4.4 Geodesics and the Christoffel symbols
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1 Clssnotes: Geometry & Control of Dynmicl Systems, M. Kwski. April 20, Geoesics n the Christoffel symbols There re mny wys in which to pproch geoesics: closest to the originl sense of this wor is the chrcteriztion of geoesics s curves tht minimize istnce in sense yet to be me precise. Closely relte is their chrcteriztion s the nlogues to the stright lines of Euclien spces. Geoesics lso govern the evolution of mechnicl systems, s the problems of minimizing length, minimizing curvture, n minimizing ction the time integrl of energy) re intimtely relte. We strt by stuying the problem of minimizing istnce, n then chnge focus to the intricte interply of geoesics n notions of curvture: One might tke the excess or eficit) of the sum of the interior ngles of tringle me up of stright sies s n intuitive mesure of the totl curvture of tringulr region). The clssicl pproch uses clculus of vritions to erive criteri necessry conitions) tht curve ˆσ : [, b] M must stisfy if its length is the lest mong ll nerby curves tht connect σ) p to σb) q. There re mny technicl issues involve, such s the smoothness of the curves uner consiertion, their omins sme intervls [, b], or vrible omins), the notion of istnce between curves, the smoothness of the functionl to be minimize, the generlity of the vritions consiere). However, for the purpose of ientifying geoesics very wek tools suffice. [[ Much more powerful tools re vilble in the context of optiml control, which lso pplies to nonsmooth problems, builing on ifferent point of view tht origintes with the Pontrygin mximum principle. One reily pprent ifference is tht it uses ifferentil equtions on the cotngent bunle, s oppose to the tngent bunle. More eeply, key ifference re much weker regulrity ssumptions on the set of missible vritions.]] The clssicl strting point for the clculus of vritions re minimiztion problems of the following form: On some subset U R 2n+1 ), given running cost L C 2 U), fin mong ll piecewise C 2 -curves γ : [, b] R n tht stisfy γ) p n γb) q the ones) tht minimize Jγ) b Lt, γt), γ t)) 203) Simple exmples in the cse of n 1 re Lt, x, y) 1 + y 2 in which cse Jγ) is the rc-length of the grph of γ, n Lt, x, y) 2π 1 + y 2 in which cse Jγ) is the re of the surfce of revolution obtine by revolving the grph of γt) bout the t-xis. For generl n 1 the integrn Lt, x, y) n y k 2 ) 1/2 yiels the length of the imge) of the curves γ : [, b] R n. k1 Our interest lies in curves tht tke vlues in Riemnnin mnifol n the integrn is function L: T M R efine by Lp, X p ) X p, X p 1/2 p. The bsic ie is tht if the functionl J is ifferentible t curve ˆγ n J ˆγ) 0 then there exist curves γ ner ˆγ such tht Jγ) < Jˆγ), i.e. ˆγ cnnot