X TM and to each vector. X p and Y. Furthermore, if. X p.) X f Y, defined by the identity

Transcription

1 PART I A RAPID COURSE IN RIEMANNIAN GEOMETRY 8 Covarat Dfferetato The obect of Part II wll be to gve a rad outle of some basc cocets of Remaa geometry whch wll be eeded later For more formato the reader should cosult Nomzu, "Le grous ad dfferetal geometry," The Mathematcal Socety of Jaa, 956; Laugwtz, "Dfferetalgeometre," Teuber 960; or Helgaso, "Dfferetal geometry ad symmetrc saces," Academc Press, 96 Let M be a smooth mafold DEFINITION A affe coecto at a ot M s a fucto whch assgs to each taget vector X TM ad to each vector feld Y a ew taget vector X f Y TM called the covarat dervatve of Y the drecto X Ths s requred to be blear as a fucto of X ad Y Furthermore, f f : M R s a real valued fucto, ad f fy deotes the vector feld ( fy ) f ( q) q Y q the f s requred to satsfy the detty X f ( fy ) ( X f ) Y + f ( ) X f Y (As usual, X f deotes the drectoal dervatve of f the drecto X ) A global affe coecto (or brefly a coecto) o M s a fucto whch assgs to each M a affe coecto f at, satsfyg the followg smoothess coo () If X ad Y are smooth vector felds o M the the vector feld X f Y, defed by the detty Note that our X f Y cocdes wth Nomzu s X Y The otato s teded to suggest that the dfferetal oerator X acts o the vector feld Y

2 must also be smooth Note that: ( X f Y ) X f Y, () X f Y s blear as a fucto of X ad Y (3) ( fx ) f Y f ( X f Y ) (4) ( X f ( fy ) ( Xf ) Y + f ( X f Y ) Coos (), (), (3), (4) ca be tae as the defto of a coecto I terms of local coordates eghborhood K defed o a coordate u,,u U M, the coecto f s determed by 3 smooth real valued fuctos Γ o U, as follows Let deote the vector feld u exressed uquely as o U The ay vector feld X o U ca be X x where the x are real valued fuctos o U I artcular the vector feld f ca be exressed as (5) f Γ These fuctos Γ determe the coecto comletely o U I fact gve vector felds X x ad Y y oe ca exad X f Y by the rules (), (3), (4); yeldg the formula f y, (6) Y x X, X Y x y, x where the symbol y, stads for the real valued fucto y, y y + Γ Coversely, gve ay smooth real valued fuctos Γ o U, oe ca defe X f Y by the formula (6) The result clearly satsfes the coos (), (), (3), (4), (5) Usg the coecto f oe ca defe the covarat dervatve of a vector feld alog a curve M Frst some deftos

3 A arametrzed curve M s a smooth fucto c from the real umbers to M A vector feld V alog the curve c s a fucto whch assgs to each t R a taget vector Vt TM c(t) Ths s requred to be smooth the followg sese: For ay smooth fucto f o M the corresodece t V should defe a smooth fucto o R As a examle the velocty vector feld vector feld alog c whch s defed by the rule Here d t f dc d c dc of the curve s the deotes the stadard vector feld o the real umbers, ad c TR t TM c : ( t) deotes the homomorhsm of taget saces duced by the ma c (Comare Dagram 9) Now suose that M s rovded wth a affe coecto The ay vector feld V alog c determes a ew vector feld c called the covarat dervatve of V The oerato V s characterzed by the followg three axoms a) D( V + W ) + DW b) If f s a smooth real valued fucto o R the D( fv ) df V + f alog c) If V s duced by a vector feld Y o M, that s f V t Y c(t ) for 3

