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Chapter 4

Manifolds, Spaces, Cotangent Spaces, Vector Fields, Flow, Integral

4.1

In Chapter 2 we defined the notion of a embed- ded in some ambient space, RN .

In order to maximize the range of applications of the the- ory of manifolds it is necessary to generalize the concept of a manifold to spaces that are not a priori embedded in some RN .

The basic idea is still that, whatever a manifold is, it is atopologicalspacethatcanbecoveredbyacollectionof open subsets, Uα,whereeachUα is isomorphic to some “standard model,” e.g.,someopensubsetofEuclidean space, Rn.

233 234 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Of course, manifolds would be very dull without functions defined on them and between them.

This is a general fact learned from experience: Geom- etry arises not just from spaces but from spaces and interesting classes of functions between them.

In particular, we still would like to “do calculus” on our manifold and have good notions of curves, tangent vec- tors, differential forms, etc.

The small drawback with the more general approach is that the definition of a is more abstract.

We can still define the notion of a on a manifold, but such a curve does not live in any given Rn,soitit not possible to define tangent vectors in a simple-minded way using . 4.1. MANIFOLDS 235 Instead, we have to resort to the notion of chart. This is not such a strange idea.

For example, a geography atlas gives a set of maps of various portions of the earth and this provides a very good description of what the earth is, without actually imagining the earth embedded in 3-space.

Given Rn,recallthattheprojection functions, pr : Rn R,aredefinedby i → pr (x ,...,x )=x , 1 i n. i 1 n i ≤ ≤

For technical reasons, in particular to ensure that par- titions of unity exist (a crucial tool in manifold theory) from now on, all topological spaces under consideration will be assumed to be Hausdorffand second-countable (which means that the topology has a countable basis). 236 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Definition 4.1.1 Given a topological space, M,achart (or local coordinate map)isapair,(U,ϕ), where U is an open subset of M and ϕ: U Ωisahomeomorphism → onto an open subset, Ω= ϕ(U), of Rnϕ (for some n 1). ϕ ≥ For any p M,achart,(U,ϕ), is a chart at p iff p U. ∈ ∈ If (U,ϕ)isachart,thenthefunctionsxi = pri ϕ are called local coordinates and for every p U,thetuple◦ ∈ (x1(p),...,xn(p)) is the set of coordinates of p w.r.t. the chart.

1 The inverse, (Ω,ϕ− ), of a chart is called a local parametrization.

Given any two charts, (Ui,ϕi)and(Uj,ϕj), if Ui Uj = ,wehavethetransition maps, j ∩ ( ∅ ϕi : ϕi(Ui Uj) ϕj(Ui Uj)and ϕi : ϕ (U ∩ U ) → ϕ (U ∩ U ), defined by j j i ∩ j → i i ∩ j j 1 i 1 ϕ = ϕ ϕ− and ϕ = ϕ ϕ− . i j ◦ i j i ◦ j 4.1. MANIFOLDS 237

i j 1 Clearly, ϕj =(ϕi )− .

j i Observe that the transition maps ϕi (resp. ϕj)aremaps between open subsets of Rn.

This is good news! Indeed, the whole arsenal of calculus is available for functions on Rn,andwewillbeableto promote many of these results to manifolds by imposing suitable conditions on transition functions.

Definition 4.1.2 Given a topological space, M,given some integer n 1andgivensomek such that k is either an integer k ≥1ork = ,aCk n-atlas (or n-atlas of class Ck), ≥,isafamilyofcharts,∞ (U ,ϕ) ,suchthat A { i i } (1) ϕ (U ) Rn for all i; i i ⊆ (2) The Ui cover M,i.e.,

M = Ui; i ! j (3) Whenever Ui Uj = ,thetransitionmapϕi (and i k ∩ ( ∅ j ϕj)isaC -diffeomorphism. When k = ,theϕi are smooth diffeomorphisms. ∞ 238 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES We must insure that we have enough charts in order to carry out our program of generalizing calculus on Rn to manifolds.

