R n α. . The funny symbol indicates DISJOINT union. Define an equivalence relation on this disjoint union by declaring v α R n α, and v β R n β

Transcription

1 Readng. Ch. 3 of Lee. Warner. M s an abstract manfold. We have defned the tangent space to M va curves. We are gong to gve two other defntons. All three are used n the subject and one freely swtches back and forth between the three. Whatever the defnton, the tangent space to M at a pont p M must enjoy the followng propertes. (1) t s a real vector space of dmenson n = dm(m) (2) t s ntrnscally attached to M. Intrnsc has two more-or-less equvalent meanngs n dfferental geometry and topology: (a) coordnate ndependent, (b): equvarant under mappngs: f F : M N s a smooth map between manfolds, then there should be defned, n a natural way, a lnear map df (p) : T p M T F (p) N between tangent spaces. The three defntons are: Defnton 1. Va curves. Defnton 2. Va dervatons actng on functons Defnton 3. Va coordnates. Defnton 1 s the one we have gven already. We recall t. Defne an equvalence relaton among the smooth curves passng through p. A tangent vector wll be an equvalence class. Ths defnton s the most geometrc of the three, but sometmes the most dffcult to compute wth. Defnton. A smooth curve c through p s a map c : ( ɛ, ɛ) M wth c(0) = p. Defnton. Two smooth curves c, c through p agree to frst order f, n some coordnate chart φ contanng p ther coordnate mages agree to frst order as curves n R n : φ c(t) = φ c(t) + O(t 2 ) Defnton. [Curve def of tangent space.] A tangent vector at p s an equvalence class of curves passng through p. The tangent space s the set of all such equvalence classes of curves. The followng lemma shows that ths defnton s ntrnsc n the sense (a). Lemma. Suppose c, c are two curves through p whch agree to 1st order n some chart φ. Then they agree to 1st order n any compatble chart. Proof. Snce c, c agree to 1st order n the chart φ we have φ c(t) = a + vt + O(t 2 ) φ c(t) = a + vt + O(t 2 ) where a = φ(p). Let ψ be another chart, and g = ψ φ 1 Then ψ c = gφ c, ψ φc(t) = g φ c(t) So that φ c(t) = g(a + vt + O(t 2 )) = g(a) + [dg(a)v]t + O(t 2 ) φ c(t) = g(a + vt + O(t 2 )) = g(a) + [dg(a)v]t + O(t 2 ) where we have used the fact that g s dfferentable, and we have used the chan rule to Taylor expand out g(a + vt + O(t 2 )). The two curves n the new coordnate system stll agree to 1st order. QED Defnton 2 of the tangent space. Va dervatons. 1

2 2 Wrte C (M) for the space of all smooth functons on M. It s forms a commutatve algebra over the reals. We can add, and multply functons, and scalar multply them. Defnton. A dervaton at p s an R lnear map C (M) R whch s a dervaton n Lebntz s sense: v[fg] = f(p)v[g] + g(p)v[f] In other words: t s a lnear functonal for whch the product rule works. Defnton. The tangent space to M at p s the space of dervatons at p. That ths defnton s ntrnsc s clear. We dd not use coordnates to defne t. It s also obvously a vector space, snce we can add dervatons and multply them by real numbers. But the dmenson of ths vector space s not obvous. Is t fnte? Maybe there are too many functons? Or s t zero? How do we know a manfold has ANY globally defned functons? Exercse. If f and g agree n a nbhd of p and v s a dervaton at p, then v[f] = v[g]. Prop. If f s smooth defned n a coord nbhd of p, then t can be smoothly extended to all of M. Pf. Bump functons. Cor. The value of v[f] s well-defned when f s defned on a nbhd of p, rather than all of M. ( Rmk. Smooth Urysohn lemma. ) Cor. to exercse. Let U be any nbhod of p and consder the space C (U) of smooth functons on To answer ths last queston wll lead us to bump functons, an mportant tool later on. Defnton 3 of the tangent space. va coordnates. Consder all charts ψ α : U α M R n contanng p, so that p U α. For each such chart ntroduce a copy of R n, denoted by R n α, and form the dsjont unon of all these vector spaces. : {α:p U α}. The funny symbol ndcates DISJOINT unon. Defne an equvalence relaton on ths dsjont unon by declarng v α R n α, and v β R n β to be equvalent: v α v β : ff d(ψ β ψα 1 )(ψ α (p))v α = v β. Then defne T p M = {α:p U α} Rn α/. Ths defnton s clearly ntrnsc accordng to noton (a) of ntrnsc, snce we have dvded out by all choce of charts. It s also clearly a vector space of dmenson n. Fx any one chart ψ α : U α M R n. Then every [v] T p M has a unque representatve v α R n α so that R N α = T p M. Pros and cons. Ths s the defnton closest to the way computatons are done. It s the uglest defnton. Bases n the varous defntons. Standard notaton. Equvalences between defntons. Let x be coordnates defned n a nbhd of p. Thus, some coordnate chart ψ : U M R n s wrtten out as ψ(q) = (x 1 (q),..., x n (q)). In all three defntons, these coordnates defne a bass, = 1,..., n for T p M. Thus a typcal element of T p M can be unquely expressed as Σv wth v R. R n α

