Ut 9. The Taget Budle ========================================================================================== ---------- The taget sace of a submafold of R, detfcato of taget vectors wth dervatos at a ot, the abstract defto of taget vectors, the taget budle; the dervatve of a smooth ma. ---------- The am of ths chater s to gve ad recocle dfferet commoly used deftos of a taget vector to a mafold. Before assg over to the abstract stuato, we shall deal wth submafolds of R. Defto. Smooth mags g :[a,b]----------l M of a terval to a dfferetable mafold (M,A) are called smooth curves the mafold. --------------- Defto A. Let M be a dfferetable mafold embedded R, x e M. A vector v s called a taget vector to M at x f there s a smooth curve -------------------- x:[-e,e]----------l M assg through x = x() such that v = x (). The taget sace T M of M at x s the set of all taget vectors to M at x. --------------- x --------------- Theorem. Let us suose that a k-dmesoal mafold M embedded R s ---------- gve a eghborhood U of x e M by a system of equaltes f =...=f =, 1 -k where f,...,f 1 -k are smooth fuctos o U such that the vectors grad f (x ),...,grad 1 f (x ) -k are learly deedet at x. The the taget sace of M at x cossts of the vectors orthogoal to grad f (x ),..., 1 grad f (x ). -k Corollary. The taget sace of a k-dmesoal submafold of R s a -------------------- k-dmesoal lear subsace of R. Proof. If x:[-e,e]----------l R s a smooth curve havg coordate fuctos x,...,x ad lyg o M, the we have 1 f (x (t),...,x (t)) = = 1,...,-k 1 for each t e [-e,e]. Dfferetatg by t we get d f d f,, --------------------(x())x () +... + --------------------(x())x () = d x 1 d x 1 whch meas that the vectors grad f (x()) ad x () are orthogoal. Now let us rove that f a vector v s orthogoal to the vectors grad f (x ), 1 < < -k, the v s a taget vector. k Let us take a smooth local arameterzato F:R ----------L M C R of M aroud the k ot x. The curve t9-----l F(F (x ) + ty), where y e R fxed, s a curve o M 1
assg through x. The seed vector of ths curve for t = s d F d F --------------------(F (x ))y +... + --------------------(F (x ))y, d x 1 d x k 1 k where y,...,y are the coordates of y. By the costructo of local 1 k arameterzatos of embedded mafolds, F s a restrcto of a k dffeomorhsm betwee oe subsets of R oto R, cosequetly, the vectors d F --------------------(F (x )) are learly deedet. We coclude, that the taget sace d x s cotaed the k-dmesoal lear subsace orthogoal to grad f (x ),...,grad f (x ) ad cotas the k-dmesoal lear subsace 1 -k d F saed by the vectors --------------------(F (x )) 1 < < k, whch meas that both lear d x subsaces cocde wth the taget sace.44444 The defto of taget vectors ca also be gve trsc terms, deedet of the embeddg of M to R. Let us defe a equvalece relato o the set Curve(M,) = {g:[-e,e]----------l M : g() = }, cosstg of curves assg through e M, by callg two curves g,g e 1 2 Curve(M,) equvalet f (xqg ) () = (xqg ) () for some chart x aroud. 1 2 The ths codto s true for ay chart (rove ths!). Defto B. A taget vector to a mafold M at the ot e M s a -------------------- equvalece class of curves belogg to Curve(M,). The set of equvalece classes s called the taget sace of M at ad deoted by T M. --------------- Gve a chart x aroud, we ca establsh a oe-to-oe corresodece m betwee the equvalece classes ad ots of R, (m = dm M), assgg to m the equvalece class of a curve g e Curve(M,) the vector (xqg) () e R. Wth the hel of ths detfcato, we ca troduce a vector sace structure o the taget sace, ot deedg o the choce of the chart. For embedded mafolds defto B agrees wth defto A. The advatage of defto B les the fact that t s alcable also for abstract mafolds, ot embedded aywhere. Defto. If x = (x,...,x ) s a chart o the mafold M aroud the 1 m ot, g e Curve(M,), the the umbers (x qg) (),..., (x qg) () are 1 m called the comoets of the taget vector rereseted by g wth ---------- -------------------- resect to the chart x. The ma dffculty of defg taget vectors to a mafold s due to the fact that a abstract mafold mght ot be embedded to a fxed fte dmesoal lear sace. Nevertheless, there s a uversal embeddg of 2
