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Differential
Geometry
and
its
Applications
www.elsevier.com/locate/difgeo
Characterization
of
exact
lumpability
for
vector
fields
on
smooth
manifolds
Leonhard Horstmeyera,∗, Fatihcan M. Atayb
a
MaxPlanckInstituteforMathematicsintheSciences,Inselstraße22,04103Leipzig,Germany
b
DepartmentofMathematics,BilkentUniversity,06800Bilkent,Ankara,Turkey
a r t i c l e i n f o a b s t r a c t
Article history:
Received26April2016 Availableonline20June2016 CommunicatedbyI.Kolář MSC: 37C10 34C40 58A30 53B05 34A05 Keywords: Lumping Aggregation Dimensionalreduction Bottconnection
Wecharacterizetheexactlumpabilityofsmoothvectorfieldsonsmoothmanifolds. Wederivenecessaryandsufficientconditionsforlumpabilityandexpressthemfrom fourdifferent perspectives,thussimplifying andgeneralizingvarious results from theliteraturethat exist forEuclidean spaces. Weintroduce a partialconnection onthepullbackbundle thatisrelatedto theBottconnection andbehaveslikea Liederivative.Thelumpingconditionsareformulatedintermsofthedifferentialof thelumpingmap,itscovariantderivativewithrespecttotheconnectionandtheir respectivekernels.Someexamplesarediscussedtoillustratethetheory.
©2016ElsevierB.V.Allrightsreserved.
1. Introduction
Dimensionalreductionisanimportantaspectinthestudyofsmoothdynamicalsystemsandinparticular inmodelingwithordinarydifferentialequations(ODEs).Oftenareductioncanelucidatekeymechanisms, finddecoupledsubsystems,revealconservedquantities,maketheproblemcomputationallytractable,orrid it from redundancies. A dimensional reduction bywhich micro state variables are aggregated into macro state variables also goes bythe name of lumping.Starting from amicrostate dynamics,this aggregation induces alumpeddynamics onthemacro statespace.Whenever anon-triviallumping,onethatisneither theidentitynormapstoasinglepoint,confersthedefiningpropertytotheinduceddynamics,onecallsthe dynamics exactlylumpable andthemapanexact lumping.
* Correspondingauthor.
E-mailaddresses:horstmey@mis.mpg.de(L. Horstmeyer),atay@member.ams.org(F.M. Atay). http://dx.doi.org/10.1016/j.difgeo.2016.06.001
Ouraim inthis paperis to provide necessary andsufficient conditionsforexact lumpabilityof smooth dynamics generated byasystem ofODEs on smoothmanifolds. To be moreprecise, letX and Y betwo smoothmanifoldsofdimensionn andm,respectively,with0< m< n.LetpX: T X→ X andpY : T Y → Y
betheirtangentbundles,whosefibers wetakeasspacesofderivations,andletv beanelementofthesmooth sectionsΓ∞(X,T X) of T X overX,i.e.smoothmapsfrom X toT X satisfyingpX◦ v = idX.Theintegral
curvesΦtofv satisfytheequation
d dt
t=sΦt(x) = v(Φs(x)) . (1)
OnalocalcoordinatepatchU ⊆ X wecanwrite(1)as ˙xi = vi(x) sothatwerecoveranODEonthatpatch. Considerasmoothsurjective submersionπ : X → Y andletΘt(x)= π◦ Φt(x). Since dim(Y )< dim(X),
themappingπ ismany-to-one,andhenceiscalledalumping.Thequestioniswhetherthereexistsasmooth dynamicsonY thatisgeneratedbyanothersystemofODEs,
d dt
t=sΘt(x) = ˜v(Θs(x))
forsomesmoothvectorfield˜v onY . Ifthatis thecase,wesaythat(1) isexactlylumpable forthemapπ.
Geometricallythis meansthatv and˜ v areπ-related[1].
ThereductionofthestatespacedimensionhasbeenstudiedforMarkovchainsbyBurkeandRosenblatt [2,3]inthe1960s.Kemenyand Snell[4]havestudieditsvariantsandcalled themweakandstrong lumpa-bility. Many conditions have been found, mostly in terms of linear algebra, for various forms of Markov lumpability [4–12]. Since Markov chainsare characterized by lineartransition kernels, mostof these con-ditions carryover directlytothe caseoflineardifference anddifferential equations. In1969Kuoand Wei studied exact [13] and approximate lumpability [14] in the context of monomolecular reaction systems, whichare systemsof linearfirstorderODEs oftheform ˙x = Ax.Theygavetwo equivalent conditionsfor exactlumpability interms of thecommutativity ofthe lumping map with theflow or withthe matrixA
respectively.Luckyanov[15]andIwasa[16]studiedexactlumpabilityinthecontextofecologicalmodeling andderivedfurtherconditionsintermsoftheJacobianoftheinducedvectorfieldandthepseudoinverseof thelumpingmap.Iwasaalsoonlyconsideredsubmersions.TheprogramwasthencontinuedbyLiand Rab-itzet al.,whowroteaseriesofpaperssuccessivelygeneralizingthesetting,butremainingintheEuclidean realm.Theyfirstconstrainedtheanalysistolinearlumpingmaps[17],wheretheyofferedforthefirsttime two constructionmethodsintermsofmatrixdecompositionsof thevectorfieldJacobian.These methods, togetherwiththeobservabilityconcept[18]fromcontroltheory,wereemployedtoarriveataschemefor ap-proximatelumpingswithlinearmaps[19].Theyextendedtheiranalysisfurthertoexactnonlinearlumpings ofgeneralnonlinearbutdifferentiabledynamics[20],providingasetofnecessary andsufficientconditions, extendingandrefining thoseobtainedbyKuo,Wei, LuckyanovandIwasa. Byconsideringthespacesthat are left invariant underthe Jacobianof the vector field,they open upa newfruitful perspective,namely thetangentspacedistributionviewpoint.
