Differentiable Manifolds Lectures

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1 Differentible Mnifolds Lectures Leonrdo Abbresci Mrch 22, First Lecture Definition 1.1. A function f defined on R n is C k for ositive integer k if l f x i1 x i2 x il exists nd is ontinuous for ny ositive integer l k. A function is C if f is C k k N Definition 1.2. A homeomorhism is continuous function with continuous inverse. Definition 1.3. A coordinte chrt U, ϕ on toologicl sce X is n oen set U X with m ϕ : U R n such tht ϕ is homeomorhism onto ϕu. Let X be toologicl sce with coordinte chrts U, ϕ, V, ψ nd function f : I R. Consider f ϕ 1 s function defined on ϕu nd differentible f ϕ 1. Question : Is f ϕ 1 differentible the sme s f ψ 1 differentible? f ϕ 1 = f ψ 1 ψ ϕ 1 f ψ 1 = f ϕ 1 ϕ ψ 1. From this we see tht the imortnt thing is tht we need to mke sire tht both ψ ϕ 1 nd ϕ ψ 1 re differentible. Definition 1.4. The coordinte chrts U, ϕ nd V, ψ re sid to be comtible if the trnsition ms re diffeomorhisms. Definition 1.5. An tls A for toologicl sce X is collection of coordinte chrts such tht ny two coordinte chrts in A re smoothly comtible. 2 Second Lecture Given toologicl sce X, coordinte chrt U, ϕ is double such tht U X is oen, ϕu is oen, nd ϕ : U R n is homeomorhism. Definition 2.1. We sy tht the coordinte chrts U, ϕ nd V, ψ re smoothly comtible if U V = or ψ ϕ 1 is diffeomorhism from ϕu V to ψu V. Definition 2.2. A diffeomorhism is homeomorhism between two mnifolds tht is differentible C whose inverse is lso differentible. Definition 2.3. An tls A for toologicl sce X is collection of coordinte chrts tht cover X such tht ny two coordinte chrts in A re smoothly comtible. 1

2 Exmle 2.4. Consider the set of ll lines through the origin in R 3. This set is clled rojective sce RP 2. Since we cn reresent ech line with more thn one vector, this set is the sme s the quotient sce R 3 \{0, 0, 0}/ x 1, x 2, x 3 λx 1, x 2, x 3. This sce is equied with the quotient toology. i.e., π : R 3 \{0, 0, 0} RP 2 is continuous. We use [x 1, x 2, x 3 ] to denote the equivlence clss of vectors x 1, x 2, x 3. On the set U 1 := {x 1 0}, we cn use the coordinte chrt ϕ 1 : U 1 R 2 x x 1, x 2, x 3 2. x 1, x3 x 1 Consider the similr, corresonding coordinte chrts U 2, ϕ 2, U 3, ϕ 3. Clim: A = {U i, ϕ i },2,3 is n tls for RP 2. We need to check tht ϕ 2 ϕ 1 1 is diffeomorhism. ϕ 1 x 1, x 2, x 3 x 2 /x 1, x 3 /x 1 ϕ 2 x 1 /x 2, x 3 /x 2 From this we see tht ϕ 2 ϕ 1 1 x 2/x 1, x 3 /x 1 = x 1 /x 2, x 3 /x 2. We choose coordintes u, v R 2 nd write ϕ 2 ϕ 1 1 in terms of u, v. Note tht U 1 U 2 = {x 1 0, x 2 0}. Using x 2 /x 1 = u nd x 3 /x 1 = v, we obtin ϕ 2 ϕ 1 1 u, v = x1, v. x 2 u This is, in fct, differentible nd homeomorhic. Definition 2.5. Two tlses re comtible or equivlent if their union is nother tls. Definition 2.6. A differentible or smooth structure on toologicl sce is n equivlence clss of tlses. This is lso clled mximl tls. Definition 2.7. A toologicl sce X is sid to be Husdorff if ny two oints cn be serted by disjoint oen sets., q X, U, V oen s.t. U, q V nd U V =. Definition 2.8. A toologicl sce X is sid to be 2nd countble if it hs countble bsis B of oen sets. Definition 2.9. A bsis B is subset of the collection of ll oen sets sometimes the toology such tht ny oen set cn be written s union of elements of B. Exmle R n is 2nd countble becuse we cn tke B to be the colltion of blls of rtionl rdii centered t rtionl oints. Exmle R n is Husdorff for two oints, q R n, tke blls of rdii 1/3 q. Remrk. A subsce of toologicl sce tht is Husdorff nd 2nd countble is one itself. Definition A differentible mnifold is toologicl sce tht is Husdorff, 2nd countble, nd hs differentible structure. 3 Third Lecture Definition 3.1. A level set of rel-vlued function f of n vribles is set of the form L c f = {x 1,..., x n fx 1,..., x n = c}. 2