4 each t, the dc s equal to f Y ( the covarat dervatve of Y the drecto of the velocty vector of c) LEMMA 8 There s oe ad oly oe oerato satsfes these three coos V whch PROOF: Choose a local coordate system for M, ad let u ( t), K, u ( t) deote the coordates of the ot c(t) The vector feld V ca be exressed uquely the form V v where v, K,v are real valued fuctos o R (or a arorate oe subset of R), ad, K, are the stadard vector felds o the coordate eghborhood It follows from (a), (b), ad (c) that Coversely, defg dv dv + + v, dc du Γ that coos (a), (b), ad (c) are satsfed f v by ths formula, t s ot dffcult to verfy A vector feld V alog c s sad to be a arallel vector feld f the covarat dervatve s detcally zero LEMMA 8 Gve a curve c ad a taget vector V 0 at the ot c(0), there s oe ad oly oe arallel vector feld V alog c whch exteds V 0 PROOF The dfferetal equatos dv +, du Γ v 0 have solutos v (t) whch are uquely determed by the tal values v (0) Sce these equatos are lear, the solutos ca be defed for all relevat values of t (Comare Graves, "The Theory of Fuctos of Real Varables," 5) The vector traslato alog c V t s sad to be obtaed from V 0 by arallel Now suose that M s a Remaa mafold The er roduct 4

5 of two vectors X, Y wll be deoted by Y X, DEFINITION A coecto f o M s comatble wth the Remaa metrc f arallel traslato reserves er roducts I other words, for ay arametrzed curve c ad ay ar P, P' of arallel vector felds alog c, the er roduct P, P' should be costat LEMMA 83 Suose that the coecto s comatble wth the metrc Let V, W be ay two vector felds alog c The d DW V, W, W + V, PROOF: Choose arallel vector felds P, K, P alog c whch are orthoormal at oe ot of c ad hece at every ot of c The the gve felds V ad W ca be exressed as v P ad w P, resectvely (where v V, P s a real valued fucto o R) It follows that Therefore V, W ad that v w dv + P + P DW dw, DW dv dw d, W + V, w v V, W +, whch comletes the roof COROLLARY 84 For ay vector felds Y, Y' o M ad ay vector X TM : X Y, Y ' X f Y, Y ' + Y, X f X Y ' PROOF Choose a curve c whose velocty vector at t 0 s X ; ad aly 83 DEFINITION 85 A coecto f s called symmetrc f t satsfes the detty ( X f Y ) ( Y f X ) [ X, Y ] (As usual, [X, Y] deotes the oso bracet [X, Y]f X(Yf) - Y(Xf) of two vector felds) Alyg ths detty to the case X, Y, 5

6 sce [, ] 0 oe obtas the relato Γ Γ 0 Coversely f Γ Γ the usg formula (6) t s ot dffcult to verfy that the coecto f s symmetrc throughout the coordate eghborhood LEMMA 86 (Fudametal lemma of Remaa geometry) A Remaa mafold ossesses oe ad oly oe symmetrc coecto whch s comatble wth ts metrc (Comare Nomzu 76, Laugwtz 95) PROOF of uqueess Alyg 84 to the vector felds,, ad settg, g oe obtas the detty g f, +, f Permutg,, ad ths gves three lear equatos relatg the three quattes f,,, f ad f, (There are oly three such quattes sce f f ) These equatos ca be solved uquely; yeldg the frst Chrstoffel detty f, ( g + g g ) The left had sde of ths detty s equal to l Γ l g l Multlyg l by the verse ( g ) of the matrx g ) ths yelds the secod Chrstoffel detty ( l l l Γ ( g + g g ) g Thus the coecto s uquely determed by the metrc g Γ λ µν λ µν g x g µν λ λρ g g ( x µσ ρν µ Γ σ νλ g + x µρ ν g νσ Γ g x σ µλ µν ρ ) 0 Coversely, defg l Γ by ths formula, oe ca verfy that the resultg coecto s symmetrc ad comatble wth the metrc Ths comletes the roof A alteratve characterzato of symmetry wll be very useful 6

7 later Cosder a "arametrzed surface" M: that s a smooth fucto s : R M By a vector feld V alog s s meat a fucto whch assgs to each R (x,y) a taget vector V TM ( x, y) s( x, y) As examles, the two stadard vector felds to vector felds brefly by x s x s ad x ad x ad y gve rse s alog s These wll be deoted y s ; ad called the "velocty vector felds" of s y For ay smooth vector feld V alog s the covarat dervatves ad y are ew vector felds, costructed as follows For each fxed y 0, restrctg V to the curve x s( x, y0 ) oe obtas a vector feld alog ths curve Its covarat dervatve wth resect to x s defed to be alog the etre arametrzed surface s x x, y ) ( 0 Ths defes x As examles, we ca form the two covarat dervatves of the two vector felds s x ad s The dervatves y D s x x ad D s y y smly the accelerato vectors of sutable coordate curves However, the mxed dervatves descrbed so smly D s x y ad LEMMA 87 If the coecto s symmetrc the D s y x are caot be D s D s x y y x PROOF Exress both sdes terms of a local coordate system, ad comute 9 The Curvature Tesor The curvature tesor R of a affe coecto f measures the extet to whch the secod covarat dervatve f ( f Z) s symmetrc ad Gve vector felds X, Y, Z defe a ew vector feld R(X,Y)Z by the detty R( X, Y ) Z X f ( Y f Z) + Y f ( X f Z) + [ X, Y ] f Z 7