possible be minimizer. For the purposes consiere here it is not necessry to evelop full notion of ifferentibility of functionls J with respect to curves γ, but it suffices to consier irectionl erivtives obtine from one-prmeter fmilies of vritions. In the bsic cse of curves in Euclien spce R n one my restrict to itive vritions. More specificlly, for two fixe C 2 -curves ˆγ, η : [, b] R n with ˆγ) p n ˆγb) q, while η) 0 n ηb) 0, consier for ε R sufficiently smll e.g. so tht for ll t [, b], γ ε t) U) the fmily of curves γ ε : [, b] R n efine by γ ε ˆγ + εη. Note tht for ll ε R, γ ε ) p n γ ε b) q If ˆγ minimizes J then necessrily for ll ε, Jˆγ) Jγ ε ). Assuming sufficient
2 Clssnotes: Geometry & Control of Dynmicl Systems, M. Kwski. April 20, smoothness of L n η, ifferentite the function ε Jγ ε ) which mps R to R. ε ε0 Jγ ε ) ε b ε b b ε0 k1 n k1 Lt, γ ε t), γ εt)) ε0 Lt, ˆγt) + εηt), ˆγ t) + εη t)) Lx kt, ˆγt), ˆγ t)) η k t) + L y kt, ˆγt), ˆγ t)) η k t) ) L y kt, ˆγt), ˆγ t)) η k t) tb t b + Lx kt, ˆγt), ˆγ t)) L y kt, ˆγt), ˆγ t)) ) η k t) k1 204) Here L x k n L y k stn for the prtil erivtives D k+1 L n D n+k+1 L. The key is the integrtion by prts in the lst step together with the observtion tht the bounry terms vnish since η) ηb) 0, i.e., since the vritions hol the enpoints fixe. Now one rgues tht if ˆγ minimizes J then this erivtive must be zero for ll C 1 -curves η : [, b] R n with η) ηb) 0. Using the subsequent lemm, n consiering specil curves η such tht ll coorinte functions η k except one re ienticlly equl to zero one estblishes tht the integrns must be ienticlly equl to zero for ech k n: Proposition 4.9 Suppose L C 2 R 2n+1 ), ˆγ C 2 [, b], R n ), n J : C 2 [, b], R n ) R is efine by Jγ) b Lt, γt), γ t)). If Jˆγ) Jˆγ + εη) for ll η C 1 [, b], R n ) with η) ηb) 0, then ˆγ must stisfy the Euler-Lgrnge equtions L x kt, ˆγt), ˆγ t)) L y kt, ˆγt), ˆγ t)) 0 for ll t [, b], k 1,..., n 205) Lemm 4.10 Suppose f C 0 [, b]). If for ll g C [, b]) with g0) gb) 0, b ft)gt) then f 0. The lemm is stte here for the cse of the integrl vnishing for ll smooth g. It is n re of continue reserch to fin ever stronger, similr conitions for vrious regulrity ssumptions. Typiclly, e.g. in control theory, one hs much weker regulrity ssumptions on L n ˆγ, while consiering more innovtive fmilies of vritions. Proof. Suppose t 0, b) is such tht ft 0 ) 0. W.l.o.g. ssume tht ft 0 ) > 0. Since f is continuous, there exists δ > 0 such tht < t 0 2δ, t 0 + 2δ < b, n such tht ft) 1 2 ft 0) for ll t [t 0 2δ, t 0 + 2δ]. There exists smooth function g C [, b]) g 0, such tht gt) 1 if t [t 0 δ, t 0 + δ] n gt) 0 if t [t 0 2δ, t 0 + 2δ]. Then b ft)gt) t0 +2δ t 0 2δ ft)gt) > t0 +δ t 0 δ ft)gt) > 0 206) yiels contriction, thus ft) 0 for ll t, b), n by continuity lso for ll t [, b].