3 Exmle 3.2. Define F x, y, z = x 2 + y 2 + z 2 nd X = {x, y, z x 2 + y 2 + z 2 = 1}. Then X is level set of F. Let F : U R be smooth m, where U R n is oen. In fct, F 1 c is lwys closed, Husdorff, nd 2nd countble. The question is when is F 1 c smooth mnifold? In other words, when cn we solve one vrible in terms of the other? F x, y, z = c = z = fx, y. Theorem 3.3. Let F : R n+m R m be smooth function. Let R n+m hve coordintes x, y such tht x R n, y R m. Fix oint, b such tht F, b = c nd c R m. If y F, b is invertible i.e. rnk D F, b = m, then there exists oen sets U, V such tht U, b V, nd unique, bijective, nd smooth function g : U V such tht g = b nd {x, gx x U} = {x, y U V fx, y = c}. Theorem 3.4. F : R n R is smooth function, c R, nd suose F x 0, x F 1 c. F 1 c. Then F 1 c is smooth n 1-dimensionl mnifold. Remrk. If smooth mnifold is connected, then ny coordinte chrt is homeomorhic to n oen subset of R n for fixed n. This n is clled the dimension of sid mnifold. Proof of Theorem 3.4. First of ll, F 1 c is Husdorff nd 2nd countble. Let { } F Ũ i = 0 U i = x Ũi F 1 c. i We know tht U i is oen in F 1 c becuse of the induced toology of R n. F 1 c is covered by these U i s becuse of our ssumtion tht F x 0 x F 1 c. In other words, F 1 c n U i. By the imlicit function theorem, U i, there exists neighborhood U of which we my ssume to be in Ũi such tht U F 1 c is the grh of function This f i is unique. We tke the collection x i = f i x 1,..., x i 1, x i+1,..., x n. F 1 c s coordinte chrts. There re two cses when two coordinte chrts overl: either they belong to the sme U i, or not. If they belong to the sme U i, the trnsition m is the identity m, which is diffeomorhism. If they belong to different U i s, we see tht the trnsition ms stisfy the criterion tht mke F 1 c mnifold: ϕ i x 1,..., x n = x 1,..., x i 1, x i+1,..., x n ϕ j x 1,..., x n = x 1,..., x j 1, x j 1,..., x n ϕ j ϕ 1 i x 1,..., x i 1, x i+1,..., x n = x 1,..., x i 1, f i x 1,..., x i 1, x i+1,..., x n, x i+1,..., x j 1, x j+1, x n U Theorem 3.5. Let U R n+m be oen. F : U R m is smooth function. Let c R m, nd F 1 c. If F 1 c, DF : R n+m R m, F 1 c is n n-dimensionl smooth mnifold. 3

4 4 Fourth Lecture Let M be smooth mnifold. Definition 4.1. f : M R is smooth function if M, U, ϕ coordinte chrt contining such tht f ϕ 1 is C. Definition 4.2. Let N be smooth mnifoldf : M N is sid to be smooth if M, U, ϕ coordinte chrt contining nd V, ψ coordinte chrt contining f such tht ψ f ϕ 1 is C. Definition 4.3. f : M N is diffeomorhism if f nd f 1 re C. Consider the sce of smooth functions on M. Question. Cn we roximte chrcteristic function by smooth functions? Question. Suose we re given k oints on M. Is there smooth function tht hs given vlue t ech of these oints? Remrk. If n nlytic function is zero on n oen set of M, then it is equl to zero if M is connected. There re smooth functions tht re not nlytic! { e 1/t t > 0 Exmle 4.4. Let fx = 0 t 0, f : R R. We see tht f n 0 = 0 n N, nd thus the tylor exnsion is identiclly zero in neighborhood of 0. However, f 0 in neighborhood of t = 0. Lemm 4.5. Existence of cut-off nd bum functions. 1 t 1 1 h C, h : R R such tht ht = 0 < ht < 1 1 < t < 2 0 t 2 1 x 1 2 H C, H : R n R such tht Ht = 0 < Ht < 1 1 < x < 2 0 x 2 Proof. Consider f2 t nd f1 t. Then, we see tht ht = f2 t f2 t + ft 1. Also, Hx = h x. We hve tht Hx is the bum function suorted in B 2 0. Definition 4.6. The suort of function g is sug = {x gx 0}. Theorem 4.7. Existence of rtition of unity. Let M be smooth mnifold, nd O is n oen cover of M. Then ϕ α C such tht ϕ : M [0, 1] where α A, ny index set nd: 1 The set of suorts {suϕ α } α A is loclly finite. i.e., M, there exists neighborhood of tht intersects finite mount of suϕ α. 2 ϕ α = 1 M. α Note tht this sum is finite becuse of 1, nd thus, we were ble to choose n rbitrry index set A. 3 α A, U O such tht suϕ α U. Exmle 4.8. R =, ,. Tke ϕ 1 t = f2 t f2 t + ft 1, ϕ ft 1 2t = f2 t + ft 1. Then we see tht this holds ll the roerties from Theorem 4.7 becuse suϕ 1 0.5, nd suϕ 2,

5 Proosition 4.9. Let M be smooth mnifold. Then for ny closed set A M nd ny oen set U contining A, there exists smooth function ϕ : M R such tht ϕ 1 on A nd suϕ U. Proof. Let U 0 = U, nd U 1 = M\A. Then U 0, U 1 is n oen cover. Then, from Theorem 4.7, there exists rtition of unity {ϕ α } such tht suϕ α U 0 or suϕ α U 1 α. Consider ϕ 1 = ϕ α = ϕ 1 0 ona. suϕ α U 1 I clim tht ϕ 0 = 1 ϕ 1 is the desired function. We see tht ϕ 0 1 on A. Also, ϕ 0 = ϕ α = suϕ 0 U 0 = U. suϕ α U 0 Theorem Suose A U M, where A is closed, U oen, nd M is smooth mnifold. f : A R is function tht cn be extended to smooth function in neighborhood of A. Then f : M R smooth such tht f A = f nd su f U. 5 Fifth Lecture Exmle 5.1. F : R n R is smooth nd c R is regulr vlue for f. Then the tngent sce of f 1 C t f 1 c is the n 1 dimensionl ffine sce ssing through nd orthogonl to f. The eqution for this tngent sce is f x = 0. We cn view directionl derivtives s tngent vectors. Given v R n, nd smooth function f defined ner is on v, the directionl derivtive is given by D v f = d dt f + tv. t=0 The directionl derivtive ssigns number to ech smooth function. D v f = d dt f + tv = v i t=0 x i f = v f. We consider the directionl derivtives s oertors which form vector sce snned by Definition 5.2. X is derivtion t if X is liner m from C R n R tht suffices Xcf = cxf Xf + g = Xf + Xg Xfg = fxg + gxf Remrk. A directionl derivtive t is derivtion t. Remrk. The set of derivtions t forms rel vector sce. x i. Theorem 5.3. The vector sce of directionl derivtives nd the vector sce of derivtions t re isomorhic to ech other s rel vector sces. Remrk. Only the locl behvior of smooth function t mtters. We should define derivtion s liner m from the sce of germs of functions t. 5