8 LEMMA 9 The value of R(X,Y)Z at a ot M deeds oly o the vectors X, Y, Z at ths ot ad ot o ther values at earby ots Furthermore the corresodece X, Y, Z R( X, Y ) Z from TM TM TM to TM s tr-lear Brefly, ths lemma ca be exressed by sayg that R s a "tesor" PROOF: Clearly R(X,Y)Z s a tr-lear fucto of X, Y, ad Z If X s relaced by a multle fx the the three terms X f ( Y f Z), Y f ( X f Z), [ X, Y ] f Z are relaced resectvely by ) fx f ( Y f Z), ) ( Yf ) f ( X f Z) + fy f ( X f Z), ) ( Yf )( X f Z) + f [ X, Y ] f Z Addg these three terms oe obtas the detty R(fX,Y)Z fr(x,y)z Corresodg dettes for Y ad Z are easly obtaed by smlar comutatos Now suose that X x, Y y ad Z z The R( X, Y ) Z R( x, y )( z x y z R(, ) Evaluatg ths exresso at oe obtas the formula ) ( R(, ) ( R ( X, Y ) Z) x ( ) y ( ) z ( ) ) whch deeds oly o the values of the fuctos x, y, x at, ad ot o ther values at earby ots Ths comletes the roof Now cosder a arametrzed surface s : R M Gve ay vector feld V alog s oe ca aly the two covarat dfferetato oerators D x ad 8 D y oerators wll ot commute wth each other LEMMA 9 D D D D s s V V R V y x x y, x y to V I geeral these PROOF: Exress both sdes terms of a local coordate system,

9 ad comute, mag use of the detty f ( f ) f ( f ) (, ) R [It s terestg to as whether oe ca costruct a vector feld P alog s whch s arallel, the sese that D D P P 0, x y ad whch has a gve value P (0,0) at the org I geeral o such vector feld exsts However, f the curvature tesor haes to be zero the P ca be costructed as follows Let P (x,0) be a arallel vector feld alog the x-axs, satsfyg the gve tal coo For each fxed x 0 let P ( x, y) be a arallel vector feld alog the curve y s( x 0, y), havg the rght value for y 0 Ths defes P everywhere alog s Clearly D y P Now the detty s detcally zero; ad D D D D s s P P R, P 0 y x x y x y D x P s zero alog the x-axs D D D mles that P 0 I other words, the vector feld P y x x arallel alog the curves y s( x 0, y) D D Sce P 0, ths mles that P x ( 0 x x,0) ad comletes the roof that P s arallel alog s] s s detcally zero; Heceforth we wll assume that M s a Remaa mafold, rovded wth the uque symmetrc coecto whch s comatble wth ts metrc I cocluso we wll rove that the tesor R satsfes four symmetry relatos LEMMA 93 The curvature tesor of a Remaa mafold satsfes: () R ( X, Y ) Z + R( Y, X ) Z 0 () R ( X, Y ) Z+ R( Y, Z) X + R( Z, X ) Y 0, (3) R ( X, Y ) Z, W + R( X, Y ) W, Z 0, 9

