Page - 408 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 425 2.3.AdaptedCoordinates For concrete expressions of the geometric constructions related to frame bundles, and for computationalpurposes, it isuseful toapplycoordinates thatareadaptedto thehorizontalbundle HFMandtheverticalbundleVFM togetherwith theirdualsH∗FMandV∗FM. Thenotationbelow followsthenotationused in, forexample, [18]. Letz=(u,ξ)bea local trivializationofT∗FM, and let (xi,uiα)becoordinatesonFMwithuiα satisfyinguα=uiα∂xi foreachα=1,. . . ,d. Tofindabasis that isadaptedto thehorizontaldistribution,definethed linearly independent vectorfieldsDj= ∂xj−Γ hγ j ∂uhγ whereΓhγj =Γ h jiu i γ is thecontractionof theChristoffel symbolsΓhij for theconnectionCwithuiα.WedenotethisadaptedframeD. Theverticaldistributioniscorrespondingly spannedbyDjβ = ∂ujβ . ThevectorsDh= dxh, andDhγ =Γhγj dx j+duhγ constitutesadualcoframeD∗. Themapπ∗ :HFM→TM is incoordinatesof theadaptedframeπ∗(wjDj)=wj∂xj. Correspondingly, thehorizontal lifthu ishu(wj∂xj)=w jDj. Themapu :Rd→TxM isgivenbythematrix [uiα] so that uv=uiαvα∂xi =uαv α. Switchingbetweenstandardcoordinatesandtheadaptedframeandcoframescanbeexpressedin termsof thecomponentmatricesAbelowtheframeandcoframeinducedbythecoordinates (xi,uiα) andtheadaptedframeDandcoframeD∗.Wehave (∂xi,∂uiα )AD= [ I 0 −Γ I ] with inverse DA(∂xi,∂uiα) = [ I 0 Γ I ] writingΓ for thematrix [Γhγj ]. Similarly, thecomponentmatricesof thedual frameD ∗ are (∂xi,∂uiα )∗AD∗= [ I ΓT 0 I ] and D∗A(∂xi,∂uiα) ∗= [ I −ΓT 0 I ] . 2.4. ConnectionandCurvature TheTMvaluedconnectionC :TM×TM→TM lifts toaprincipalconnectionTFM×TFM→ VFMon theprincipalbundleFM. C can thenbe identifiedwith thegl(n)-valuedconnection form ωonTFM. The identificationoccursbythe isomorphismψbetweenFM×gl(n)andVFMgivenby ψ(u,v)= ddtuexp(tv)|t=0 (e.g., [19,20]). Themapψ isequivariantwithrespect to theGL(n)actiong →ug−1 onFM. Inorder toexplicitly see the connection between theusual covariant derivative∇ : Γ(TM)×Γ(TM)→ Γ(TM) on M determinedbyC andC regardedasaconnectionontheprincipalbundleFM, following[19],we let s :M→TMbealocalvectorfieldonM; equivalently,s∈Γ(TM) isalocalsectionofTM. sdetermines amap sFM : FM→Rd by sFM(u)= u−1s(π(u)); i.e., it gives thecoordinatesof s(x) in the frameu atx. Thepushforward (sFM)∗ :TFM→Rdhas in its ithcomponent theexteriorderivatived(sFM)i. Let noww(x)bea local sectionof FM. The compositionw◦(sFM)∗ ◦hw : TM→ TM is identical to the covariant derivative∇·s : TM→ TM. The construction is independent of the choice ofw becauseof theGL(n)-equivarianceof sFM. The connection formω canbe expressedas thematrix (sFM1 ◦hw, . . . ,sFMd ◦hw)whenletting sFMi (u)= ei. The identificationbecomesparticularlysimple if thecovariantderivative is takenalongacurvext onwhichwt is thehorizontal lift. In thiscase,wecanlet st=wt,isit. Then, s FM(wt)=(s1t , . . . ,s d t) T, and w−1t ∇x˙ts=(sFM)∗(hwt(x˙t))= ddt(s1t , . . . ,sdt)T ; (1) i.e., the covariant derivative takes the form of the standard derivative applied to the frame coordinates sit. 408