Theconnectiontocontroltheoryhasbeenmadeexplicitin[21].Coxsonnotesthatexactlumpabilityisan extremecaseofnon-observability,wherethelumpingmapisviewedastheobservable.Shespecifiesanother necessaryandsufficientconditionbystatingthattherankoftheobservabilitymatrixoughttobeequalto therankofthelumpingmapitself.Thegeometrictheoryofnonlinearcontrolisoutlinedin,e.g.,[22].There, IsidoriconsiderssetsofobservableshiwithvaluesinR andtheirdifferentialsdhi.Hediscusseshowtoobtain
themaximal observable subspaceinan iterativefashion,where one consecutivelyconstructs distributions that are invariant under the vector field and contain the kernel of the dhi [22, p. 69]. This distribution
is constructed by meansof Lvdhi, the Lie derivativesof dhi. Although this theory is notconcerned with
preciselythekernelofthedhiandtheLiederivativesLvdhi arejustlinearcombinationsofdhi.(Weobtain
similar results,butallowforgeneralmaps,thatarenotnecessarilyR-valued.)
In this paper we tie together all these strands into one geometric theory of exact lumpability. The conditions obtainedby Iwasa, Luckyanov, Coxson,Li,Rabitz, and Toth are containedinthis framework. Instead of considering thedistribution spanned by the differentialof the lumping map, as is donein [20] although not explicitly, we consider the vertical distributionwhich is defined by the kernel of the differ-ential. We begin by stating the mathematical setting in Section2.1. We then define the notion of exact smooth lumpability and provide two elementary propositions in terms of commutative diagrams in Sec-tion 2.2. InSection2.3 we characterize exactlumpability intermsof thevertical distribution and partial connections onit. InSection3we investigatesomeproperties ofexactlumpings and illustrate them with examples.
2. Characterizationoflumpability
2.1. Preliminaries
As above, let X and Y be two smooth manifolds of dimension n and m and pX : T X → X and
pY : T Y → Y betheirtangentbundles,respectively.Thedifferentialofasmoothmanifoldmapπ : X → Y
at point x is aR-linearmap Dπx : TxX → Tπ(x)Y . For wx ∈ TxX the vectorDπxwx can be definedvia
its action as aderivation Dπxwx[f ] = wx[f◦ π] on smooth test functionsf ∈ C∞(X,R). We use square
brackets to enclosethe argument of the derivation. Themap π isa submersion if Dπx is surjective with
constantrankforallx∈ X.Wedenotebyπ−1T Y thepullbackbundlewhosefibersatx areTπ(x)Y .There
are twobundlemaps associatedto thedifferential.Thefirstoneisamanifold mapDπ : T X → T Y which
respects the vector bundle structure and satisfies pY ◦ Dπ = π ◦ pX. The second oneis avector bundle
homomorphism over thesamebase Dπ : T X → π−1T Y . This latteroneinduces aC∞-linearmap onthe vector fieldsDπ : Γ∞(X,T X)→ Γ∞(X,π−1T Y ).Allofthese aredenotedbyDπ andthecontext willtell them apart.Onecanonlydefineavectorfieldw on˜ Y wheneverthereexists auniquevector Dπxw(x) for
allx∈ π−1(y) andally∈ Y .
AsmoothregulardistributionS isasmoothsubbundlelocallyspannedbysmoothandlinearindependent vectorfields[1,23].Thedistributionker Dπ =
x∈Xker Dπxcanbeshowntobesmooth,where
denotes
disjoint union. This follows from the existenceof a smooth local coframe (c.f. [1]) spanned by m smooth
1-forms (dπ1,. . . ,dπm) that annihilate ker Dπ. The distribution ker Dπ is regular if and only if π is a
submersion. Anintegral submanifoldW of S isan immersedsubmanifold ofX such thatT W ⊆ S|W. It
hasthemaximalintegralsubmanifoldpropertyifT W = S|W andW isnotcontainedinanyotherintegral
submanifold. Following Sussmann and Stefan [24,25], S is integrable if every point of X is contained in an integralsubmanifoldwiththemaximal integralsubmanifoldproperty. Frobeniustheorem statesthata regulardistributionisintegrableifandonlyifthespaceofitssectionsisclosedundertheLiebracket,i.e.,
S is involutive. Thedistribution ker Dπ is by construction anintegrable distributionwhere {π−1(x)}x∈X
are themaximal integralsubmanifoldsofmaximaldimension.
Let v andw betwo vectorfields where v generatestheflow Φ.TheLie derivativeofw inthedirection
v isdefinedby Lvw := d dt t=0DΦ−tw◦ Φt. (2)
TheLie derivativeLv : Γ∞(X,T X)→ Γ∞(X,T X) isaderivation ontheC∞-moduleofvector fields.One
can also show [1] that Lvw = Jv,wK, where J·,·K : Γ∞(X,T X)× Γ∞(X,T X) → Γ∞(X,T X) is the Lie
A linear connection on avector bundle E → X is a map ∇E : Γ∞(X,T X)× Γ∞(X,E) → Γ∞(X,E)
whichistensorialinthefirstargumentandforanyv∈ Γ∞(X,T X) themap∇Ev :=∇E(v,·) isaderivation
on Γ∞(X,E). A partial connection over asubbundle S ⊂ T X is a map ∇˚E : Γ∞(X,S)× Γ∞(X,E) →
Γ∞(X,E).Anotablepartial connectionistheBottconnection[26]definedoveranintegrablesubbundleS
onthequotientbundleQ= T X/S.Letρ bethecorrespondingquotientmap;thentheconnectionisdefined by ˚ ∇Q w v= ρqw, ρ−1vy, (3)
where theright inverseρ−1v= v+ w picksoutsmoothlyanarbitraryrepresentativeoftheequivalence class,withw ∈ ker ρ.SincetheLiebracketisbilinearandS isinvolutive,thisisindependentofthechoicew
andthuswelldefined.Onlythetermthatislinearinw survivestheprojectionbyρ andsotherequirements ofaveritableconnectionaresatisfied.Thepartialconnectioncanbecompletedtoafullconnection[27].For example,onecouldintroduceaRiemannianmetricwhichsplitsT X = S⊕S⊥anddecomposesg = gS⊗gS⊥. ThecorrespondingLevi-Civita connection∇ restricted¯ to Q completes˚∇Q to ametricconnection:
∇Q
w= ˚∇Qw+ ¯∇|Qw .
