φR ( φX, ξ ) ξ ) + 2 A + 2( A ∗ ) . Using (6.16) in (6.15) we have0 = R ( X, ξ ) ξ − φR ( φX, ξ ) ξ + R ∗ ( X, ξ ) ξ − φR ∗ ( φX, ξ ) ξ +( ∇ ∗ ξ A ) X + ( ∇ ∗ ξ A ) X − φ ( ∇ ∗ ξ A ) φX − φ ( ∇ ∗ ξ A ) φX (6.17) +2( A + ( A ∗ ) ) . Beacause of Corollary 10 we have(6.18) φ ( ∇ ∗ ξ A ) φX + φ ( ∇ ∗ ξ A ) φX = ( ∇ ξ A ) X + ( ∇ ∗ ξ A ∗ ) X − g ( ξ, ∇ ξ A + ∇ ∗ ξ A ∗ X ) ξ. A short calculation leads to(6.19) g ( ξ, ∇ ξ A + ∇ ∗ ξ A ∗ X ) = 0 . We finally find(6.20) φ ( ∇ ∗ ξ A ) φX + φ ( ∇ ξ A ∗ ) φX = ( ∇ ξ A ) X + ( ∇ ∗ ξ A ∗ X ) X. Combining (6.17) with (6.20) we get0 = R ( X, ξ ) ξ − φR ( φX, ξ ) ξ + R ∗ ( X, ξ ) ξ − φR ∗ ( φX, ξ ) ξ (6.21) +( ∇ ∗ ξ ( A X − A ∗ )) X − ( ∇ ξ ( A − A ∗ )) X +2( A + ( A ∗ ) ) . Using the equality A − A ∗ = 2 K ξ in (6.21) we obtain (6.11).Taking into account φ -basis and (6.11), we readily find S ( ξ, ξ ) + S ∗ ( ξ, ξ ) = − tr ( A + ( A ∗ ) ) . (cid:3) References [1] S.Amari,
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