Seite - 79 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 Letωbean invariantvolumeelementonG/K inanaffine local coordinatesystem { x1,x2,...,xn } inaneighborhoodofo: ω=Φ ·dx1∧ ...∧dxn (128) WecanwriteX∗=∑ i χi ∂ ∂xi anddeveloptheLiederivativeof thevolumeelementω: LX∗ω=(LX∗Φ) .dx1∧ ...∧dxn+∑ j Φ.dx1∧···∧LX∗dxj∧···∧dxn= ( X∗Φ+ ( ∑ j ∂χj ∂xj ) Φ ) dx1∧ ...∧dxn (129) Since thevolumeelementω is invariantbyG: LX∗ω=0⇒X∗Φ+ ( ∑ j ∂χj ∂xj ) Φ=0⇒X∗logΦ=−∑ j ∂χj ∂xj (130) ByusingAX∗Y∗=−DY∗X∗,wehave:( D ∂ ∂xi (AX∗) )( ∂ ∂xj ) =D ∂ ∂xi ( AX∗ ( ∂ ∂xj )) −AX∗ ( D ∂ ∂xi ∂ ∂xj ) =−D ∂ ∂xi D ∂ ∂xj ( ∑ k χk ∂ ∂xk ) =−∑ k ∂2χk ∂xi∂xj ∂ ∂xk (131) ButasD is locallyflatandX∗ isan infinitesimalaffinetransformationwithrespect toD: D ∂ ∂xi (AX∗)=0⇒ ∂ 2χk ∂xi∂xj =0 (132) TheKoszul formandcanonicalbilinear formaregivenby: α=∑ i ∂logΦ ∂xi dxi=DlogΦ (133) Dα=∑ i,j ∂2logΦ ∂xi∂xj dxidxj=DdlogΦ (134) LX∗α=LX∗DlogΦ=DLX∗logΦ=DX∗logΦ=−D ( ∑ j ∂χj ∂xj ) =−∑ ,j ∂2χj ∂xi∂xj dxi=0 (135) Then,LX∗α=0∀X∈ g. ByusingX∗logΦ=−∑ j ∂χj ∂xj ,wecanobtain: α(X∗)= (DlogΦ)(X∗) ⇒ LX∗α=0 DX∗logΦ=−∑ j ∂χj ∂xj (136) ByusingAX∗Y∗=−DY∗X∗,wecandevelop: AX∗ ( ∂ ∂xj ) =−D ∂ ∂xj X∗=−∑ i ∂χi ∂xj ∂ ∂xi (137) As f(X)=AX∗,o andq(X)=X∗o: Tr(f(X))=Tr(AX∗,o)=−∑ i ∂χi ∂xi (o)=α(X∗0)=α0(q(X)) (138) Ifweuse thatLX∗α=0∀X∈ g, thenweobtain: (Dα)(X∗,Y∗)=(DY∗α)(X∗)=−(AY∗α)(X∗)=−AY∗ (α(X∗))+α(AY∗X∗)=α(AY∗X∗) (139) 79