Thisissometimescalled anadapted connection. 2.2. Lumpabilityandcommutativity
Inthis sectionwe statetwo necessary andsufficient conditionsforexactlumpability. Henceforthπ is a smooth surjective submersionand v ∈ Γ∞(X,T X) is asmoothvector field generating theflow Φ: TX ⊆
R× X → X, where TX :={(Tx,x) : Tx ⊆ R,x ∈ X} isthe domain of theflow and Tx contains anopen
intervalaround0.WedenotebyΦx:Tx→ X theintegralcurveswithstartingpointx,andbyΦt:Xt→ X
theflow mapparametrized bytime,with Xt:={x∈ X : t∈ Tx} beingthedomain ofdefinition.We start
bygivingaprecisedefinitionoflumpability.
Definition1(Exactsmoothlumpability).Thesystem
d dt
t=sΦ = v◦ Φs (4)
is called exactly smoothly lumpable (henceforth exactly lumpable) for π iff there exists asmooth vector field˜v∈ Γ∞(Y,T Y ) suchthatthedynamicsofΘ= π◦ Φ isgovernedby
d dt
t=sΘ = ˜v◦ Θs. (5)
The Picard–Lindelöf theorem guarantees aunique solution of (4) for sufficiently small times for all x,
sincev issmoothandinparticularLipschitz. ItexistsforalltimesofdefinitionTx⊆ R.Formallyequation
(4)shouldbeunderstoodasthepushforwardofthesection ∂t∂ onTX byΦ:
d dt t=sΦ : = DΦs ∂ ∂t ,
and likewise for (5). Theflow of the vector fieldv˜∈ Γ∞(Y,T Y ) isdenotedby Φ :˜ TY → Y ,where again
TY :={(Ty,y): (−,)⊆ Ty⊆ R,y∈ Y } isthedomain oftheflow.A priorithereisnoconnectionbetween
Proposition1.Thesystem(4)isexactlylumpable forπ iffthereexistsasmoothvectorfieldv˜∈ Γ∞(Y,T Y ) such that
Dπxv(x) = ˜v◦ π(x) (6)
forallx∈ X.
Proof. ConsiderthetimederivativeofΘ:
d dt t=0Θx= D(π◦ Φx ) 0 ∂ ∂t = Dπxv(x).
Byexactlumpability,Θ isgeneratedby(5),so dtdt=0Θx= ˜v◦Θ0(x)= ˜v◦π(x).Therefore,exactlumpability
implies(6).Ontheotherhand,ifwedemand(6)forallx andinparticularforΦs(x),then
DπΦs(x)v(Φs(x)) = ˜v◦ π ◦ Φs(x) .
Therighthandsideequalsv˜◦ Θs(x) andthelefthandsideequals dtdt=sΘt(x),whichimpliesexact
lumpa-bility. 2
Remark1.Alternatively,wecansaythat(4)isexactlylumpableforπ iffthere existsasmoothvectorfield ˜
v∈ Γ∞(Y,T Y ) suchthatv and˜ v areπ-related.Proposition 1canbeformulatedasacommutativediagram
Y T Y T X X ˜ v v Dπ π
whichreads ˜v(π(x))= Dπxv(x) forallx∈ X.
Proposition 2. The system (4) is exactly lumpable for π iff for all y ∈ Y the time domain Ty = Tx is
independent ofthechoice x∈ π−1(y),and
˜
Φt◦ π(x) = π ◦ Φt(x) (7)
forallx∈ X andalltimest∈ Tπ(x).
Proof. One implicationisobtainedbytakingtime derivativesonbothsidesof (7)att= 0 andusingthat ˜
v isthegeneratorofΦ.˜ Thisgivesriseto(6) andbyProposition 1impliesexactlumpability.Ontheother hand, by the definition of exact lumpability, the curve Θx is an integral curve to v for˜ any x. There is
another integralcurve Φ˜π(x)for v which˜ at t= 0 coincides with Θx. Bytheuniquenessof integralcurves
they must coincide,so Φ˜π(x)(t)= Θx(t) for all t∈ Tx and all x.Since they are thesameintegral curves,
Tπ(x)=Tx forallx.Thisproves theproposition. 2
Remark2.Proposition 2canalso becastintoacommutativediagram
Y Y X X ˜ Φt Φt π π
2.3. Lumpabilityandthevertical distribution
Inthissectionwediscusssomerelationsbetweenexactlumpability,invariantdistributions,andtheBott connection. Thelumping map π : X → Y givesriseto a subbundleker Dπ ⊆ T X ofthe tangentbundle. Thisiscalledthevertical distribution,whichisintegrablebyconstructionandρ: T X→ T X/ker Dπ isthe corresponding quotientmap.Westartwithabasicproposition.
Proposition3.Thedistributionker Dπ isinvariantundertheflowΦ iffthespaceofsectionsΓ∞(X,ker Dπ)
isinvariantunder Lv.