10 (4) R ( X, Y ) Z, W R( Z, W ) X, Y PROOF: The sew-symmetry relato () follows mmedately from the defto of R Sce all three terms of () are tesors, t s suffcet to rove () whe the bracet roducts [X,Y], [X,Z] ad [Y,Z] are all zero Uder ths hyothess we must verfy the detty X f ( Y f Z) + Y f ( X f Z) Y f ( Z f X ) + Z f ( Y f X ) Z f ( X f Y ) + X f ( Z f Y ) 0 But the symmetry of the coecto mles that Y f Z Z f Y [ Y, Z] 0 Thus the uer left term cacels the lower rght term Smlarly the remag terms cacel ars Ths roves () To rove (3) we must show that the exresso < R(X,Y)Z,W > s sew-symmetrc Z ad W Ths s clearly equvalet to the asserto that R ( X, Y ) Z, Z 0 for all X,Y,Z Aga we may assume that [X,Y] 0, so that <R(X,Y)Z,Z> s equal to X f ( Y f Z) + Y f ( X f Z), Z I other words we must rove that the exresso Y f ( X f Z), Z s symmetrc X ad Y Sce [X,Y] 0 the exresso YX < Z,Z > s symmetrc X ad Y Sce the coecto s comatble wth the metrc, we have X Z, Z X f Z, Z hece YX Z, Z Y f ( X f Z), Z + X f Z, Y f Z But the rght had term s clearly symmetrc X ad Y Therefore Y f ( X f Z), Z s symmetrc X ad Y; whch roves roerty (3) Proerty (4) may be roved from (), (), ad (3) as follows Formula () asserts that the sum of the quattes at the vertces of shaded tragle W s zero Smlarly (mag use of () ad (3)) the sum of the vertces of each of the other shaded tragles s zero 0

11 Addg these dettes for the to two shaded tragles, ad subtractg the dettes for the bottom oes, ths meas that twce the to vertex mus twce the bottom vertex s zero Ths roves (4), ad comletes the roof 0 Geodescs ad Comleteess Let M be a coected Remaa mafold DEFINITION A arametrzed ath γ : I M, where I deotes ay terval of real umbers, s called a geodesc f the accelerato vector feld D Thus the velocty vector feld γ s a geodesc, the the detty d shows that the legth s detcally zero D,, 0 must be arallel alog γ If /, of the velocty vector s costat alog γ Itroducg the are-legth fucto s( t) + costat ths statemet ca be rehrased as follows: The arameter t alog a geodesc s a lear fucto of the arc-legth The arameter t s actually equal to the arc-legth f ad oly f I terms of a local coordate system wth coordates u, K,u a curve t γ ( t) M determes smooth fuctos

12 u ( t), K, u form ( t) The equato D for a geodesc the taes the d u + du du Γ ( u, K, u ) 0, The exstece of geodescs deeds, therefore, o the solutos of a certa system of secod order dfferetal equatos More geerally cosder ay system of equatos of the form r d u r r r du F( u, ) Here u r stads for ( K ) ad F r stads for a -tule of u,,u fuctos, all defed throughout some eghborhood U of a ot r r ( u, v R ) C EXISTENCE AND UNIQUENESS THEOREM 0 There exsts a eghborhood W of the ot ( u r, v r ) ad a umber ε > 0 so that, r r for each ( u0, v0 ) W the dfferetal equato r d u r r r du F( u, ) r has a uque soluto t u(t) whch s defed for t < ε, ad satsfes the tal coos r r r du r u( 0) u0, ( 0) v0 Furthermore, the soluto deeds smoothly o the tal coos I other words, the corresodece r r r u, v, t) u( ) ( 0 0 t from W ( ε, ε ) to R s a C fucto of all + varables PROOF: Itroducg the ew varables v du ths system of secod order equatos becomes a system of frst order equatos: r du r v r dv r r r F( u, v) The asserto the follows from Graves, "Theory of Fuctos of Real Varables," 66 (Comare our 4) Alyg ths theorem to the dfferetal equato for geodescs, oe obtas the followg

13 LEMMA 0 For every ot 0 o a Remaa mafold M there exsts a eghborhood U of 0 ad a umber ε > 0 so that: for each U ad each taget vector v TM wth legth < ε there s a uque geodesc satsfyg the coos γ v : (,) M γ v (0), v (0) v PROOF If we were wllg to relace the terval (-,) by a arbtrarly small terval, the ths statemet would follow mmedately from 0 To be more recse; there exsts a eghborhood U of 0 ad umbers ε, ε > 0 so that: for each U ad each v TM wth v <ε there s a uque geodesc γ v ( ε,ε ) M : satsfyg the requred tal coos To obta the sharer statemet t s oly ecessary to observe that the dfferetal equato for geodescs has the followg homogeety roerty Let c be ay costat If the arametrzed curve t γ (t) s a geodesc, the the arametrzed curve wll also be a geodesc t γ (ct) Now suose that ε s smaller tha ε ε The f v < ε ad t < ote that v / ε < ε ad ε < ε t Hece we ca defe γ (t) to be γ ε ε t) Ths roves 0 v v / ( Ths followg otato wll be coveet Let v TM q be a taget vector, ad suose that there exsts a geodesc satsfyg the coos γ :[0,] M γ (0) q, (0) v The the ot γ () M wll be deoted by exq ( v) ad called 3