6 Definition 5.4. Given R n, germ of function t is ir f, U with U such tht f is differentible in U. We define n equivlence reltion f, U g, v if f g on U V. Lemm 5.5. Let c denote constnt function with vlue c. Then Xc = 0 for ny derivtion. Proof. Xc = Xc 1 = cx1 = X1 = X1 1 = 1X1 + 1X1 = 2X1 X1 = 0 Lemm 5.6. Suose f is differentible in some neighborhood U of R n. B ε U nd differentible functions g i in B ɛ such tht fx = f + g i xx i i in B ε nd g i = x i f. Then, ε > 0 such tht Proof of Theorem 5.3. Any directionl derivtive is derivtion. Thus, it suffices to show tht the inclusion m is n isomorhism. It is obvious to see tht this inclusion m is liner. We wnt to show tht if f, v i x i f = 0, then v i = 0 for ech i. This would show tht the kernel is zero, nd thus the inclusion m is injective. x j { 0 if i j Note: x i = 1 if i = j. Let f = xj. Then we see tht 0 = v i xj x i = v j. Since is true for ny j, we hve tht indeed, v = 0, nd the inclusion m is injective. Now to get subjectiveness, we need to sk ourselves the question: If derivtion X corresonds to directionl derivtive v i x i, wht re the v i s with resect to X? We define them in the following wy: v i x i x j = v j = Xx j. Tking these Xx j s, I clim tht Xf = Xx i x i f f C. From Lemm 5.6, We hve tht Xf = X f + g i xx i i. 6

7 From Lemm 5.5 nd the roduct rule tht we defined, we hve Xf = X f + g i xx i i = = = Xg i xx i i i i Xg i x + g i Xx i X i Xx i x i f Definition 5.7. Let M be differentible mnifold, nd M. The tngent sce of M t, T M, is the vector sce of ll derivtions X t, where X : C M R nd Xfg = fxg + gxf. Remrk. Let F : M N be differentible m, nd f : N R be differentible function. Then, we cn ull-bck f by F, i.e. f F = F f : M R. Suose X T M is derivtion t M. Then we cn ush-forwrd X by F, i.e. F X is derivtion t F = F X T F N, where 6 Sixth Lecture F Xf := XF f = Xf F. Recll tht for M differentible mnifold, M, we define T M s the rel vector sce of derivtions t. X is derivtion t if X : C R is liner nd Xfg = fxg + gxf. For R n, T R n is the rel vector sce snned by x i. For coordinte chrt U, ϕ on M, we cn identify U with ϕu nd tke x i = ϕ 1 x i s bsis for T M. Question. Wht if two coordinte chrts U, ϕ nd V, ψ overl? Let x i be the coordintes for the imge of ϕ nd x j be the coordintes of the imge of ψ. We hve lredy estblished tht t, we cn identify x i. Notice however, tht we cn write x j = ψ ϕ 1 j x 1,..., x n. Then, from the chin rule, x i = ψ ϕ 1 j j=1 x i = ψ ϕ 1 j x i x j x j Einstein - Summtion Convetion. 7

8 Definition 6.1. Let F : R n R m be differentible. If g : R m R nd X is derivtion t R n, then F X is derivtion t F R m. This is defined s F Xg = X F g = Xg F. }{{} Pullbck Question. In terms of locl derivtions, wht is F x i g? Denote y α = F α x 1,..., x n Well from definition, we know tht F x i g = x i gf = x i gf 1 x 1,..., x n,..., F m x 1,..., x m gf F α = y α x i. This then imlies tht F x i = F α x i y α. F In the cse where U, ϕ nd V, ψ re coordinte chrts, consider F : U, ϕ V, ψ. Then we cn write it s F x i = ψ F ϕ 1 α x i y α. F Definition 6.2. T M clled the tngent bundles of M s set is the disjoint union of tngent sces: T M = T M. Question. Wht is the toology on T M? M Exmle 6.3. On R 2 there is the usul toology. However, we cn find different toology from the metric dx 0, y 0, x 1, y 1 = y 0 + x 1 x 0 + y 1. Then consider the sequence 1/n, 1. Then in this converges in this metric to 0, 1! Exmle 6.4. Consider S 2 = {x, y, z x 2 + y 2 + z 2 = 1} R 3. Then we see tht { } T S 2 = x, y, z, u, v, w x2 + y 2 + z 1 = 1. ux + vy + wz = 0 Exmle 6.5. If c is regulr vlue of f, nd f 1 c R n, then we hve tht { T f 1 c =, x f } 1 c R 2n. f x = 0 Note. We need to find n intrinsic definition of tngent bundles! Note tht there is lwys rojection m π : T M M. We will define differentible structure to mke π smooth. 8