3 Clssnotes: Geometry & Control of Dynmicl Systems, M. Kwski. April 20, To provie feeling for the Euler-Lgrnge equtions it is encourge to work few clssicl exmples from clculus of vritions. Exercise 4.22 Evlute the Euler-Lgrnge equtions for the problem of minimizing the rclength of the grph mong ll C 2 -functions γ : [, b] R n tht stisfy f) p, n fb) q for fixe enpoints p, q R n. Hint: Use Lt, x, y) 1 + y 2.) Crefully rgue why minimizer must exist, why it is unique, n then ientify the minimizing function. Exercise 4.23 Evlute the Euler-Lgrnge equtions for the problem of minimizing the rclength mong ll C 2 -curves γ : [, b] R n tht stisfy f) p, n fb) q for fixe enpoints p, q R n. Hint: Use Lt, x, y) y.) Crefully rgue why minimizer must exist, why it is unique, n then ientify the minimizing function. Exercise 4.24 Evlute the Euler-Lgrnge equtions for the problem of minimizing the re of the surfce of revolution obtine by revolving the grph of C 2 -function γ : [, b] R tht stisfy f) p, n fb) q bout the t-xis. Hint: Use Lt, x, y) 2πx 1 + y 2.) Crefully rgue why minimizer must exist, why it is unique, n then ientify the minimizing function by integrting the ifferentil equtions. The solution is commonly expresse in terms of the hyperbolic cosine. Similr clssicl exmples involve the shpe of physicl chin hnging uner its own weight between two points the solution is lso ctenry), n the curve of fstest escent uner grvity brchystochrone). These exercises suggest tht it my be worth to spen some extr effort to circum-nvigte the unplesnt ricl in the integrn of bove problems. Inee, s the subsequent nlysis shows, this is inee preferble pproch. The next exercise will give some sense for wht is going on. Exercise 4.25 Evlute the Euler-Lgrnge equtions for the problem of minimizing the integrl of the energy mong ll C 2 -curves γ : [, b] R n tht stisfy f) p, n fb) q for fixe enpoints p, q R n, i.e. use the cost Lt, x, y) y 2.) Compre the resulting ifferentil equtions, n their solutions with those of minimizing rc-length s in exercise xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx Exercise 4.26 Euler-Lgrnge equtions when integrn oes not epen on x Exercise 4.27 Euler-Lgrnge equtions when integrn... xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
4 Clssnotes: Geometry & Control of Dynmicl Systems, M. Kwski. April 20, On Riemnnin mnifols, the notion of itive vritions γ ε ˆγ + εη generlly oes not mke sense. Hence, for fixe curve ˆγ : [, b] M one my consier fmilies of vritions of the form γ : ε 0, ε 0 ) [, b] M such tht for ll ε ε 0, ε 0 ), γε, ) ˆγ), γε, b) ˆγb), n such tht for ll t [, b], γ0, t) ˆγ. At this point the ro forks: A clen geometric tretment woul evelop the nlogous theory for cost functions L C 2 [, b] T M) leing to ifferentil equtions on T T M. This pproch involves little more bstrction, n quite bit extr nottion tht reflects the inherent symmetries of the 4n-imensionl mnifol T T M, e.g. ech T Xq T M contins two copies of T q M. We here follow more peestrin pproch by working in coorinte chrts which is sufficient for the locl results tht we esire, which is consierbly fster but less elegnt. In prticulr, this route emns tht one py ttention to curves leving chrts n entering neighboring chrts. It is left s n exercise to work out the etils for eriving the Euler- Lgrnge equtions for generl, not necessrily itive vritions, in the setting of curves in Euclien spce which in some sense is first step towrs the geometric lterntive ro. Exercise 4.28 Work out n nlogous clcultion to 204) tht le to proposition 4.9, for the more generl cse which uses not necessrily itive vritions γ : ε 0, ε 0 ) [, b] R n s iscusse bove. Hint: Now η is replce by ε ε0 γ n key step uses tht t ε γ ε t γ. in progress Chnge to moern presenttion, long the lines of H. J. Sussmnn n J. C. Willems, 300 Yers of Optiml Control: from the Brchystochrone to the Mximum Principle, IEEE Control Systems Mgzine, 1997, pp rutgers.eu/~sussmnn/ppers/systems-mgzine-brch.ps.gz in progress