each dfferetable mafold to a fte dmesoal lear sace. Let us deote by F (M) the lear vector sace of smooth fuctos o M, ad by F (M) the dual sace of F (M) that s the sace of lear fuctos o F (M), ad cosder the embeddg of M to F (M) defed by the formula [()](f) = f(), where e M, f e F (M). Havg embedded the mafold M to F (M), we ca defe taget vectors to M to be elemets of the lear sace F (M). Defto. Let M be a dfferetable mafold, e M. We say that a lear fucto D e F (M) defed o smooth fuctos o M s a dervato at the ot f the equalty holds for every f,g e F (M). D(fg) = D(f)g() + f()d(g) Each curve g e Curve(M,) defes a dervato at the ot by the formula D (f) = (fqg) (), where f e F (M). D s the seed vector g () g () of the curve qg F (M). Sce two curves defe the same dervato ff they are equvalet, there s a oe-to-oe corresodece betwee the equvalece classes of curves ad the dervatos obtaed as D g () g. for some Defto C. A taget vector to a mafold M at the ot e M s a -------------------- dervato of the form D, where g e Curve(M,). g () The taget sace T M of M at the ot s the set of dervatos D --------------- g () alog curves M assg through = g (). Theorem. The taget sace to a dfferetable mafold M at the ot e ---------- M cocdes wth the sace of dervatos o F (m) at, whch s a lear sace havg the same dmeso as M has. Lemma 1. If f e F (M) s a costat fucto ad D s a dervato at a ot e M, the D(f) =. Proof. Because of learty, t s eough to show that D(1)=, where 1 the costat 1 fucto o M. But we have s D(1) = D(1 1) = D(1) 1() + 1() D(1) = 2 D(1). 44444 3
Lemma 2. If two fuctos f,g e M cocde o a eghborhood U of e M ad D s a dervato at the D(f) = D(g). Proof. Sublemma. If x e R ad B(x,e) s a fxed oe ball about t, the there --------------- exsts a smooth fucto h:r ----------L R such that h(y) s equal to 1 f y e B(x,e/2), ostve f y e B(x,e) ad zero f y m B(x,e). Defe the fucto h of the real varable t by the formula 2 & -(1-t ) e f t e (,1) h (t) = { 7 otherwse. It s a good exercse to rove that h s a smooth fucto o R. Set h (y) := h (4NyN /e), let c deote the characterstc fucto of the ball 1 B(x,3e/4), ad defe the fucto h as follows 2 h (y) := c(z)h (y-z)dz. 2 1 R If we ut h(y) = h (y)/h (x) the we get a desrable fucto. 2 2 Now let us rove the lemma. Usg the costructo above we ca defe a smooth fucto h o M whch s zero outsde U ad such that h() = 1. I ths case h(f-g) s the costat fucto o M. Thus we have = D() = D(h(f-g)) = D(h) (f()-g()) + h() D(f-g) = D(f) - D(g).44444 Remarks. ---------- ) The sublemma shows that the mag :M----------L F (M) above s deed a cluso. If $ q are dstct ots of M, the there s a smooth fucto h o M such that [()](h) = h() = 1 $ [(q)](h) = h(q) =. ) We ca exted a dervato D at a ot o fuctos f defed oly a eghborhood U of by takg a smooth fucto h o M such that h s zero outsde U ad costat 1 a eghborhood of ad uttg D(f) := ~ D(f), where ( ~ f(x)h(x) for x e U f(x) = { for x m U. 9 By lemma 2 ths exteso of D s correctly defed. Lemma 3. Let f:b----------l R be a smooth fucto defed o a oe ball B C R aroud the org. The there exst smooth fuctos g 1< < o B such that f(x) = f() + S x g (x) for x = (x,...,x ) e B 1 =1 ad d f g () = --------------------(). d x 4