Proof. ker Dπ isinvariantundertheflowif(DΦt)x(ker Dπ)x⊆ (ker Dπ)Φt(x) forallx,t whereitisdefined. SinceΦtisadiffeomorphism,thisconditionisequivalent to(DΦ−t)Φt(x)(ker Dπ)Φt(x)⊆ (ker Dπ)x. So,for any w∈ Γ∞(X,ker Dπ), wehave(DΦ−t)w◦ Φt∈ Γ∞(X,ker Dπ). Takingtime derivativesandevaluating
at0,weobtainthattheLiederivative(2)ofv in thedirectionofw isagainasectionofker Dπ. 2 Wewouldliketodefine aderivativeofthedifferentialDπ tofindfurtherconditions.
Definition 2(Covariant derivative of the differential). Let ∇H be a connection on H = (π◦ Φ)−1T Y ⊗
T∗(X× R) andv∈ Γ∞(X,T X) withflow Φ.Then
L∇
vDπ :=∇H∂
∂t D(π◦ Φ)
0 (8)
isthecovariantderivativewithrespectto∇HofthedifferentialDπ inthedirection ∂ ∂t = (0,
∂
∂t)∈ T (X ×R).
Thecovariantderivativetakestheplaceof d
dt andensuresthatthemapD(π◦ Φt): T X→ (π ◦ Φt)−1T Y
isdifferentiatedproperlyandcovariantly.ItisworthnotingthatthisobjectbehaveslikeaLiederivativeas wewillseein(13),butsinceDπ isnotatensoronecannotdefine aproperLiederivative.Nevertheless,we willusethesimilarnotation.
We shall make the connection to the Lie derivative more apparent. Let V → X be a vector bundle,
L : T X → V a vector bundle homomorphism, and θ : X → X a diffeomorphism. Then there exists an inducedlinearmapθL: T X→ V ofL:
θL := L◦ Dθ−1 .
AnalogouslytotheLie derivative(2)of sectionsonthetangentbundle,wecanthendefine(8)as
L∇ vDπ :=∇H∂ ∂t Φ−tDπ◦ Φt0 withrespect to∇H.
InDefinition 2oneneedsto specifyacovariantderivative. Thisis ofcourse unfortunate,becausethere aremany options.Howeverit turnsoutthatweare fortunatenevertheless,becausethere isagoodchoice whichturnsouttobecloselyrelatedtotheBottconnection.Givenaconnection∇E onE→ Y andamap
π : X→ Y , thereisaunique[28] connectionπ∗∇π−1E onπ−1E→ X,calledthepullbackconnection
π∗∇πv−1E(s◦ π) =∇EDπvs◦ π,
defined for sections s ∈ Γ∞(Y,E) and extended locally to arbitrary sections aca(s
a◦ π) ∈ π−1E by
linearity, where ca ∈ C∞(X,R) for alla. Given atensor product bundle H = H
1⊗ H2, connections ∇H1
and∇H2 onH
∇H(s
1⊗ s2) =∇H1s1⊗ s2+ s1⊗ ∇H1s2, (9)
wheres1ands2aresectionsonH1andH2,respectively.Forthenextpropositionwerequiretheconnections
to betorsionfree.Recall that∇ iscalledtorsionfreeif∇vw− ∇wv =Jv,wK.
Lemma 3.Letg : M→ N and ∇¯T N beatorsion-free connection onT N . Then
g∗∇¯gw−1T NDg v− g∗∇¯gv−1T NDg w = Dgqw, vy, (10)
where v,w aresections onT M .
Proof. Seepage6of[28]. 2
Proposition4.Let∇¯T Y and∇¯T∗(X×R)betorsion-freeconnectionsand∇¯Hthetensorproductconnection(9).
Then ¯ ∇H ∂ ∂t D(π◦ Φ) 0 w = π ∗∇¯π−1T Y w (Dπv) . (11)
Proof. The proof follows [28]inthe firstpart. Withsomeabuseof notation, weuse w(x,t)= (w(x),0)∈
T(x,t)(X⊗ R) and ∂t∂ = (0,1)∈ T(x,t)(X⊗ R).Then ¯ ∇H ∂ ∂tD(π◦ Φ) 0w = π ∗∇¯(π◦Φ)−1T Y ∂ ∂t D(π◦ Φ)w 0− D(π ◦ Φ) ¯∇ T (X×R) ∂ ∂t w (12)
Thesecondtermvanishesbecause∇¯T (X×R)= ¯∇T X⊕T Randw and ∂t∂ areorthogonal.NowweuseLemma 3 with M = X× R,N = Y ,andg = π◦ Φ,as wellasthefactthat∇¯T Y istorsion free,to obtain(p. 6[28])
¯ ∇T Y D(π◦Φ)∂ ∂tD(π◦ Φ)w − ¯∇ T Y D(π◦Φ)wD(π◦ Φ) ∂ ∂t = D(π◦ Φ) r∂ ∂t, w z .
This vanishesbecausew doesn’tdepend ont.Thepullbackofthisequationallowsusto rewrite(12)as ¯ ∇H ∂ ∂tD(π◦ Φ) 0w =π ∗∇¯(π◦Φ)−1T Y w D(π◦ Φ) ∂ ∂t 0 =π∗∇¯πw−1T YDπv.
Thelast termisinprincipleoverT (X× R) butafterhaving sett= 0 wecanomittheTR part. 2
Lemma 3 and Proposition 4 show the analogy between Lv∇¯Dπ and the Lie derivative for torsion-free connections. Uponsubstitutionof(8)into (11),equation (10)reads
π∗∇¯πv−1T YDπw = (L∇v¯Dπ)w + DπLvw, (13)
whichshouldbecompared to
Lv dπ, w = Lvdπ, w + dπ, Lvw ,
where π : X→ R isareal-valuedfunction, dπ isadifferentialone-form,and ·,· : T∗X× T X → R isthe naturalpairingoftangentandco-tangentvectors.