3 3 Bass n defnton 1. Consder the coordnate x -curve γ. By ths we mean the curve n R n parallel to the x axs and through the pont correspondng to p: thus:γ (t) s defned by x j (t) = x j 0, x (t) = x 0 + t, where x 0 are the coordnates of p. Then means the equvalence class of the curve ψ 1 (γ ). Bass n defnton 2. If f C (M) then f ψ 1 : R n R can be wrtten (by a slght abuse of notaton) as f(x 1,..., x n ). The dervaton s the usual partal dervatve: [f] = f Bass n defnton 3. s the vector whose representatve n the chart for ψ = ψ α, s the vector e = (0,... 0, 1, 0,...) R n α. Equvalences of defntons. Def 1 to 2. To defne a dervaton v from a curve c we set v[f] = ( d dt t=0(f(c(t)). Note: R M c R f Coordnate computatons shows c v s well-defned on the level of equvalence classes: f c 1 c 2 then ( d dt t=0(f(c 1 (t)) = ( d dt t=0(f(c 1 (t)). A coordnate computaton also shows that the map c (0) v s lnear. If v = Σv we compute, usng the chan rule that ndeed: v[f] = Σv [f] as t should. Defnton 2 corresponds to the drectonal dervatve of vector calculus. Def 1 to Def 3. We went from 1 to 3 when we showed that the equvalence class of def 1 was chart-ndependent. (The lemma followng the explanaton of def 1: wrte the curve n a coord chart. Its frst order dervatve s the tangent vector as represented n that chart. ) Conversely, gven a chart φ α : U α R n = R n α wth P 0 correspondng to p, the nverse mage under φ α of the curve P 0 + tv R n α s the curve whose equvalence class represents v. Def 3 to 2. The vector v = (v 1,... v n ) R n α represents the dervaton Σv 1. More on Def 2. Why s the space of dervatons based at p spanned by the It s not even clear that ths space of dervatons s fnte-dmensonal..? Defne bump functons. Dscuss bump fns. Extendng fns. Why we can thnk of x as fns on all of M. Impossblty for cx mfds. 2. Dervatons. Maxmal deals. Cotangent space. We state some algebrac consequences of the defnton. Let v be a dervaton at p, and let m p C (M) be the subalgebra of functons vanshng at p. Fact 1. v[c] = 0, c a constant functon. Proof: 1 = 1 1, so v[1] = 1v[1] + 1v[1] = 2v[1], mplyng that v[1] = 0. Now v[c] = cv[1]. Consequence. v[f] = v[f f(p)]. 3. Fact 2. If h = m 2 p then v[h] = 0. Proof h m 2 p means h = fg for some f, g m p. Then v[h] = f(p)v[g]+g(p)v[f] = snce f(p) = g(p) = 0.

4 4 Proposton 0.1. v defnes a lnear map m p /m 2 p R and s unquely determned by ths map. Proof: By fact 1 the value of v s determned by ts values on m p. Any lnear map L : E F between vector spaces descends canoncally to a quotent map E/ker(L) F, and, f we know ker(l), then L s unquely determned by ths quotent map. 3. Fact 3. m p /m 2 p s a fnte-dmensonal vector space of dmenson n. Proof of Fact 3. The n coordnates are the 1st order Taylor expanson of a functon. In detal, use coordnates x, = 1,..., n centered at p. NOTE: Centered at p means that x (p) = 0.) Then Fact 2 transfers to the same statement regardng m 0 C (R n ). Expand f n terms of the x : f = Σa x + O( x 2 ). We must show that the remander term O( x 2 ) s n m p. Ths s easly seen from Lemma 0.2 (Hadamard s lemma. ). Let f be a smooth functon vanshng at 0 R n. Then we can wrte f = Σx g (x 1,..., x n ) where the g are smooth functons satsfyng g(0) = f 0 Proof of Hadamard. Wrte f(x) = f(0) + d dt out the ntegral to obtan f(x) = Σx g (x) where g (x) = 1 f(tx)dt Use f(0) = 0 and expand f (tx)dt. QED (also see, Lee, appendx). End of proof of Fact 3. Hadamard s lemma asserts that by subtractng off the lnear Taylor expanson, Σa x from f we obtan a functon n m 2 0. Defnton. m p /m 2 p s called the space of covectors at p or the cotangent bundle at p and s denoted by T p M. Then the proposton asserts that the tangent space T p M (vewed as dervatons) s canoncally dual to T p M. Dscusson. There s a long tradton of recoverng and better understandng a manfold, varety, topologcal space,... by way of algebras of functosn on t. the correspondence p m p C (M) embeds M as a famly of maxmal deals wthn C (M). Perhaps the most strkng s n the subject of C -algebras, the theorem called the Gelfand-Namark theorem. Let M be a compact Hausdorff space and A the space C 0 (M, C) of contnous complex-valued functons on M, It s a commutatve algebra over C. The sup norm, and the operaton f f, gves A the structure of what s known as a C -algebra. Gelfand-Namark Theorem. Every commutatve C algebra s C 0 (M, C) for some compact Hausdorff space M. The space M n the Gelfand-Namark theorem s bult out of multplcatve lnear functonals and forms the spectrum of A. Conversely, gven M, the assocated multplcatve lnear functonal s evaluaton at p: f f(p), a map from A C. The kernel of ths map corresponds to our m p. Algebrac geometry. The subject begns by studyng the zero locuses of polynomal functons. These zero locuses are called varetes. Varetes are very often manfolds, and are always manfolds at most of ther ponts lke our cone or cross. An essental theme n algebrac geometry s the traffc back and forth between the varety and the algebra of polynomal functons defned on the varety. The varety 0

5 5 s reconstructed out of the algebra, as the space of maxmal deals n the algebra. Our defnton of the cotangent space and tangent space work n the algebrac settng, even over arbtrary rngs (!) and are called the Zarsk (co)tangent space.