5 Clssnotes: Geometry & Control of Dynmicl Systems, M. Kwski. April 20, Bck to Riemnnin mnifols, pply the clculus of vritions rgument in one ech chrt t time. It will turn out helpful lter to llow the reference curves to only be piecewise C 2, n lso llow more flexibility with regr to the enpoints. Specificlly, suppose M is Riemnnin mnifol n p q M re two points joine by piecewise C 2 -curve ˆσ : [, b] M, i.e. ˆσ) p n ˆσb) q. Suppose tht t 0 < t 1 <... t N b re times such tht the restriction of ˆσ to ech of the intervls t r 1, t r ) is C 2. W.l.o.g. one my ssume tht for ech subintervl [t r 1, t r ] there is chrt u r, U r ) such tht ˆσ[t r 1, t r ]) U r. To voi excessive nottion we o not introuce itionl symbols to enote the sequence u r, U r ) of chrts, n simply write u, U) - but cre must be tken lter s e.g. there will be functions g ij tht re ifferent from chrt to chrt. As we shll justify lter, inste of minimizing length, we shll investigte the problem of minimizing the integrl of the energy σ, σ. Writing γ u σ : [t r 1, t r ] uu) R n, the problem is to minimize the sum of the integrls sum over ll subintervls) Eσ [tr 1, t r] ) tr tr t r σ t), σ t) σt) t r u 1 ) γ t), γ t) γt) tr t r i,j1 g ij σt)) γi γ j 207) In generl, consier vritions tht preserve the piecewise C 2 -nture of the clss in which we re looking for extremls, σ : [, b] ε 0, ε 0 ) M, such tht for ll ε ε 0, ε 0 ), σ, ε) p n σb, ε) q. 208) n for ech ε ε 0, ε 0 ) fixe, n ech 1 k N, the restriction of the curve t σt, ε) M to the intervl t r 1, t r ) is C 2. More generlly, one coul hve tht the subintervls themselves epen on the prmeter, i.e. t r 1 ε), t r ε)), which in similr pplictions my le to stronger necessry conitions for optimlity. But there is no nee for this here.) On ech of the subintervls, rgue s before: integrte by prts n obtin the nlogue of the Euler Lgrnge equtions for this energy functionl. However, in generl the bounry terms nee no longer vnish t the enpoints of the subintervls. Thus one expects tht the finl necessry conition contins lso vrious terms evlute t these times t r. However, to simplify mtters, one my in first step only consier vritions tht re supporte on single, fixe subintervl, i.e., for ll t [, b] \ t r 1, t r ), n ll ε ε 0, ε 0 ), σt, ε) σt, 0) ˆσt). Note tht by only consiering such specil kins of vritions one potentilly obtins weker necessry conition for extremls i.e., if some erivtive hs to be zero for ll vritions, then it must, fortiori, be zero for the specil clss. Two lst nottionl items before proceeing to the clcultion: Assuming tht u, U) is chrt such tht σ[t r 1, t r ] ε 0, ε 0 ) U, write γ u σ, n for 1 i, j n write g ij,. u i u j Note tht fully written out in terms of the vribles t n ε, the integrn in the energy integrl is σ t, ε), σ t, ε) σt,ε) g ij σt, ε) γi γj t, ε) t, ε). 209) using tht for every t, ε), σ t, ε) n i1 ij1 ui σ)t, ε) u i σt,ε). In prticulr, the g ij re evlute long the imge of) σ, wheres the first n secon erivtives re of the coorinte functions γ i u i σ.