9 Theorem 6.6. Let M be n n-dimensionl mnifold. There exists differentible structure on T M such tht the rojection m π : T M M is differentible, nd T M is 2n-dimensionl mnifold. Proof. Suose U, ϕ is coordinte chrt on M, nd x i, i = 1,..., n re coordintes on ϕu. Tke π 1 U, ϕ s coordinte chrt for T M. Any element in π 1 U is derivtion t U. But we know tht ny derivtion is of the form v i x i. We define the m ϕ s: v i x i x ϕ 1,..., x n, v 1,..., v n ϕu R }{{}}{{} ϕu R n ϕ : π 1 U ϕu R n R 2n. It is bijection onto its imge becuse its inverse cn be written exlicitly by x, v v i x i. ϕ 1 x When two coordinte chrts U, ϕ, V, ψ overl, then π 1 U, ϕ nd π 1 V, ψ overl. We wnt to show tht ψ ϕ 1 is smooth m. The sets ϕ π 1 U π 1 V = ϕu V R n nd ψ π 1 U π 1 V = ψu V R n re both oen in R 2n. We see tht the trnsition m ψ ϕ 1 x 1,..., x n, v 1,..., v n = ψ = x j v i xj = vi xi x i x 1,..., x n, v 1 xi xj x,..., vj xi x j x Is clerly smooth. Choosing countble cover {U i } of M by coordinte domins, we obtin countble cover of T M by coordinte domins. Note tht two oints in the sme fiber of π lie in one chrt. If X t nd Y t q lie in different fibers, there exist disjoint coordinte domins U i, U j for M such tht U i nd q U j becuse M is Husdorff, nd then the sets π 1 U i nd π 1 U j re disjoint neighborhoods contining, X nd q, Y resectively. Thus, T M is 2n-dimensionl mnifold. We see tht π is smooth becuse its coordinte reresenttion with resect to chrts U, ϕ for M nd π 1 U, φ for T M is πx, v = x. 7 Seventh Lecture Rec of lst lecture: Let M be C mnifold, nd let T M = M T M be the tngent bundle t. Then from lst clss, we estblished tht there exists differentible structure on T M tht mkes T M C mnifold. Definition 7.1. A C vector field Y t is C m Y : M T M such tht π Y = Id M. i.e., Y T M. In locl coordinte chrts, we hve tht Y = Y i x i. And Y i re C functions on coordinte chrts. Consider U V where U, ϕ, V, ψ re the usul coordinte chrts on M. Then we hve tht Y = Y i x i = Ỹ j x j. Then, from the chin rule, we hve tht Y i xj x i = Ỹ j. n.. 9

10 Exmle 7.2. On R 2 \0, 0, Let Y 1 = x y + y x Y 2 = x y y y 1 Ỹ 1 = x x2 + y 2 y + y x 1 Ỹ 2 = x x2 + y 2 y x. y We see tht Y 1, Y 2 cn be extended into 0, 0 nd become vector fields on R 2, but not Ỹ1, Ỹ2. Consider the stereogrhic rojection ϕ : S 2 \1, 0, 0 R 2. We cn use ϕ 1 to ush forwrd Y 1 nd Y 2 so tht they become vector fields on S 2 \1, 0, 0. We notice tht ϕ 1 Y 1 is tngent to longitude line. We lso see tht Y 2 is tngent to ltitude line. ϕ 1 Let T M be the set of ll C vector fields on M. Agin, this is rel vector sce. We cn lso multily C vector field by C funtion. C M hs ring structure. Then we see tht T M cn be considered s module over C M. i.e., vector sce over rings. All of this mens tht if Y T M, f C M = fy T M. We sy tht T M cn be identified with the vector sce of globl derivtions on M. A globl derivtion is liner m Y : C M C M such tht Y fg = fy g + gy f. Any globl derivtion comes from globl C vector fields. Now suose X, Y re C vector fields nd thus globl derivtions. Question. Is Y X globl derivtion? Y Xfg? = f Y Xg + g Y Xf But since we hve tht We notice tht = Y gxf + fxg = Y gxf + g Y Xf + Y fxg + f Y Xg XY fg = XgY f + g Y Xf + XfY g + f XY g, [X, Y ]fg = XY Y Xfg = gxy Y Xf + fxy Y Xg. And thus we see tht if X, Y T M, then X, Y T M but [X, Y ] = XY Y X T M. This is binry oertion on T M tht stisfies Linerity: [X + by, Z] = [X, Z] + b[y, z]. b Skew symmetry: [X, Y ] = [Y, X]. c Jcobi identity: for X, Y, Z T M, then This mkes T M lie lgebr. [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]] = 0. Locl Formul: For X = v i, nd Y = wj, we see tht xi xj [X, Y ] = v i wj vi wj xi x i x i. 10

11 8 Eighth Lecture Recll tht for mnifold M, X is smooth vector field if nd only if X is globl derivtion: X : C M C M liner, nd Xfg = fxg + gxf. XY Y X = [X, Y ] is smooth vector field. [, ] mkes J M lie lgebr. Suose F : M N is smooth. Recll tht F : T M T F. M F N f R. For Y T M, we see tht F Y f = Y f F T F N. Exmle 8.1. γ : ε, ε M. If we hve tngent vector in ε, ε, t, then we see tht γ = γ t t is tngent vector of γt. We see tht γ is only defined in the imge of γ. t Definition 8.2. Y is vector field on M. Z is vector field on N. Y nd Z re sid to be F -relted t if F Y = ZF. Theorem 8.3. Suose V 1, V 2 T M nd W 1, W 2 T N. Suose V i nd W i re F -relted for i = 1, 2. Then [V 1, V 2 ] is F -relted to [W 1, W 2 ]. In rticulr, if F is diffeomorhism, then F cn be defined on T N. Then W i = F V i. nd the sttement is equivlent to F [V 1, V 2 ] = [F V 1, F V 2 ] = F is n isomorhism. Question. Wht is the geometric mening of [X, Y ]? We re going to interret it s the lie derivtive of Y long X. Exmle 8.4. M = R n, X = V is constnt vector field. Y is nother smooth vector field on R n. Then we see tht Y + tv Y Y Y tv D v X = lim = lim. t 0 t t t Definition 8.5. A C m ϕ : ε, ε M M is flow generted by vector field X if ϕt, : M M is diffeomorhism if t ε, ε nd ϕ0, =. i.e., ϕ0, : M M is the identity. We see tht ϕ t, = Xϕt,. t We write ϕ t = ϕt,. Exmle 8.6. Let M = R 2. Xx, y = y x + x. Wht is the flow generted by X? Tke =, b y on R 2. We look for curve ϕt, where is fixed. We write it s γt = xt, yt. Then we hve tht ϕ0, x0 =, y0 = b. Also we see tht γ t = x t, y t = yt, xt. { x t = yt x0 = y t = xt y0 = b = xt = cost b sint yt = b cost + sint This is for fixed, b. Since we wnt this for n rbitrry, we hve tht ϕt, x, y = x cost y sint, y cost + x sint x cost sint x ϕ t =. y sint cost y ϕ t x + iy = e it x + iy We see tht this corresonds with rottion! We cn check tht ϕ t is diffeomorhism. Exmle 8.7. Xx, y = x x + y y 11