ThelinearmapL∇v¯Dπ : T X→ π−1T Y isavectorbundlehomomorphismandthekernelkerL∇v¯Dπ isa smooth distribution,whichcanbe checkedbyviewingLv∇¯Dπ asadifferentialone-form: Oneachpullback patch U∩ π−1V ⊆ X withlocal coordinatesψ : V˜ ⊆ Y → Rm,oneconstructslocally aset ofone-forms
σa:= (L∇v¯Dπ)a= d(Dπv)a+ ¯Γabc(Dπv)cdπb (14) where a,b,c are the indices of the local coordinates and Γ¯a
bc is the Christoffel symbol of ∇.¯ Here and in
the remainderof the article, we usethe convention thatrepeated indices aresummed over, unless stated otherwise. Since π has full rank, (σ1,. . . ,σm) spans a smooth m-dimensional local co-frame. We have
σa,w = ((L∇¯
vDπ)w)a;so,this co-frameannihilatesvectorsinkerL
¯
∇ vDπ.
ThemotivationfortheDefinition 2partlystemsfromthefollowingtwo propositions:
Proposition 5. The distribution ker Dπ is invariant under the flow Φt iff the space of sections
Γ∞(X,ker Dπ)⊆ Γ∞(X,kerL∇v¯Dπ).
Proof. By Proposition 3, the distribution ker Dπ is invariant under the flow Φt iff the space of sections
Γ∞(X,ker Dπ) isinvariantunderLv.By(13),ifw∈ Γ∞(X,ker Dπ) then(L∇v¯Dπ)w = 0 ⇐⇒ Lvw = 0. 2
AslightlystrongerversionthatimpliesProposition 5isthefollowing.
Proposition6.The distributionker Dπ isinvariantunder theflow Φtiffker Dπ⊆ ker L∇v¯Dπ.
Proof. ker Dπ isinvariantundertheflowif(DΦt)x(ker Dπ)x⊆ (ker Dπ)Φt(x) forallx,t whereitisdefined. So, (Dπ)Φt(x)(DΦt)xwx = 0 for wx ∈ (ker Dπ)x, or in other words (Dπ)Φt(x)(DΦt)x maps (ker Dπ)x to (ker Dπ)Φt(x) sothatD(π◦ Φt)w remains0foranyw∈ ker Dπ.Ininfinitesimaltermsthismeansthatthe covariantderivative(8)vanishes,∇¯H
∂ ∂t D(π◦ Φt)w0= (L ¯ ∇ vDπ)w = 0 onw. 2
Wewouldnowliketodefineapartialconnectiononthepullbackbundleπ−1T Y oversectionsofker Dπ. Thenextpropositionestablishesanisomorphismthatwillhelpusdefinethepartial connection.
Proposition7.There isavectorbundleisomorphismϕ: π−1T Y → T X/ker Dπ.
Proof. We shallshow thaton eachfiberϕx : Tπ(x)Y → TxX/ker Dπx is avectorspace isomorphism. Let
˜
v ∈ Tπ(x)Y . We fix local coordinatesand denote theJacobian of π by Mia = ∂π
a
∂xi. There exists a unique pseudoinverse [29] M+ such that M+M : T
xX → (ker M)⊥ is an orthogonal projection and M M+ =
idTπ(x)X.Weshowthatϕx: ˜v→
M+v˜isone-to-oneandonto.Supposeϕxv = ϕ˜ xv˜,thenM+v˜−M+v˜= w
andw∈ ker M.ApplyingM yieldsv = ˜˜ v.Toshowsurjectivity,weconstructv = M˜ v,whichistheelement thatmapstov. Soϕx isclearlyafiberwise isomorphismandϕ isavectorbundleisomorphism.Infact,
ϕ−1◦ ρ = Dπ (15)
isthedifferential. 2
Definition3.Wedefinethepartialconnection ˚ ∇π−1T Y : Γ∞(X, ker Dπ)× Γ∞(X, π−1T Y )→ Γ∞(X, π−1T Y ) by ˚ ∇π−1T Y w v := Dπ˜ q w, vy, (16)
Definition 3indeedsatisfiestherequirementsofaconnection:Letf ∈ C∞(Y,R) beatestfunctiononY .
Recall thatDπw[f ]:= w[f◦ π]; so,
Dπqw, vy[f ] = w[v[f◦ π]] − v[w[f ◦ π]] . (17) Ifw∈ Γ∞(X,ker Dπ) thenthesecondtermvanishes.Thefirsttermislinearinw andaderivationinDπv.
Proposition8.Theconnection definedin(16)isrelatedtotheBottconnection (3)throughthecommutative diagram π−1T Y T X/ ker Dπ T X/ ker Dπ π−1T Y ϕ ϕ ˚ ∇T X/ ker Dπ ˚ ∇π−1T Y where T X/ker Dπ = Q in(3). Proof. By(15), ϕ˚∇πw−1T Y˜v = ϕ◦ ϕ−1◦ ρqw, ρ−1(ϕ(˜v))y= ˚∇T X/ ker Dπw ϕ(˜v). Therefore, ϕ˚∇π−1T Y w v = ˚˜ ∇ T X/ ker Dπ
w ϕ(˜v) foranyw∈ Γ∞(X,ker Dπ). 2
Proposition 9.Let ∇¯T Y be a torsion-free connection on T Y . Then π∗∇¯π−1T Y completes thepartial
con-nection (16).