6 Clssnotes: Geometry & Control of Dynmicl Systems, M. Kwski. April 20, As before, for the Euler Lgrnge equtions, s before integrte by prts tr ε Eσ ε ) [tr 1, t r] ε0 tr t r 1 k1 i,j1 t r i,j1 ) 1 gij γ i 2 u k σ ε g ij σ) γi ε0 γ j j1 γ j ) ) g ik σ) γi ε ) γε k 210) ε0 For lter use, recor tht in generl in this clcultion one woul hve for ech subintervl t r 1, t r ) the itionl bounry terms j,l1 g ik σ) γi ) tr ε γ ε. 211) ε0 t r 1 Agin use lemm 4.10 to obtin for 1 k n now consiering only vritions whose support is contine in [t r 1, t r ]) the following equtions. For nottionl simplicity one my from now on gin write σ ) n γ ) for σ, 0) n γ, 0) i,j1 ) gij γ i u k σ γ j g ik u l γ l σ)γi + g ik σ) 2 γ i 2 212) i,l1 i1 This is customrily rewritten in the more symmetric form i1 g ik σ) 2 γ i i,j1 gij u k g ik u j g ) ) jk γ i u i σ γ j 213) Interpret this first erivtive is equl to zero conition s system of n liner equtions in the n unknowns 2 γ i, 1 k n. To solve for the secon erivtives, multiply by the inverse of 2 the mtrix G g ij σ) n ij1, i.e. 2 γ k 2 l1 δ k i 2 γ i 2 g kl σ)g il σ) 2 γ i 2 214) i,l1 to obtin the geoesic equtions, for 1 k n, 0 2 γ k i,j,l1 g kl σ) + g il u j + g jl u i g ij u l ) ) γ i σ These re usully bbrevite by introucing the Christoffel symbols of the first kin [ij, l] 1 gil 2 u j + g jl u i g ) ij u l γ j 215) 216) n the Christoffel symbols of the secon kin Γ k ij g kl [ij, l] l1 l1 g kl 1 2 gil u j + g jl u i g ) ij u l. 217)
7 Clssnotes: Geometry & Control of Dynmicl Systems, M. Kwski. April 20, Using these bbrevitions the geoesic equtions become for 1 k n, 0 2 γ k 2 + Γ k ij σ) γi in progress i,j1 γ j. 218) revisit the piecewise smooth problem, trck own the corner conitions somewhere: write out Γ k ij for some exmples - surfces in R3. Refer to MAPLE worksheets Proposition 4.11 Suppose σ : [, b] M is piecewise smooth n criticl point of the energy functionl E b. Then σ is smooth. in progress Proposition 4.12 If σ : [, b] M is criticl point for the energy E, i.e. it stisfies 218) then it is prmeterize proportionlly to rc-length, i.e. σ, σ :, b) R is constnt. Proof. This is strightforwr clcultion - minly requiring ttention to the severl summtion inices. Suppose tht σ : [, b] M is criticl point for the energy functionl E b, i.e. it stisfies 218), n let u, U) be chrt bout t 0, b). For t sufficiently close to t 0 let γ u σ. For nottionl convenience omit the rgument of g ij which techniclly is to evlute t σt)), n clculte: Continuing the common nottionl buse or simplifiction, in the following clcultion write g ij where correctly one shoul use g ij σ. σ, σ m i,j1 j1 i,j1 j1 i,j,l1 γ j l1 ij1 gij u l γl i,l1 γ j m l1 γ j i,l1 g ij γi γj γ i gij u l γl gij u l γl gij u l γl γ j ) +2 g ij 2 γ i γ j ) 2 γ i ) γ i ) γ i ) 2g ij 2g ij 2 m r,s1 m k,r,s1 grj r,s1 Γ i rs γr g ik grk u s gij u l g ij u l g lj u i + g ) il u j γi γ j γ l γ s + g sk u r u s + g sj u r g rs u j 0. g ) rs u k γr ) γr γ s γ s
8 Clssnotes: Geometry & Control of Dynmicl Systems, M. Kwski. April 20, Now return to the originl problem of minimizing the length of the curve joining two points p n q, i.e. consier the functionl L b : σ b σ t), σ t) 219) The clculus of vritions is essentilly s before, but in orer to stisfy the ifferentibility hypotheses in the erivtion of the Euler-Lgrnge equtions first only consier curves tht re immersions, i.e σ t) 0 for ll t [, b]. To outline the bsic steps, informlly write... F xt), ẋt)) for the integrn in the minimiztion problem. The