Proof. Sce 1 1 d f(tx) d f f(x) - f() = ----------dt = S x --------------------(tx) dt = d x dt =1 1 1 d f d f = S x --------------------(tx) dt, we may take g (x) = --------------------(tx) dt. 44444 d x d x =1 Now we are ready to rove the theorem. Let us take a dfferetable mafold (M,A) ad a chart x defed a eghborhood of e M. = (x,...,x )ea 1 Defe the dervatos d () as follows d fqx # $ d () (f) := ----- (x()). 3 4 d x We rove that the dervatos d () form a bass the sace of dervatos at. They are learly deedet sce f we have S a d () =, =1 the alyg ths dervato to the -th coordate fucto x we get d x S a --------------------(x()) = a =. d x =1 O the other had, f D s a arbtrary dervato at, the we have D = S D(x )d (). =1 Ideed, let f e F (M) be a arbtrary smooth fucto o M ad aly lemma 3 to fqx aroud x(). We obta fuctos g defed aroud x() such that d fqx f = f() + S (x -x ())g qx ad g (x()) = ----- (x()). d x =1 I ths case however we have D(f) = D(f()) + S D((x -x ())) g (x()) + (x ()-x ()) D(g qx) = =1 d fqx # $ S D(x ) ----- (x()) = S D(x ) d () (f). d x 3 4 =1 =1 To fsh the roof, we oly have to show that every dervato at the ot ca be obtaed as a seed vector of a curve assg through. Defe the curve g :[-e,e]----------l M by the formula g (t) := x (x() + (ta,...,ta )). 1 d The obvously the seed vector g () s ust S a --------------------(). 44444 d x =1 5
The taget budle --------------- The uo of the taget saces of M at the varous ots, u T M, has a em atural dfferetable mafold structure, the dmeso of whch s twce the dmeso of M. Ths mafold s called the taget budle of M ad s deoted by TM. A -------------------- ot of ths mafold s a vector D, taget to M at some ot. Local coordates o TM are costructed as follows. Let x = (x,...,x ) be a chart 1 o M the doma U of whch cotas, ad D(x ),...,D(x ) be the comoets 1 of D the bass d (). The the mag D9-----L(x (),...,x (),D(x ),...,D(x )) 1 1 gve a local coordate system o u T M C TM. The set of all local eu 8 coordate systems costructed ths way forms a C -comatble atlas o TM, that turs TM to a dfferetable mafold. Exercse. Check the last statemet. --------------- The mag :TM----------L M whch takes a taget vector D to the ot e M at whch the vector s taget to M s called the atural roecto. The ---------- verse mage of a ot e M uder the atural roecto s the taget sace T M. Ths sace s called the fber of the taget budle over ---------- the ot. -------------------- The dervatve of a ma --------------- Defto. Let f:m----------l N be a smooth mag betwee the dfferetable mafolds (M,A), (N,B),ad let e M. The dervatve of f at the ot s the lear ma of the taget saces f : T M ----------L T N, f() whch s gve the followg way. Let D e T M ad cosder a curve g :[-e,e]----------l M wth g()=, ad seed vector D. The f (D) s the taget vector rereseted by the curve fqg. Proosto. The dervatve f s correctly defed (does ot deed o ----- the choce of g) ad s lear. Proof. We derve a formula for f usg local coordates whch wll both arts of the roosto clearly. show Let x = (x,...,x ) ad y = (y,...,y ) be local coordates a 1 m 1 eghborhood of e M ad f() e N resectvely. 6
If the comoets of D the bass d () corresodg to the chart x are { a : 1 < < m } the we have (x qg) () = a. Observe, that a deeds ~ oly o D. The comoets { b :1 < < } of f (D) the bass d (f()) geerated by the chart y ca be comuted by the formula b = (y qfqg) (). ~ Deote by f the -th coordate fucto of the mag yqfqx,.e. ~ f = y qfqx. The we have ~ b = (y qfqg) () = [(y qfqx )qxqg] () = [f q(xqg)] () = ~ ~ m d f m d f d x d x =1 =1 = S -------------------- (x())(x qg) () = S -------------------- (x())a, whch shows that f (D) deeds oly o D ad that f s a lear mag the ( ) ( ) ~ matrx of whch the bases {d ()} ad {d (f())} s 9 9 & ~ d f -------------------- (x()).44444 d x 1<<m 7 81<< 7