Proof. Let w∈ Γ∞(X,ker Dπ).By(10)wehave
π∗∇¯πw−1T YDπv = Dπqw, vy= ˚∇πw−1T YDπv,
and thereforeπ∗∇¯π−1T Y = ˚∇π−1T Y + π∗∇¯π−1T Y
(ker Dπ)⊥. 2
Wenow connectallofthese conceptstoexactlumpability.
Theorem 4.Thesystem(4) isexactlylumpable forπ iffΓ∞(X,ker Dπ) isinvariantunder Lv.
Proof. First we show thatexactlumpabilityimplies theinvariance of Γ∞(X,ker Dπ) under Lv. Byexact
lumpability,weknowfrom(6)thatthereisavectorfield˜v suchthatv[f◦ π]= ˜v[f ]◦ π foranytestfunction
f ∈ C∞(Y,R).Substitutingthisconditioninto (17)yields
Dπqv, wy[f ] = v[w[f◦ π]] − w[˜v[f] ◦ π]
Theright handsideequals v[Dπw[f ]]− Dπw[˜v[f]].Sothelefthandsidevanishesforw∈ Γ∞(X,ker Dπ). SecondlyweshowthatexactlumpabilityisimpliedbytheinvarianceofΓ∞(X,ker Dπ) underLv.Wewant
to definethevector fieldv as˜ asmoothfunctionofy suchthat˜vπ(x)= Dπxv(x) forallx∈ X.Thiswould
implyexactlumpabilitydueto(6).IfDπxv(x) isconstantalongthefibersx∈ π−1(y),then˜v iswelldefined
everywheremodulosmoothness,sinceπ issurjective.Weconsideravectorfieldw∈ Γ∞(X,ker Dπ) tangent to the fibers. ByProposition 9 the covariant derivative π∗∇¯π−1T Y
w Dπv = ˚∇π
−1T Y
w Dπv = DπJw,vK = 0
Itremainstoshowthat˜v isasmoothfunctionofy.Thisisthecaseifforanysmoothcurve˜γy: (−,)→
Y thecomposition˜v◦ ˜γy isasmoothfunctionintime.Butanysuchcurvecanbeviewedasthecomposition
of π withacurve γx : (−,)→ X, where π(x) = y.Since for any γx the equality ˜v◦ π ◦ γx = Dπ v◦ γx
holds,andsincetherighthandsideisacompositionofsmoothfunctionsandisthusalsosmooth,itfollows that˜v mustbesmooth. 2
Corollary5.The system(4) isexactlylumpableforπ iff ker Dπ isinvariantundertheflow Φ. Proof. Thisfollowsimmediatelyfrom Proposition 3. 2
Corollary6.The system(4) isexactlylumpableforπ iff ker Dπ⊆ ker L∇v¯Dπ.
Proof. ThisfollowsfromProposition 6. 2
Wemaketheconnectiontocontroltheorybyintroducingthe2-observabilitymap:
O2:= Dπ L∇¯ vDπ : T X→ π∗T Y ⊕ π∗T Y , asthemapping v→ (Dπ ⊕ L∇v¯Dπ)(v⊕ v)
Then-observabilitymapOn : T X→ nπ∗T Y isdefinedanalogouslywithhigher-orderLiederivatives.In
thelinearcase,where ˙x = v(x)= Ax andπ(x)= Cx,wehaveDπ = C,L∇v¯Dπ = CA,andO2=
C CA
; furthermore,Onisjustthestandardobservabilitymatrixfamiliarfromlinearcontroltheory[30],wherethe
systemiscalledobservable if rankOn= n.
Proposition10. The system(4)isexactlylumpable forπ iffrankO2= rank Dπ.
Proof. We consider thesituation locally. Let ψ : V˜ ⊆ Y → Rm be local coordinates ona patch V ⊆ Y ,
indexed bya,b andψ : U ∩ π−1V → Rn coordinateson apullback patch indexedby i.The rankofO2 is
equaltotherankof Dπ ifandonlyif
(L∇v¯Dπ)ai =
b
φab(Dπ)bi (18)
withsmoothcoefficientfunctionsφa
b.Noww∈ ker Dπ impliesw∈ ker L∇v¯Dπ,whichimpliesexact
lumpabil-itybyProposition 6.Ontheotherhand,consideringthelocalcoordinateform(14)ofL∇v¯Dπ anddemanding thesystemtobeexactlylumpable,
(L∇v¯Dπ)ai = ∂ ∂xi(˜v a◦ π) + ¯Γa bc(˜vc◦ π) ∂πb ∂xi = b ∂ ˜va ∂yb + ¯Γ a bcv˜c ◦ π (Dπ)b i,
Corollary 7.Thesystem (4)isexactlylumpable ifflocally:
m
b=1
(Dπ)b∧ d (Dπv)a= 0 ∀ a ∈ {1, . . . , m}.
Proof. Proposition 10statesthatthelocal condition(18)isnecessary andsufficientforexactlumpability. So, thevectors(Dπ)aand (L∇¯
vDπ)b arelinearlydependent.However,from (14)itis seenthatthe second
summand of (L∇v¯Dπ)b is already proportional to (Dπ)a, with the proportionality constant given by the
Christoffel symbol.Hence,onlythefirstsummandd(Dπv)ahastobecheckedforlineardependence. 2 3. Propertiesandexamples
Wenextdiscusssomepropertiesofexactlylumpablesystemsandillustrate themwithexamples.Avery prominent class of submersions are fiber bundles π : X → Y , and our examples are fiber bundle maps mostly over the 2-sphere Y = S2. We begin by relating lumpability to the theory of integrable systems.
Recall thatafirstintegral forthedynamicsv isafunctionI : X→ R suchthatv[I]= 0. Proposition 11.Any systemwith afirstintegralI of rank1isexactlylumpable.