Euler-Lgrnge equtions evlute to ) F x 1 F y Fx F ) y + F Fx 1... ) 3... F y ) + F F ) 220) Exercise 4.29 Write out the etils to erive the rel) geoesic eqution, i.e. the conition for curve σ to be criticl point of the length functionl L b. The squre root in the enomintor plys no role. The next two terms re exctly the sme s in the minimiztion of the energy functionl. The lst term is in the problem of minimizing length) recognize s σ t), σ t) σ t), σ 221) t) In the cse tht the spee of σ is constnt, i.e. when σ is prmeterize proportionlly to rc-length then this term vnishes ienticlly, n thus: Proposition 4.13 If σ is criticl point of the energy functionl E b, then it is lso criticl point of the length functionl L b. Conversely, suppose σ is criticl point of L b, n, in prticulr, σ t) 0 for ll t [, b]. Let σ be the reprmeteriztion of σ by rc-length, i.e. σ σ s 1 where s: t L t σ [,t] ) t σ τ), σ τ) 1 2 τ. 222) The totl length T L b σ) of σ clerly is the sme s the totl length L T σ). Thus σ is lso criticl point of the length functionl but generlly for the ifferent intervl [, t]. Hence it lso stisfies the Euler-Lgrnge equtions for the length functionl but ue to the constnt spee the lst term 221) vnishes ienticlly. Consequently: Proposition 4.14 If σ is criticl point of the length functionl n in prticulr σ t) 0 for ll t), then σ is lso criticl point of the energy functionl. These two proposition justify tht one generlly simply works with the energy functionl s oppose to the messier length functionl except for one techniclity: A length minimizing curve between two points p n q might conceivbly be smooth, but hve corner t which the spee must tke the vlue zero):
9 Clssnotes: Geometry & Control of Dynmicl Systems, M. Kwski. April 20, Proposition 4.15 Suppose σ : [, b] M is piecewise smooth curve with σ) p n σb) q, n σ hs corner t t 0, b), i.e. the left n right hn limits σ t 0 0) σ t 0 +0) o not gree. Then for every ε > 0 n ll δ > 0 sufficiently smll, there exists curve γ : [, b] M which grees with σ on [, b] \ t 0 δ, t 0 + δ), which is smooth on [t 0 δ, t 0 + δ] n whose length L b γ) is strictly less thn L b σ). Exercise 4.30 Prove the proposition. [[The ssertion is rther intuitive n irect construction guie by intuition oes go through however there re lots of technicl etils which provie for rther long rgument. Since the rgument is of locl nture it mkes sense to work in coorintes, n use the metric u 1 ), on uu) R m. The obvious construction of shortcuts voiing the corner is strightforwr, but the estimtes of the length t the ens where the short-cut smoothly joins σ require creful ttention to etil. ]]. So fr it is cler tht for every initil point p M n every initil velocity X p there exists unique geoesic σ with σ0) p n σ 0) X p, which is efine on some intervl ε, ε) with ε > 0. In orer to evelop suitble tools to prove e.g. minimlity, n to fcilitte lter chrcteriztions of curvture, it is esirble to shift the vribility n obtin escriptions tht show more uniformity. The key steps will be to consier open sets of initil points, n to obtin fixe time intervls on which the geoesics re efine t the cost of limiting the initil velocity). In the en, it is most elegnt to only consier the evlution of geoesics t time t 1, while letting the initil point n velocity rnge over some subsets of the tngent bunle T M. We procee in severl smll steps. Theorem 4.16 For every p M there exist neighborhoo U of p n r 0 > 0 such tht for every q U n every X q T q M with X q < r 0 there exits unique geoesic