14 the exoetal of the taget vector v The geodesc γ ca thus be descrbed by the formula γ ( t) ex ( tv) q Lemma 0 says that exq ( v) s defed rovdg that v s small eough I geeral, exq ( v) s ot defed for large vectors v However, f defed at all, exq ( v) s always uquely defed DEFINITION The mafold M s geodescally comlete f γ ( t) ex ( tv) s defed for all q M ad all vectors q v TM q Ths s clearly equvalet to the followg requremet: For every geodesc segmet ossble to exted γ 0 to a fte geodesc γ : R M γ : [ a, b] M t should be We wll retur to a study of comleteess after rovg some local results Let TM be the taget mafold of M, cosstg of all ars (,v) 0 wth M, v TM We gve TM the followg C structure: f ( K ) s a coordate system a oe set M u,,u every taget vector at U the q M ca be exressed uquely as t + L+ t, where u q The the fuctos K, u,,u K costtute a coordate system o the oe set TM t,,t Lemma 0 says that for each ( q, v) ex ( v) M the ma s defed throughout a eghborhood V of the ot Furthermore ths ma s dfferetable throughout V Now cosder the smooth fucto q TU (,0) TM F : V M M defed by F( q, v) ( q,ex ( v)) We clam that the Jacoba of F at the ot q ( 0,0) s o-sgular I fact, deotg the duced coordates o ( U U M M by u, K, u, u, K, u ), we have 4

15 F +, u u u F t u I I Thus the Jacoba matrx of F at (,0) has the form, ad 0 I hece s o-sgular It follows from the mlct fucto theorem that F mas some eghborhood V' of (,0) TM dffeomorhcally oto some eghborhood of (, ) M M We may assume that the frst eghborhood V', cossts of all ars (q,v) such that q belogs to a gve eghborhood U' of ad such that v < ε Choose a smaller eghborhood W of so that the followg LEMMA 03 For each a umber ε > 0 so that: F ( V ') W W The we have rove M there exsts a eghborhood W ad () Ay two ots of W are oed by a uque geodesc M of legth < ε () Ths geodesc deeds smoothly uo the two ots (Ie, f t ex q ( tv), 0 t, s the geodesc og q ad q, the the ar ( q,v) (3) For each q W the ma TM deeds dfferetably o ( q, q )) ex q mas the oe ε -ball TM q dffeomorhcally oto a oe set U q W REMARK Wth more care t would be ossble to choose W so that the geodesc og ay two of ts ots les comletely wth W Comare J H C Whtehead, Covex regos the geometry of aths, Quarterly Joural of Mathematcs (Oxford) Vol 3, (93), 33-4 Now let us study the relatosh betwee geodescs ad arc-legth THEOREM 04 Let W ad ε be as Lemma 03 Let γ :[0,] M be the geodesc of legth < ε og two ots of W, ad let ω :[0,] M be ay other ecewse smooth ath og the same two ots The, 0 0 dω, where equalty ca hold oly f the ot set ω ([0,]) cocdes wth 5

16 γ ([0,]) Thus γ s the shortest ath og ts ed ots The roof wll be based o two lemmas Let q γ (0) ad let U q be as 03 LEMMA 05 I traectores of hyersurfaces U q, the geodescs through q are the orthogoal {ex ( v) : v TM, v costat} q q PROOF Let t v(t) deote ay curve TM q wth v ( t) We must show that the corresodg curves t ex ( r0v( t)) q U q, where 0 < 0 r < ε, are orthogoal to the radal geodescs r exq ( rv( t0 )) I terms of the arametrzed surface f gve by we must rove that for all (r, t) Now r f ( r, t) ex ( rv( t)), 0 r ε, q f f, 0 r t f f D f f f D f,, +, r t r r t r r t The frst exresso o the rght s zero sce the curves r f (,t) are geodescs The secod exresso s equal to f D f f f,, 0, r t r t r r f sce v( t) Therefore the quatty r deedet of r But for r0 we have f ( 0, t) exq (0) q, f f, s r t f hece ( 0, t) 0 Therefore t comletes the roof f f, s detcally zero, whch r t 6