Proof. Sincerank DI = 1,thequotientmapπ = I issubmersive.Thereexistsavectorfieldv = 0 on˜ Im(I) suchthatDIv = v[I]= 0= ˜v◦ I.Thus,v isexactlylumpableforI. 2
Remark8.Proposition 11alsoholdstrueifwerelaxtheconditionthatexactlumpingshavetobesubmersive and allowfortarget manifoldsthathaveboundaries or aresingularinother waysbutcanneverthelessbe endowedwith asmoothstructure.
InordertoillustrateProposition 11,weconsiderasanexamplethegeodesicflowonthe2-sphere,whichis generatedbyavectorfieldonthetangentbundleT S2.WeembedT S2 → R6by(x,v)→ (X,V )∈ R3×R3, togetherwiththerequirementthattheEuclideandotproductsforX andV satisfyX·X = 1 andX·V = 0.
Then, d dtXi = Vi d dtVi = −(V · V )Xi (19)
generatesthegeodesicflow[31].ThereisastationarysubmanifoldΩ={(X,V )∈ T S2: V = 0}.
We will use Proposition 11 to show that the geodesic flow (19) on T S2\Ω is exactly lumpable for
I : T S2→ R, given by I(X,V ) = V · V . First we note that I is a first integral to (19), which can
eas-ilybeseenbydifferentiatingI withrespecttotimeandusingX·V = 0.Thegeodesicflowcanbeviewedas aHamiltonianflowwhoseenergyisgivenby 12V · V .TherankofI is1,exceptonthestationary submani-fold Ω,whereitequals 0.Hence,I issubmersiveonT S2\Ω and satisfiesv[I]= 0.Therefore,thedynamics is exactlylumpableforI byProposition 11.
Asaconsequenceoftheenergyconservation,thegeodesicflowisjustconsideredononeenergyshell,say
V · V = 1;so iteffectivelytakesplaceontheunittangentbundleU T S2→ S2.
Proposition 12.Any dynamicsv isexactlylumpable forthequotientmapπ : X→ X/Φ totheorbit space.
Proof. The kernelofπ issimplythedistributionspannedbyv.ThisistriviallyinvariantundertheflowΦ generated by v,sinceDΦsv = v◦ Φs bydefinitionand v = DΦt∂t∂
0.Exact lumpabilitythen followsfrom
Fig. 1. Wechooselocalcoordinates(x,y,α)∈ ψ(U),whereU istheunittangentbundlerestrictedtothenorthpoleN⊂ S2.The
functionψ actsbystereographicprojectiononthe2-sphereandmapstheunittangentvectorv toanangleα∈ [0,2π),whichisthe angleenclosedbythex-directionandthepushforwardofv underthestereographicprojection.Wedepictfibersoftheprojection
π intherangeπ/2≤ α≤ 3π/2 fromtwodifferentperspectives,indicatingalsotheflowfieldin(a).Thelongitudesandlatitudes ofthesphereareseenonthebottomofthefiguresforreference.
To exemplify this proposition, we now consider the geodesic flow on the unit tangent bundle of the 2-sphere U T S2.Weclaimthatitis exactlylumpable forthecrossproduct(X,V )→ X × V ∈ S2 and use
theaboveproposition to showthis.
Thereisanisomorphism[31]betweentheunittangentbundleU T S2andSO(3),givenby(X,V )→ M,
whereMi1 = Xi,Mi2= Vi,andMi3= (X× V )i,orincompressednotationM = (X|V |X × V ).So,forany
p∈ S2 this matrixmapsto another pointy = M · p∈ S2, andthere isacollectionof lumpingcandidates
indexed by p. We choosep = (0,0,1) and calculatethe vector fieldinduced by π(X,V ) = M (X,V )p =
X× V : 3 i=1 ∂π ∂Xi d dtXi+ 3 i=1 ∂π ∂Vi d dtVi= (0, 0, 0) .
Thus,thedynamics(19)onU T S2liesinthekernelofDπ.Butπ issurjectiveontoS2,ithasconstantrank,
and dim ker Dπ = 1.So, thevectorfieldand hencetheflowis parallelto thefibersandeverypoint onS2
correspondstoaflowlineofthegeodesicflow.ThisisillustratedinFig. 1.ByProposition 12,π isanexact lumping.
We next discuss the relation of lumpability to the symmetries of the system. We shall show that the properactionof aLie groupthatiscompatiblewiththevectorfieldresultsinanexactlumping;however, theconverseisnottrue.LetG beafiniteLiegroupwithLiealgebrag.WedenotebyA: G→ Diff(X) the leftactionoftheLieGrouponX anda: g→ Γ∞(X,T X) thecorrespondingactionoftheLiealgebra.The actiononthewholealgebraisdenotedbyD = a(g).
Proposition 13. If D isinvariant under Lv and G actsproperly andfreely, then v is exactlylumpable for
Proof. By the quotientmanifold theorem [1] thequotient map of aproper and freeLie groupaction is a submersionandthequotientspacehasanaturalsmoothmanifoldstructure.Thevectorfieldsthatgenerate the action are annihilated by the differential of the quotient map; therefore, D = Γ∞(X,ker Dπ) andso Proposition 4 impliesexactlumpability. 2
The conversestatementtoProposition 13is nottrue.Givenavector fieldv and alumpingπ, thelevel setsneednotbeorbitsofaproperandfreeLiegroupaction.TheintegrabledistributionofafreeLiegroup action is spanned byits linearly independent generatorsmaking D a finitely generated submoduleof the sections of T X. There are many integrable distributions that are not finitely generated and thus do not stemfromaLieGroupaction.Anysectionofsuchadistributiongivesrisetoalumpingthatdoesnotstem from aLie groupaction.
