Page - 174 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Definition31. Adualpair is aquadruple (M,g,D,D∗)where (M,g) is aRiemannianmanifold,DandD∗ areKoszul connections inMwhichare related to themetric tensor gby Xg(Y,Z)= g(DXY,Z)+g(Y,D∗XZ) ∀X,Y,Z. Werecall thataRiemanniantensor isanondegeneratesymmetricbilinear2-form. The dualistic relation between linear connections plays a central role in the information geometry [17,18,47,48]. Definition32. Let (M,g)beaRiemannianmanifold. (1) Adualpair (M,g,D,D∗) is calledaflatpair if the connectionDisflat, vizR∇=0. (2) Aflatpair (M,g,D,D∗) is calledaduallyflatpair if both (M,D)and (M,D∗)are locallyflatmanifolds. Givenadualpair(M,D,D∗) letussetA=D−D∗.Herearetherelationshipsbetweenthetorsion tensorsTD andTD ∗ (respectively therelationshipbetweenthecurvature tensorsRD andRD ∗ ) g(RD(X,Y) ·Z,T)+g(Z,RD(X,Y) ·T)=0, g(TD(X,Y),Z)−g(TD∗(X,Y),Z)= g(Y,A(X,Z))−g(X,A(Y,Z)). Proposition3. Givenaflatpair (M,g,D,D∗), the followingassertionsare equivalent. (1) BothDandD∗ are torsion free. (2) D is torsion free andAis symmetric, viz A(X,Y)=A(Y,X). (3) D∗ is torsion free and the metric tensor g a is KV cocycle of the KV algebraA∗ of the locally flat manifold (M,D∗). (4) Theflatpair (M,g,D,D∗) is aduallyflatpair. Proof. Letusprove that1 implies (2) IfbothTD andTD ∗ vanish identically thenA is symmetric,vizA(X,Y)=A(Y,X). Letusprove that (2) implies (3). SinceD is aflat connection, (2) implies that both the torsion tensor and the curvature tensor of D vanish identically. Then (M,D) is a locally flatmanifoldwhoseKV complex is denoted by (C∗(A,R),δKV). Usingthedualistic relationof thepair (M,g,D,D∗)oneobtains the identity δKVg(X,Y,Z)= g(A(X,Y)−A(Y,X),Z)= g(TD∗(X,Y),Z), therefore (2) implies (3). Letusprove that (3) implies (4). The assertion (3) implies that (M,D∗) is a locally flat manifold. Since g is δKV-closed D is torsionfree. Thereby (M,g,D,D∗) isaduallyflatpair. Letusprove that (4) implies (1). This implicationderivesdirectly fromthedefinitionofduallyflatpair. 174