17 Now cosder ay ecewse smooth curve ω : [ a, b] U [ q] q Each ot ω (t) ca be exressed uquely the form exq ( r( t) v( t)) wth 0 < r(t) < ε, ad v ( t), v( t) TM q LEMMA 06 The legth b a dω s greater tha or equal to r( b) r( a), where equalty holds oly f the fucto r(t) s mootoe, ad the fucto v(t) s costat Thus the shortest ath og two cocetrc shercal shells aroud q s a radal geodesc PROOF Let f ( r, t) ex ( v( t)), so that ω ( t ) f ( r( t), t) The q dω f f r' ( t) + r t Sce the two vectors o the rght are mutually orthogoal, ad sce f r, ths gves ω t r' ( t) f + t r' ( t) f dv where equalty holds oly f 0 ; hece oly f 0 Thus t b a dω b a r' ( t) r( b) r( a) where equalty holds oly f r(t) s mootoe ad v(t) s costat Ths comletes the roof The roof of Theorem 04 s ow straghtforward Cosder ay ecewse smooth ath m from q to a ot q' ex ( rv) ; q U q where 0 < r < ε, v ( t) The for ay δ > o the ath ω must cota a segmet og the shercal shell of radus δ to the shercal shell of radus r, ad lyg betwee these two shells The legth of ths segmet wll be r δ ; hece lettg δ ted to 0 the legth of ω wll be r If ω ([0,]) does ot cocde wth 7

18 γ ([0,]), the we easly obta a strct equalty Ths comletes the roof of 04 A mortat cosequece of Theorem 04 s the followg COROLLARY 07 Suose that a ath ω : [0, l] M, arametrzed by arc-legth, has legth less tha or equal to the legth of ay other ath from ω (0) to ω (l) The ω s a geodesc PROOF Cosder ay segmet of ω lyg wth a oe set W, as above, ad havg legth < ε Ths segmet must be a geodesc by Theorem 04 Hece the etre ath ω s a geodesc DEFINITION A geodesc γ :[ a, b] M wll be called mmal f ts legth s less tha or equal to the legth of ay other ecewse smooth ath og ts edots Theorem 04 asserts that ay suffcetly small segmet of a geodesc s mmal O the other had a log geodesc may ot be mmal For examle we wll see shortly that a great crcle arc o the ut shere s a geodesc If such a arc has legth greater tha π, t s certaly ot mmal I geeral, mmal geodescs are ot uque For examle two atodal ots o a ut shere are oed by ftely may mmal geodescs However, the followg asserto s true Defe the dstace ρ (, q) betwee two ots, q M to be the greatest lower boud for the arc-legths of ecewse smooth aths og these ots Ths clearly maes M to a metrc sace It follows easly from 04 that ths metrc s comatble wth the usual toology of M COROLLARY 08 Gve a comact set umber K M there exsts a δ > 0 so that ay two ots of K wth dstace less tha δ are oed by a uque geodesc of legth less tha δ Furthermore ths geodesc s mmal; ad deeds dfferetably o ts edots PROOF Cover K by oe sets W α, as 03, ad let δ be small eough so that ay two ots K wth dstace less tha δ le a commo W α Ths comletes the roof Recall that the mafold M s geodescally comlete f every geodesc segmet ca be exteded dftely THEOREM 09 (Hof ad Row) If M s geodescally comlete, the ay two ots ca be oed by a mmal geodesc PROOF Gve, q M wth dstace r > 0, choose a 8