TheHopffibrationoverS2,
S1 → S3−→ Sπ 2,
illustratesProposition 13.WeusetheformulationoftheHopfmapintermsofthequaternionsH= (R4, ,∗),
whichisthevectorspaceR4togetherwithaninvolution∗:H→ H andanalgebraproduct· ·:H×H→ H.
Leta= (a0,a1,a2,a3) andb= (b0,b1,b2,b3) betwoelements inH.Then isdefinedby
(a b)0=a0b0− ajbj
(a b)i=a0bi+ aib0+ ijkajbk,
where theindices i,j,k runover{1,2,3} andijk is theLevi-Civita symbol. Itistotally antisymmetricin
itsindices. Theinvolutionactsas (a0,a1,a2,a3)→ (a0,−a1,−a2,−a3).The3-sphereS3 canbeembedded
into H by UH= {x ∈ H : ||x|| = 1}. To each unitquaternion x ∈ UH one can associate an element in
SO(3), actingonpurelyimaginaryquaternions u∈ IH={a∈ H: a0= 0}∼=R3 by
u→ Rx(u) = x u x∗∈ IH.
Onecanshow thatthemappingx→ Rx isasmooth, nondegenerate,two-to-one, surjectiveassignment of
any x toanelementof SO(3) andthatS3isinfactthedoublecoverofSO(3).Hencethere isacollection
of submersionsπu: S3→ S2 indexedbyvectorsu∈ S2thatactlikeπu(x)= Rx(u).Choosingu= (0,0,1)
andsettingπ = πuweget
πi(x) = (x20− xjxj)δi3+ 2x0ij3xj+ 2x3xi (20)
as oneexampleofaHopfmap.Alternatively,onecandescribethismapasthequotientofaU (1) actionon
S3∼= UH.WeusetheabbreviationI= (1,0,0,0),I = (0,1,0,0),J = (0,0,1,0),andK = (0,0,0,1).They
satisfythequaternionalgebraI I = J J = K K = I J K =−I.TheU (1) action
(eKt, x)→ eKt (x0+ Kx3) + e−Kt J (x2− Kx1) (21)
is generatedbythevectorfieldw(x)= (−x3,x2,−x1,x0).Wenow showthatπ isthequotientmap ofthe
U (1)-action (21).Thedifferentialof(20)isgivenby
(Dπ)iμ= 2(x0δμ0− xjδμj)δi3+ 2(xjδμ0+ x0δμj)ij3+ 2δμ3xi+ 2δμix3.
A calculation reveals that Dπw = 0, and w spans ker Dπ since π is a submersion and ker Dπ is one-dimensional.
Having introducedthe lumping map π : S3 → S2 intheframework ofquaternions and theLie algebra
action,generatedbyw,wenowproceedwiththeexample.Thereisacollectionofvectorfieldsvc(x)= c x,
indexedbyc∈ IH,givenby
(vc)μ(x) =−δμ0cjxj+ δμjcjx0+ δμjjklckxl,
whichisexactlylumpableforπ asin(20).WewillnowshowthatthisfollowsfromProposition 13.TheLie groupU (1) iscompact; so,itsactionis properand,sincew is nowherevanishing,itisalso free.Wecheck whetherLvcw∈ Γ∞(X,ker Dπ): [w, vc]α= wμ ∂(vc)α ∂xμ − (vc )μ ∂wα ∂xμ = + (x1c2− x0c3− x2c1+ x0c3− 3klxkcl)δα0 − (x0c2+ 2klxlck)δα1+ (x0c11klxlck)δα2+ (xjcj− x0c0)δα3 − (x3cj− jk1x2ck+ jk2x1ck− jk3x0ck)δαj = 0 .
SoweinvokeProposition 13whichimplieslumpability.Infact, (Dπvc)i(x) = 2ijkcjπk(x).
Thelumpeddynamicsforthevectorfieldthatgeneratesquaternionrotationsvc=dtd0etc x= c x under
thequotientmapπ is ˜vc(y)= 2c× y.Clearlyitrunstangenttothespheresincev˜c· y = 0 fory∈ S2.
Proposition14. Exactlumpingspreserve invariantsets.
Proof. LetA beaforward(resp.,backward)invariantset,i.e.forallt≥ 0 theflowpreservestheinvariantset ΦtA⊆ A (resp.,Φ−tA⊆ A).Afteraprojectionwiththelumpingmap,π◦ΦtA⊆ πA (resp.,π◦Φ−tA⊆ πA).
Invokingthelumping conditionfrom Proposition 2yields ˜
Φt◦ πA ⊆ πA (resp., ˜Φ−t◦ πA ⊆ πA);
so,πA isaforward(resp.,backward)invariantset ofΦ˜t. 2
Thispropertycanbeexploitedtodetermineinvariantsetsofthedynamicsbyfindingthestationarypoints ofa1-dimensionalexactlumping.Weconcludewithafinalexamplewhichalsoillustratesthisfeature.For asetofrealcoefficientsai whicharenotallzero,thelogisticdynamics
˙xi = xi(1− ajxj), i = 1, . . . , n,
has two invariant sets Ω0 = {ajxj = 1} and Ω1 = {x = 0} that are preserved under the lumping map
π(x)= ajxj.Withvi= xi(1− ajxj) wecalculate
Dπv(x) = ∂π
∂xi
vi(x) = aixi(1− ajxj)
andfindthatv(y)˜ = y(1− y) isthelumpeddynamics.HencebyProposition 14,πΩ0andπΩ1areinvariant
Acknowledgement
The research leadingto these resultshas receivedfunding from the European Union’sSeventh Frame-workProgramme(FP7/2007-2013)undergrantagreementno.318723(MATHEMACS).L.H.acknowledges fundingbytheMaxPlanckSocietythroughtheIMPRSscholarshipandfurthersupportbytheBRCPe.V. References
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