Page - 414 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 425 ∂ ylζ gij=Wij,lζ , ∂ylζg ijβ =−Wih,lζΓ jβ h −WihΓ jβ h,lζ , ∂ ylζ giαj=−Γiαh,lζW hj−ΓiαhW hj ,lζ , ∂ ylζ giαjβ =Γiαk,lζW khΓ jβ h +Γ iα kW kh ,lζΓ jβ h +Γ iα kW khΓ jβ h,lζ , Γiαh,lζ = ∂ylζ ( Γihku k α ) = δζαΓihl , W ij ,lζ = δilujζ+δ jluiζ . Combiningtheseexpressions,weobtain x˙i=Wijξj−WihΓjβh ξjβ , X˙iα=−ΓiαhWhjξj+ΓiαkWkhΓ jβ h ξjβ ξ˙i=WhlΓ kδ l,iξhξkδ− 1 2 ( Γhγk,iW khΓkδh +Γ hγ k W khΓkδh,i ) ξhγξkδ ξ˙iα =Γ hδ k,iα WkhΓkδh ξhγξkδ− ( Whl,iαΓ kδ l +W hlΓkδl,iα ) ξhξkδ− 1 2 ( Whk,iαξhξk+Γ hδ k W kh ,iαΓ kδ h ξhγξkδ ) . 4.3.AccelerationandPolynomials forC Wecan identify the covariant acceleration∇x˙t x˙t of curves satisfying theMPPequations, and hencenormalMPPsthroughtheir framecoordinates. Let (ut,ξt) satisfy (13). Then,ut isahorizontal liftofxt=π(ut)andhenceby(1), (3), (10), and(15), u−1t ∇x˙t x˙t= d dt ⎛⎜⎝ξ(hut(ute1))... ξ(hut(uted)) ⎞⎟⎠= ⎛⎜⎝ξ˙(hut(ute1))... ξ˙(hut(uted)) ⎞⎟⎠+ ⎛⎜⎝ξ(∂thut(ute1))... ξ(∂thut(uted)) ⎞⎟⎠ =− ⎛⎜⎜⎝ ξ(∂hut(ute1) hut(π∗(u˙t)) ... ξ(∂hut(uted) hut(π∗(u˙t)) ⎞⎟⎟⎠+ ⎛⎜⎜⎝ ξ(∂hut(π∗(u˙t)) hut(ute1)) ... ξ(∂hut(π∗(u˙t)) hut(uted)) ⎞⎟⎟⎠ = ⎛⎜⎝ξ(ψ(ut,Rut(ute1,π∗(u˙t))))... ξ(ψ(ut,Rut(uted,π∗(u˙t)))) ⎞⎟⎠ . (16) The fact that the covariant derivative vanishes for classical geodesic leads to a definition of higher-orderpolynomials throughthecovariantderivativebyrequiring (∇x˙t)kx˙t=0forakthorder polynomial (e.g., [26,27]).Asdiscussedabove, comparedtoclassicalgeodesics, curvessatisfyingthe MPPequationshaveextrad2 degreesof freedom,allowingthecurves to twistanddeviate frombeing geodesicwith respect toCwhile still satisfying thehorizontality constraint on FM. Thismakes it natural toask ifnormalMPPsrelate topolynomialsdefinedusingC. ForcurvessatisfyingtheMPP equations,using(16)and(15),wehave u−1t (∇x˙t)2x˙t= d dt ⎛⎜⎝ξ(ψ(ut,Rut(ute1,π∗(u˙t))))... ξ(ψ(ut,Rut(uted,π∗(u˙t)))) ⎞⎟⎠= ⎛⎜⎝ξ(ψ(ut, d dtRut(ute1,π∗(u˙t)))) ... ξ(ψ(ut, ddtRut(uted,π∗(u˙t)))) ⎞⎟⎠ . Thus, ingeneral,normalMPPsarenotsecondorderpolynomials in thesense (∇x˙t)2x˙t=0unless thecurvatureRut(utei,π∗(u˙t)) is constant in t. Forcomparison, in theRiemanniancase, avariational formulationplacingacostoncovariant acceleration[28,29] leads tocubicsplines (∇x˙t)2x˙t=−R(∇x˙t x˙t,xt,)x˙t . 414