12 Theorem 8.8. A C vector field X of comct suort gives smooth fmily of diffeomorhisms tht re defined for t R. sux = { M X 0}. Here, the Existence nd uniqueness deend on the system of first order ODE. Smooth deendence of the ODE solutions deends on the initil dt nd the vector field. Uniqueness of solution of the ODE system imlies ϕ t ϕ s = ϕ t+s. ϕ t forms 1-rmeter grou of diffeomorhisms. We see tht ϕ t ϕ t = ϕ 0 =. in rticulr, ϕ t : T ϕ tm T M. Definition 8.9. We define the Lie Derivtive s Proosition [X, Y ] = L X Y. 9 Ninth Lecture Y ϕ t Y ϕ t L X Y = lim. t 0 t Where do vector fields come from? They come from velocity of smooth flows! If ϕ : ε, ε M M, is smooth m, then we hve tht ϕ is vector field on M. t Recll. On R n, there exists grdient vector field ssocited with ny smooth function f: f f = x 1,..., f x n. Question. Given smooth functions f on smooth mnifold M, is f smooth vector field? Answer. No! Proof. Recll tht given locl coordintes of two overling chrts x i nd x j, on the overl we cn write derivtion s v = v i x i = ṽj x j = ṽ j xi x j x i. Here, we got the lst line from the chin rule. Then, this imlies tht on the overl, we cn write v i = ṽ j xi x j. Recll tht v is derivtion if nd only if it is smooth vector field. Now on R n, this imlies we need f x i = f x j x j x i i = 1,..., n. However, notice tht x j x i x i x k = xj x k = δj k. Thus, we hve tht f is not smooth vector field in generl mnifold. 12

13 Definition 9.1. A rel vector bundle of rnk k over smooth mnifold M is ir E, π where E is smooth mnifold of dimension n + k, nd π is smooth rojection m π : E M, such tht 1 M, π 1 is k-dimensionl rel vector sce. 2 There is existence of locl triviliztion: M, there exists neighborhood U of nd diffeomorhism Φ : π 1 U U R k such tht the digrm commutes. π 1 U Φ U R k π π 1 Here we hve tht π = π 1 Φ π 1 is sent to {} R k by Φ. i.e., it sends fiber to fiber. 3 q U, the restriction of Φ to π 1 q is liner isomorhism between π 1 q nd {q} R k. U Exmle 9.2. Consider T M = the tngent bundle. Then we hve tht Φ : π 1 U U R k where Φ is defined by v i x i, v i. Furthermore, we hve tht when chnging coordintes, v i = ṽ j xi x j. The Jcobin for this trnsformtion is liner isomorhism. Definition 9.3. A section of E, π is smooth m σ : M E such tht π σ = Id M. This corresonds to globl sections. I cn obviously define locl section by σ : U M E. Note. A locl triviliztion gives locl sections of the bundle. Exmle 9.4. Consider E = T M. Then we hve tht Φ : π 1 U U R n where Φ is defined by v i x i, v i. Then I clim v 0 R n, Φ 1, v 0 gives locl section. In fct, if v 0 = 1, 0,..., 0, then we see tht Φ 1, v 0 = x 1. Definition 9.5. A vector bundle E, π is trivil if globl triviliztion exists. i.e., there exists diffeomorhism Φ : E M R k defined by π 1 q {q} R k such tht Φ π 1 q is liner isomorhism between π 1 q nd {q} R k. Exmle 9.6 Product Bundles. Consider smooth mnifold M. Then n exmle of rnk-k vector bundle over M is M R k. Here, we hve π = π 1 : M R k M s its smooth rojection. We see tht M R k is trivil becuse we hve the identity m Id : M R k M R k defined s the globl triviliztion. Another exmle of roduct bundle follows: Tke S 1 R. We hve this is trivil bundle over S 1. Exmle 9.7. The totl sce of n infinite Mobius stri is non-trivil rnk 1 vector bundle over S 1. Note. If bundle is trivil, then there exists globl section tht is nowhere zero. Tke ny non-zero vector v 0 R k, nd consider Φ 1, v 0. Since Φ is liner isomorhism on ech fiber, Φ 1, v 0 0 π 1. But since we hd tht the bundle ws trivil, then there exists globl section tht tkes M to this rticulr Φ 1, v 0. Definition 9.8. Suose two locl triviliztions overl we hve overling neighborhoods U, V M. Then on π 1 U π 1 V, we obtin Ψ Φ 1 : U V R k U V R k. We hve Ψ Φ 1 restricts to liner isomorhism between fibers. Notice tht Ψ Φ 1, v =, f, v. Now, f, v is liner isomorhism U V. However, notice tht Ψ Φ 1 is smooth. This imlies tht f, v = τv = τ is smooth s vries. The longshot is tht we hve τ is liner isomorhism from R k R k tht deends smoothly on U V. Thus, we get smooth m from U V GL k R. 13