σ : 2, 2) M such tht σ0) q n σ 0) X q ). Proof. By virtue of the existence n uniqueness theorem for initil vlue problems there exists for every p M some neighborhoo U of p n some r 0, t 0 > 0 such tht for every q U n every X q T q with X q < r 0 there exists unique geoesic γx q, ) efine on the intervl t 0, t 0 ) such tht γx q, 0) q n γ X q, 0) X q. Fix ny ε < 2r 0 t 0. If q U, X q T q M with X q < ε n t < 2, then Y q 1 2t 0 X q stisfies Y q < 1 2t 0 ε < r 0 n 1 2 t 0 t < t 0 Hence one my efine for X q < ε n t < 2, σx q, t) γy q, 1 2 t 0 t). 223) Definition 4.8 Let V T M be the set of ll initil points/velocities q, X q ) such tht there exists geoesic σ σ q,xq) : [0, 1] M with σ0) q n σ 0) X q. Define the mp exp: V M, clle exponentil by expx q ) exp q X q ) σ q,xq)1). Since the geoesic flow s solution of smooth initil vlue problem is smooth, it is cler tht exp is C mp. Since the geoesic eqution is secon orer ifferentil eqution, there is little surprise tht the tngent mp of exp comes into ply. It mps suitble subset of T T M) to T M. It is common prctice to use bbrevite nottion s the 4m-imensionl mnifol T T M) contins some upliction tht is rrely of importnce. The following mkes n effort to be precise, without being overly pentic but it my help to just recognize tht ifficulties my simply ue to nottionl problems. For ny fixe point p M the mp exp p mps some subset of T p M to M. At ny X p T p M the tngent mp exp p ) Xp mps T Xp T p M) to T expp X p)m. In the specil cse tht X p 0 0 p, clerly exp p X p ) p n the tngent mp exp p ) 0p mps T 0p T p M) to T p M.
10 Clssnotes: Geometry & Control of Dynmicl Systems, M. Kwski. April 20, Lemm 4.17 The mp exp p ) 0p : T 0p T p M) T p M is the ientity mp uner the nturl ientifiction of the vector spces T 0p T p M) with T p M. Proof. Fix X p T p M n for r > 0 sufficiently smll consier the curve σ : r, r) T p M efine by σt) t X p. Note tht σ 0) X p T p M T 0p T p M). Then exp p σt)) exp p tx p ) n hence exp p ) 0p X p ) exp p σt)) X p T p M. 224) t0 Theorem 4.18 [lmost verbtim from Spivk 2 n e., p.454] For every p M there exist neighborhoo W n ε 0 > 0 such tht Any two points q n q in W re joine by unique geoesic of length less thn ε. Let γq, q ) T q M be the unique tngent vector with γq, q ) < ε such tht exp q γq, q )) q. Then γ C W W, T M). For ech q W the mp exp q mps the open bll {X q T q M : X q < ε} iffeomorphiclly onto n open set U q W. Proof. not relly hr, min ifficulty is nottionl involving objects in T T M. The min observtion is tht the mp Φ mpping neighborhoo of p, 0 q ) in T M into M M must be one-to-one on some sufficiently smll neighborhoo. This is immeite from the observtion tht the tngent mp F p,0p) is the ientity from T p,0p)t M to T p,p) M M) T p M T p M.... to be complete... Definition 4.9 Suppose p M n r 0 > 0 is such tht exp p is efine for ll X p T p M with X p < r 0. Then for r < r 0 the imge of the set {X p T p M : X p + r} uner the mp exp p is clle the geoesic sphere bout p of rius r. Exercise 4.31 Use computer lgebr system, possibly combine with numericl pckge to explore the shpe of geoesic spheres for bsic mnifols, such s the grphs of z x 2 + y 2 n z 1/1 + x 2 + y 2 ). Py specil ttention for which mgnitues of the initil velocities the spheres re imbee spheres, n for which vlues they evelop folovers. [[Compre the existing MAPLE worksheets n exten the nimtions provie therein.]] Proposition 4.19 Guss lemm) Suppose p M. Then for t > 0 sufficiently smll the geoesics through p re orthogonl to the submnifols {exp p X p ): X p T p M, X p t}. Proof. Let r 0 > 0 be so tht exp p is efine n one-to-one on the bll {X p T p M : X p r 0 }. Fix ny r 0, r 0 ] n consier ny curve γ : ε 0, ε 0 ) T p M for some ε 0 > 0) with γε) r for ll ε ε 0, ε 0 ). Define the mp Φ: ε 0, ε 0 ) [0, 1] M by Φε, t) exp p t γε)). 225) For ech fixe vlue of ε the curve t Φε, t) hs energy E 1 0Φε, )) 1 0 D 2 Φε, t) r 2 r 2 226)