19 eghborhood U as Lemma 03 Let U S deote a shercal shell of radus exsts a ot δ < ε about Sce S s comact, there ex ( ), v, 0 q δv o S for whch the dstace to q s mmzed We wll rove that ex ( rv) q Ths mles that the geodesc segmet t γ ( t) ex ( tv), 0 t r, s actually a mmal geodesc from to q The roof wll amout to showg that a ot whch moves alog the geodesc γ must get closer ad closer to q I fact for each t [ δ, r] we wll rove that (t) ρ (γ ( t), q) r t Ths detty, for t r, wll comlete the roof Frst we wll show that the equalty (δ ) s true Sce every ath from to q must ass through S, we have ρ (, q) M( ρ(, s) + ρ( s, q)) δ + ρ( 0, q) Therefore s S ρ ( 0, q) r t Sce γ ( ), ths roves (δ ) 0 δ Let t [ δ, ] deote the suremum of those umbers t for whch 0 r (t) s true The by cotuty the equalty ( t 0 ) s true also If t 0 < r we wll obta a cotradcto Let S' deote a small shercal shell of radus δ ' about the ot γ t ) ; ad let S' be a ot of S' wth mmum dstace from q (Comare Dagram 0) The ( 0 ' 0 q hece ρ ( γ ( t0 ), q) M( ρ( γ ( t0 ), s) + ρ( s, q)) δ ' + ρ( ' 0, q), s S ' () ρ ', q) ( r t ) ' ( 0 0 δ We clam that ' 0 s equal to γ ( t 0 + δ ' ) I fact the tragle equalty states that ρ (, ' ) ρ(, q) ρ( ' 0, q) t0 + 0 δ 9 '

20 (mag use of ()) But a ath of legth recsely ' 0 t + ' from to 0 δ ( t 0 s obtaed by followg γ from to γ ), ad the followg a mmal geodesc from γ ( t 0 ) to ' 0 Sce ths broe geodesc has mmal legth, t follows from Corollary 07 that t s a (ubroe) geodesc, ad hece cocdes wth γ Thus γ ( t 0 +δ ') ' 0 Now the equalty () becomes ( t 0+ δ ' ) ρ ( γ ( t 0 + δ '), q) r ( t0 + δ ') Ths cotradcts the defto of t 0 ; ad comletes the roof COROLLARY 00 If M s geodescally comlete the every bouded subset of M has comact closure Cosequetly M s comlete as a metrc sace (e, every Cauchy sequece coverges) PROOF If X M has dameter d the for ay X the ma ex : TM M mas the ds of radus d TM oto a comact subset of M whch (mag use of Theorem 09) cotas X Hece the closure of X s comact Coversely, f M s comlete as a metrc sace, the t s ot dffcult, usg Lemma 03, to rove that M s geodescally comlete For detals the reader s referred to Hof ad Row Heceforth we wll ot dstgush betwee geodesc comleteess ad metrc comleteess, but wll refer smly to a comlete Remaa mafold FAMILIAR EXAMPLES OF GEODESICS I Eucldea -sace, R, wth the usual coordate system x, K, x ad the usual Remaa metrc dx dx + L+ dx dx we have Γ 0 ad the equatos for a geodesc γ, gve by t ( x ( t), K, x ( t)) become d x 0 0, whose solutos are the straght les Ths could also have bee see as follows: t s easy to show that the formula for arc legth dx / cocdes wth the usual defto of arc legth as the least uer boud of the legths of scrbed olygos; from ths defto t s clear that straght les have mmal legth, ad are therefore geodescs

21 The geodescs o tersectos of S are recsely the great crcles, that s, the S wth the laes through the ceter of PROOF Reflecto through a lae I : S S whose fxed ot set s S E s a sometry C S S Let x ad y be two ots of C wth a uque geodesc C' of mmal legth betwee them The, sce I s a sometry, the curve I(C') s a geodesc of the same legth as C betwee I(x) x ad I(y) y Therefore C' I(C') Ths mles that C' C Fally, sce there s a great crcle through ay ot of ay gve drecto, these are all the geodescs S Atodal ots o the shere have a cotum of geodescs of mmal legth betwee them All other ars of ots have a uque geodesc of mmal legth betwee them, but a fte famly of o-mmal geodescs, deedg o how may tmes the geodesc goes aroud the shere ad whch drecto t starts By the same reasog every merda le o a surface of revoluto s a geodesc The geodescs o a rght crcular cylder Z are the geeratg les, the crcles cut by laes eredcular to the geeratg les, ad the helces o Z PROOF If L s a geeratg le of Z the we ca set u a sometry I : Z L R by rollg Z oto R :

22 The geodescs o Z are ust the mages uder them I of the straght les R Two ots o Z have ftely may geodescs betwee (Ed of Part II)