Page - 411 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 425 where [uαi ] is the inverseof [u i α]andWij= δαβuαi u β j . DefinenowW kl= δαβukαulβ, so thatW irWrj= δij andWirWrj= δ j i.Wecanthenwrite themetricgFMdirectlyas gFM(ξhDh+ξhγD hγ)=WihξhDi, (11) because 〈w,gFM(ξ)〉= 〈 w,WjhξhDj 〉 = WijwiWjhξh = wiξi = ξhDh(wjDj) = ξ(w). One clearly recognizes thedependenceon thehorizontalH∗FMpart ofT∗FMonly, and the fact that gFM has image inHFM. Thesub-Riemannianenergyofanalmosteverywherehorizontalpathut is lFM(ut)= ∫ gFM(u˙t, u˙t)dt; i.e., the lineelement isds2=WijDiDj inadaptedcoordinates. Thecorrespondingdistance isgivenby dFM(u1,u2)= inf{lFM(γ) |γ(0)=u1,γ(1)=u2}. Ifwewish toexpressgFM incanonical coordinatesonT∗FM,wecanswitchbetweentheadapted frameandthecoordinates (xi,uiα,ξi,ξiα). From(11),gFMhasD,D∗ components DgFM,D∗= [ W−1 0 0 0 ] . Therefore,gFMhas the followingcomponents in thecoordinates (xi,uiα,ξh,ξhγ) (∂xi,∂uiα )gFM,(∂x,∂uiα) ∗= (∂xi,∂uiα )ADDgFM,D∗ D∗A(∂xi,∂uiα) ∗= [ W−1 −W−1ΓT −ΓW−1 ΓW−1ΓT ] orgijFM=W ij, g ijβ FM=−WihΓ jβ h , g iαj FM=−ΓiαhWhj, andg iαjβ FM=Γ iα kW khΓ jβ h . 3.2. CovarianceandNonholonomicity The metric gFM encodes the anisotropic weighting given the frame u, thus up to an affine transformationmeasuringtheenergyofhorizontalpathsequivalentlytothenegativelog-probabilityof samplepathsofEuclideananisotropicdiffusionsas formallygiven in (4). Inaddition, therequirement thatpathsmust stayhorizontal almost everywhereenforces thatC(u˙t) = 0a.e., i.e., thatno change of the covariance ismeasured by the connection. The intuitive effect is that covariance is covariantly constantas seenby theconnection. Globally, curvatureofCwill imply that thecovariancechanges whentransportedalongclosedloops,and torsionwill implythat thebasepoint“slips”whentravelling alongcovariantly closed loopsonM. However, thezeroacceleration requirement implies that the covariance isasclose tospatiallyconstantaspossiblewith thegivenconnection. This isenabledbythe parallel transportof the frame,anditensures that themodelcloselyresembles theEuclideancasewith spatiallyconstantstochasticgenerator. With non-zero curvature of C, the horizontal distribution is non-integrable (i.e., the brackets [Hi,Hj]arenon-zeroforsome i, j). Thisprevents integrabilityof thehorizontaldistributionHFM in thesenseof theFrobenius theorem. In thiscase, thehorizontal constraint isnonholonomicsimilarly to nonholonomicconstraintsappearingingeometricmechanics(e.g., [24]). Therequirementofcovariantly constantcovariance thusresults inanonholonomicsystem. 3.3. RiemannianMetrics onFM If the horizontality constraint is relaxed, a relatedRiemannianmetric on FM can bedefined bypulling back ametric on gl(n) to eachfiber using the isomorphism ψ(u, ·)−1 : VuFM → gl(n). 411