14 10 Tenth Lecture Recll. A smooth vector bundle over smooth mnifold M of rnk k is smooth mnifold E, π where π is smooth rojective m. M is clled bse. Here, ech fiber E = π 1 is vector sce of dim K. Also, there exists locl triviliztion. i.e., for ech oint, there exists locl diffeomorhism on neighborhood U of defined by Φ : π 1 U U R k. Here, Φ sends fiber π 1 q, q U to fiber {q} R k. Finlly, Φ π 1 q : π 1 q {q} R k is liner isomorhism. A section σ is smooth m M σ E such tht π σ = Id M. Definition We cn define the zero section of E s the globl section ζ : M E defined by ζ = 0 E M. Definition 10.2 Trnsition M. When we hve two overling locl triviliztions, π 1 U π 1 V 0, we hve tht Ψ Φ 1 : U V R k U V R k sends {q} R k {q} R k nd is liner isomorhism. Becuse Ψ Φ 1 is diffeomorhism, this liner isomorhism vries smoothly s q vries. This gives smooth m U V GL k R. Note. A locl triviliztions gives locl section. Tke ny v 0 R k nd consider Φ 1, v 0. In the cse of tngent sces, we see tht v i Φ, v i x i. Here, Φ 1, 1, 0,..., 0 = x 1. Definition Consider the bsis e 1,..., e k for R k. Then the elements {Φ 1, e i },...,k form bsis for ech E = π 1. {Φ 1, e k } is locl frme. Definition Suose π : E M, π : E M re two smooth vector bundles. A bundle m from E to E is ir of C ms F : E E nd f : M M such tht F covers f, or the digrm commutes: E F E π f M M nd F E : E E f is liner m. We cn define smooth bundle isomorhism by relcing C ms bove with diffeomorhisms nd liner ms with liner isomorhisms. Question. Cn we clssify vector bundles u to bundle isomorhisms? Exmle A C m f : M N induces bundle m on the tngent bundle by f. π T M π 1 M F f T N N π 2 Then we know tht F X = f X T f N. We know by the chin rule tht f x i = f α x i y α. Exmle In the cse where M = N nd f is the identity, then ny bundle m induces m on smooth sections. Denote the smooth globl sections of E by ΓE, nd the ones of E by ΓE. We hve the following commuttive digrm: 14

15 F E E π π M We see tht F induces m nlogous to the ush forwrd F : ΓE ΓE defined by Fσ = F σ = F σ. We see tht F σ is section of E, nd it is smooth by comosition. Becuse F is liner on fiber to fiber, the resulting m F is liner over R. In fct, it stisfies strong linerity roerty. A m F : ΓE ΓE is sid to be liner over C M if for ny smooth functions u 1, u 2 C M, nd smooth sections σ 1, σ 2 ΓE, then It is clled module over C M. Fu 1 σ 1 + u 2 σ 2 = u 1 Fσ 1 + u 2 Fσ 2. Question. Does every liner m F : ΓE ΓE come from bundle m F : E E? Theorem Suose E nd E re smooth vector bundles over M. A liner m F : ΓE ΓE comes from the bundle m F : E E if nd only if F is liner over C M. We consider ΓE nd ΓE s modules over C M. We require the F is module morhism. Proof. If F : ΓE ΓE is from bundle m F, we show tht it is liner. We know tht F σ = F σ. Let u be smooth function. Then we see Fuσ = F uσ = F uσ = uf σ = ufσ = ufσ. Thus, we see tht F is liner. Now ssume F : ΓE ΓE liner over C M, we show tht it must come from bundle m F : E E. To define F on n v E, tke section σ ΓE such tht σ = v = F v = F v = F σ = Fσ. Wht if we choose different σ such tht σ = σ. We need to show Fσ = F σ if σ = σ, or if σ = 0 = Fσ = 0. If σ = 0, we write it s σ = u σ such tht u = 0 = Fσ = Fu σ = ufσ = ufσ = 0. Thus, we see tht F is liner in C M. 11 Eleventh Lecture In this lecture we re going to introduce dul sces nd work out some of their roerties. We then ly this to vector bundles. Definition Let V be finite dimensionl rel vector sce. V = {ω ω : V R is liner}. V is rel vector sce. Tke ny bsis {E i },...,n of V. Consider the dul bsis {E i },...,n of V defined by Proosition {E i },...,n is bsis. E i E j = δ i j. Proof. We first need to show tht they sn. Given ny ω V, tke X V, nd define it by X = X i E i. Then we see tht ωx = ωx i E i = X i ωe i from linerity of ω. Notice tht E i X = E j X i E i = X i E j E i = X i. This kills off ll the other terms. Then we see tht ωx = ωe i E i X = ω = ωe i E i X = ωe i E i. So we see tht it sns. Now we check tht the elements in the set re linerly indeendent. Consider the cse where i E i = 0. 15