11 Clssnotes: Geometry & Control of Dynmicl Systems, M. Kwski. April 20, Consiering Φε, ) s one-prmeter fmily of smooth vritions of the geoesic σ Φ0, ), the first vrition formul yiels 0 ε E0Φε, 1 )) D 1 Φ0, 1), D 2 Φ0, 1) D 1 Φ0, 0), D 2 Φ0, 0). 227) ε0 Since the ltter term is clerly zero, this yiels tht σ r) D 2 Φ0, 1) D 1 Φ0, 1). The result follows from the fct tht ny tngent vector in the tngent spce to the submnifol {exp p X p ): X p T p M, X p t} is of the form D 1 Φ)0, 1) for some curve γ s bove. Corollry 4.20 Suppose p M n r 0 > 0 is such tht the exp p mps the bll B 0 r 0 ) iffeomorphiclly onto some open neighborhoo U of p. Suppose σ : [, b] U \ {p} is piecewise smooth curve. Let r : [, b] [0, r 0 ) n θ : [, b] {X p T p M : X p 1} be the unique curves such tht exprθ) σ. Then L b σ rb) r) with equlity only if θ is constnt n r is monotoniclly incresing or monotoniclly ecresing. Pictorilly, equlity mens tht σ grees up to reprmeteriztion) with ril geoesic tht connects two geoesic spheres. Proof. firly strightforwr Corollry 4.21 Suppose p M n r 0 > 0 is such tht the exp p mps the bll B 0 r 0 ) iffeomorphiclly onto some open neighborhoo U of p. Suppose σ : [, b] U \ {p} is piecewise smooth curve. Let r : [, b] [0, r 0 ) n θ : [, b] {X p T p M : X p 1} be the unique curves such tht exprθ) σ. Then L b σ rb) r) with equlity only if θ is constnt n r is monotoniclly incresing or monotoniclly ecresing. Exercise 4.32 guie explortions with MAPLE first provie intuitive n technicl efinitions of conjugte point n envelopes from clculus of vritions then explore them in ction. see lso mple worksheets Exercise 4.33 Use combintion of computer lgebr system to generte the coe for the geoesic equtions) n numericl ifferentil equtions solver e.g. MATLAB) to nimte the evolution of the geoesic spheres, i.e the imges of the spheres {X p T p M : X p c} for incresing vlues of the rius c uner the exp mp. A goo strting point might be the grph of the function f : x, y) ε sinx) siny) for suitble prmeter ε. [[First experiments suggest very rpi evelopment of mny folovers. Choosing smller vlues for ε will prouce clmer ynmics. In the en, ny such irect pproch my soon become imprcticl n much more vnce methos, e.g. involving viscosity solutions for solving ssocite prtil ifferentil equtions might be more pproprite... specultions... enjoy!]] Definition 4.10 A Riemnnin mnifol M is clle geoesiclly complete if every geoesic cn be extene to geoesic efine on the whole rel line, ). Theorem 4.22 Hopf-Rhinow) A Riemnnin mnifol is geoesiclly complete if n only if it is complete w.r.t. its ntive metric). Any two points p n q in geoesiclly complete mnifol cn be joine by geoesic of miniml length.
12 Clssnotes: Geometry & Control of Dynmicl Systems, M. Kwski. April 20, Proof. Use exp mp n erive contriction from ssuming tht there is finite time beyon which geoesic cnnot be extene. Obvious Cuchy sequences in other irection. [[Spivk pp , Boothby pp ]] in progress Work out geoesics on sphere n hyperbolic spce Poincre hlf-plne emphsize: constnt curvture in clcultions)
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