16 Does this imly tht i = 2 = = n = 0? Aly this eqution to ech bsis vector. Then we see tht i E i E j = j = 0. Definition V = V. Definition If A : V W is liner m, the dul m A : W V is liner m defined s follows: for ω W, we hve A ωx = ωax. We cn esily check tht this is liner. Definition 11.5 Cobundle or Dul Bundle. Consider rnk k vector bundle on differentible mnifold M. We know π 1 = E is k dimensionl vector sce. We define locl triviliztions Φ : π 1 U U R k s diffeomorhism. Φ restricts to liner isomorhism between π 1 nd {q} R k. Then we define E = E, M where π : E M is the obvious rojection m. Given U M, we consider π 1 U nd define locl triviliztions s follows: π 1 U U R k E ω, ω E 1,..., ω E n, where {E 1,..., E k } is the locl frme coming from locl triviliztion of E: E i = Φ 1, e i. Exmle We ly this rocedure to the tngent bundle nd denote the dul bundle by T M, nd section of T M is clled 1-form. The sce of smooth sections of T M is denoted by ΓT M. Note. Given smooth function on M, the grdient f =,..., x i x n is not section of the tngent bundle. However, it is section of the contngent bundle, or 1-form. In fct, it is better to consider the totl differentil df = f x i dxi, i.e., dfx = Xf. Remrk. D v f = f v. Since this oertion is liner, it cn be considered s liner functionl. Question. How does section of T M trnsform? Recll for the tngent bundle, Φ T M = M T M π U R n M Here, we hve tht Φ : π 1 U U R k is defined by v i x i, v 1,..., v n, nd Φ 1, e i = x i s our locl frme. 16

17 Question. Wht re the dul locl frme to x i? We re looking for something of the form ω j x i = δ j i. Tke x j nd define dx j X = Xx j. We ly this to dx j x i = δ j i = x i. Then we hve dx i,..., dx n form locl frme of T M on U. Then we cn define locl triviliztion s follows: Notice tht they trnsform differently from 12 Twelfth Lecture In this clss we re going to go over tensor roducts! π 1 U U R k v i dx j, v 1,..., v n.. So now we cn define this bstrct thing df s xi df = f x i dxi. Definition Let V 1,..., k, W be finite dimensionl vector sces. F : V 1 V k W is multiliner if v 1,..., v i 1, v i+1,..., v k V 1 V i 1 V i+1 V k, the m V i W defined by v v 1,..., v i 1, v, v i+1,..., v k is liner. Remrk. If W = R, then we cll F multiliner function. If V 1 = = V k = V, then the collection of multiliner functions V 1 V k R is denoted by T k V nd n element of which is clled covrint k-tensor. Exmle T 1 V = V, T 2 V is just the collection of biliner forms on V. Exmle On R n, the dot roduct is n element of T 2 R n. The determinnt function is det : R n R n R, so it is n element of T n R n. On R 3, the cross roduct is not becuse it is vector vlued. Definition If ω, m T 1 V, we define ω m T 2 V by ω mv 1, v 2 = ωv 1 mv 2. More generlly, if S T k V, T T l V, then S T T k+l V is defined by S T v 1,..., v k+l = SV 1,..., V k T V k+1,..., V k+l. Note. In generl S T T S. Let ω, η T 1 V. Then ω ηv 1, v 2 = ωv 1 ηv 2 η ωv 1, v 2 = ηv 1 ωv 2. However, we do hve ssocitivity! i.e., S T U = S T U. 17

18 Theorem The set {v i1 v i k 0 i 1 i k n} is bsis for T k V. Definition We define the free roduct s { N } F V W = i v i, ω i i R, N N, v i, ω i V W. Exmle An element of F R 2, R 2 would be 31, 2, 3, , 2, 1, 2 13, 0, 5, Thirteenth Lecture Recll. The cotngent bundle T M is dul to the tngent bundle T M M. Recll tht if F : M N is smooth m, then the ush forwrd F : T M T F N is defined for For X T M, we hve tht F Xf = Xf F. Recll. A : V W is liner, then the dul m A : W V is liner. So we consider the dul m F : T F N T M. Is defined by ω T F N F ω T M. When we re cting on vector field X T M, we hve F ωx = ωf X where F is the ush forwrd. Recll. F : T M F F N cn t be globlized in the sense tht F : ΓT M ΓT N is not defined. However, F : T F N T M cn be globlized. i.e., J : ΓT N ΓT M is well defined. i.e., ny one form ω on N cn be defined s J ω s one form on M. Exmle γ : [, b] M nd suose ω is one form on M in locl coordintes ω = ω i dx i nd γt = x 1 t,..., x n t. Then γ is one-form on [, b]. We choose coordintes on [, b]. We bsiclly hve to nswer γ ω =? dt. Consider Then we see tht d d γ ω = ω γ dt dt x i = ω t x i = ω j dx j x i = w i x i t. t x i γ ω = ω i x 1 t,..., x n t x i t dt. Consider in generl, consider F : M N, with locl coordintes x i,...,m on M nd yα α=1,...,n on N. Now suose ω = ω α y 1,..., y n dy α, in locl coordintes. Suose F is given by y α = F α x 1,..., x m. Then we see tht F ω = ω α F 1 x 1,..., x m,..., F n x 1,..., x m F α x j dxj. 1 Proosition The differentil commutes with the ull bck. i.e, consider F : M N nd let f be smooth function on N. Then F df = df F. Proof. Tke X T M. Then, F d f X = df F X = F Xf But this is true for ny X, so we get tht F df = df F. = df F X. 18

19 Definition Now we define line integrls! Suose [, b] R is comct intervl, nd ω is smooth convector field on [, b] this mens tht the comonent function of ω dmits smooth extension to some neighborhood of [, b]. Let t denote the stndrd coordinte on R, then ω cn be written s ω t = ft dt, for some smooth coordinte function f : [, b] R. We define the integrl of ω over [, b] s [,b] ω = b ft dt. 2 Lets consider the more generl cse. If γ : [, b] M is smooth curve segment nd ω is smooth convector field on M, we define the line integrl of ω over γ to be b ω = γ ω γ = b ω i x 1 t,..., x n t xi t dt. We should check tht this is indeendent of coordinte chrts. i.e, ω i = xi t = ω x j j t. Exmle Let ω be smooth covector field on [, b] R. Let ϕ : [c, d] [, b] be n incresing diffeomorhism mening tht t 1 < t 2 imlies ϕt 1 < ϕt 2. Let s denote the stndrd coordinte on [c, d] nd t on [, b]. Then we hve ϕ s = fϕsϕ s ds. Then [c,d] ϕ ω = d Similrly, if ϕ ws decresing diffeomorhism, c fϕsϕ s ds = [c,d] b ϕ ω = ω. [,b] ft = Proosition Let M be smooth mnifold, γ : [, b] M be iecewise smooth curve segment, F : M N be smooth m, nd η ΓN. Then F η = η. γ Proof. The roof follows from the definitions. Assume γ is smooth curve segment whose imge is contined in the domin of single smooth chrt. F η = γ F η = γ η j F df j γ = [,b] b η j F γ F γ [,b] F γj t Let F γ = γ be curve on N. Then we see tht the lst line bove becomes b η j F γ F γj t dt = b b = = F γ dt. [,b] η j γ γj t dt γ η = η η. γ ω. 19

20 Recll tht this ws in the cse where the imge of γ ws contined in single smooth chrt. Suose now tht γ is n rbitrry smooth curve segment. By comctness there exists finite rtition = 0 < 1 < < k = b of [, b] such tht γ [ i 1, i ] is contined in the domin of single smooth chrt for ech i = 1,..., k, so we ly the comuttion bove on ech sub intervl. Finlly, if γ is only iecewise smooth, we simly ly the sme rgument on ech subintervl on which γ is smooth on nd dd everything u. Definition If γ : [, b] M nd γ : [c, d] M re iecewise smooth curve segments, we sy tht γ is rermetriztion of γ if γ = γ ϕ for some diffeomorhism ϕ : [c, d] [, b]. If ϕ is n incresing function, we sy tht γ is forwrd rermetriztion. If ϕ is decresing, it is clled bckwrds one. Proosition Suose M is smooth mnifold, ω ΓM, nd γ is iecewise smooth curve segment in M For ny rmetriztion γ of γ, we hve { γ ω = ω if γ is forwrd rermetriztion γ ω if γ is bckwrd rermetriztion. γ Proosition If γ : [, b] M is iecewise smooth curve segment, the line integrl of ω over γ cn lso be exressed s the ordinry integrl b ω = ω γt γ t dt. γ Proof. Suose tht γ is smooth nd tht its imge is contined in the domin of single smooth chrt. Writing the coordinte reresenttions of γ nd ω on this smooth chrt s x 1 t,..., x n t nd ω i dx i, we hve ω γt γ t = ω i γt dx i γ t = ω i γt γ i t. Recll the formul F ω = ω j F df j. Alying this on the ull bck γ ω, we hve γ ω t = ω i γt dγ i t = ω i γt γ i t dt = ω γt γ t dt. Then we hve tht by definition of our integrl, ω = γ ω = γ [,b] b ω γt γ t dt. Recll tht this ws in the cse where the imge of γ ws contined in single smooth chrt. Suose now tht γ is n rbitrry smooth curve segment. By comctness there exists finite rtition = 0 < 1 < < k = b of [, b] such tht γ [ i 1, i ] is contined in the domin of single smooth chrt for ech i = 1,..., k, so we ly the comuttion bove on ech sub intervl. Finlly, if γ is only iecewise smooth, we simly ly the sme rgument on ech subintervl on which γ is smooth on nd dd everything u. Theorem 13.9 Fundmentl Theorem for Line Integrls. Suose γ : [, b] M is iecewise smooth curve segment on M. Let f C M. Then df = f γb f γ. γ Proof. First suose tht γ is smooth. Then, by Proosition 13.8, b df = df γt γ t b dt = f γ t dt = f γb f γ. γ Now suose tht γ is merely iecewise smooth, let = 0 < < k = b be the end oints of the subintervls on which γ is smooth. We ly the clcultion bove to ct segment nd sum them u: k df = fγ i fγ i 1 = fγb fγ. γ 20

21 Definition A smooth covector field ω on smooth mnifold M is sid to be exct on M if there is function f C M such tht ω = df. f is sid to be the otentil for ω. 14 Fourteenth Lecture Consider the tensor bundle nd its following globl sections Γ k T M ω. We hve tht k T M is loclly given by A i1,i 1,...,i k dx i1 dx i k, where 1 i 1 i k n, where n is the dimension of M. Recll. k V = T k V = {L : V V R}. }{{} k coies The elements of this set re clled covrint k-tensors. L is multiliner function. There re two distinguished clsses of multiliner sces: symmetric, nd lternting. Definition A lement ω T 2 V is symmetric if ωx, Y = ωy, X. We sy tht ω T k V is symmetric if ω X 1,..., X k = ω Xσ1,..., X σk, where σ is n element in the grou of ermuttions S k. Exmle If A is n n n mtrix, we hve tht A T + A/2 is symmetric. In fct, is skew symmetric. A = A + AT 2 + A AT 2 Definition ω T 2 V. We define its symmetriztion s SymωX, Y = 1/2 ωx, Y + ωy, X. For ω T k V, we define it s Symω X 1,..., X k = 1 k! σ S k ω Xσ1,..., X σk. Definition T k v Σ k V = {ω T k V ω is symmetric }. It is esy to check tht Symω Σ k V. Definition We cn lso define the symmetric roduct, which is sort of the the symmetriztion of the tensor roduct. Let ω T k V, η T l V. Then we define ω η = Sym ω η. This imlies tht ω η Σ k+l V becuse ω η T k+l V. In rticulr, lets look t the cse when ω, η T 1 V. Then we hve ω ηx, Y = Sym ω ηx, Y = 1 ω ηx, Y + ω ηy, X 2 = 1 ωxηy + ωy ηx. 2 So now, for ω V, η V we hve tht ω η = 1/2ω η + η ω 2 V is n element tht corresonds with n element in Σ 2 V. 21