For this, we must be able to add new charts whenever necessary, provided that they are consistent with the pre- vious charts in an existing atlas.

Technically, given a Ck n-atlas, ,onM,foranyother chart, (U,ϕ), we say that (U,ϕ)isA compatible with the 1 1 k atlas iffevery map ϕi ϕ− and ϕ ϕi− is C (whenever U UA = ). ◦ ◦ ∩ i ( ∅

Two atlases and , on M are compatible iffevery chart of one isA compatibleA with the other atlas.

This is equivalent to saying that the union of the two atlases is still an atlas. 4.1. MANIFOLDS 239 It is immediately verified that compatibility induces an equivalence relation on Ck n-atlases on M.

In fact, given an atlas, ,forM,thecollection, ,of all charts compatible withA is a maximal atlas inA the A equivalence class of charts compatible with . " A

Definition 4.1.3 Given some integer n 1andgiven some k such that k is either an integer k ≥1ork = , a Ck-manifold of n consists of≥ a topological∞ space, M,togetherwithanequivalenceclass, ,ofCk n-atlases, on M.Anyatlas, ,intheequivalenceclassA is called a differentiable structureA of class Ck (and Adimension n)onM.WesaythatM is modeled on Rn. When k = ,wesaythatM is a smooth manifold. ∞ Remark: It might have been better to use the terminol- ogy abstract manifold rather than manifold, to empha- size the fact that the space M is not a priori a subspace of RN ,forsomesuitableN. 240 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES We can allow k =0intheabovedefinitions.Condition (3) in Definition 4.1.2 is void, since a C0-diffeomorphism j is just a homeomorphism, but ϕi is always a homeomor- phism.

In this case, M is called a topological manifold of di- mension n.

We do not require a manifold to be connected but we require all the components to have the same dimension, n.

Actually, on every connected component of M,itcanbe shown that the dimension, nϕ,oftherangeofeverychart is the same. This is quite easy to show if k 1butfor k =0,thisrequiresadeeptheoremofBrouwer.≥

What happens if n =0?Inthiscase,everyone-point subset of M is open, so every subset of M is open, i.e., M is any (countable if we assume M to be second-countable) set with the discrete topology!

Observe that since Rn is locally compact and locally con- nected, so is every manifold. 4.1. MANIFOLDS 241

Figure 4.1: A nodal cubic; not a manifold In order to get a better grasp of the notion of manifold it is useful to consider examples of non-manifolds.

First, consider the curve in R2 given by the zero locus of the equation y2 = x2 x3, − namely, the set of points M = (x, y) R2 y2 = x2 x3 . 1 { ∈ | − } This curve showed in Figure 4.1 and called a nodal cubic is also defined as the parametric curve x =1 t2 − y = t(1 t2). − 242 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES

We claim that M1 is not even a topological manifold. The problem is that the nodal cubic has a self-intersection at the origin.

If M1 was a topological manifold, then there would be a connected open subset, U M1,containingtheorigin, O =(0, 0), namely the intersection⊆ of a small enough open disc centered at O with M1,andalocalchart, ϕ: U Ω, where Ωis some connected open subset of R (that→ is, an open interval), since ϕ is a homeomorphism.

However, U O consists of four disconnected com- ponents and− Ω {ϕ}(O)oftwodisconnectedcomponents, contradicting the− fact that ϕ is a homeomorphism. 4.1. MANIFOLDS 243

Figure 4.2: A Cuspidal Cubic Let us now consider the curve in R2 given by the zero locus of the equation y2 = x3, namely, the set of points M = (x, y) R2 y2 = x3 . 2 { ∈ | }

This curve showed in Figure 4.2 and called a cuspidal cubic is also defined as the parametric curve x = t2 y = t3. 244 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Consider the map, ϕ: M R,givenby 2 → ϕ(x, y)=y1/3. 2/3 1 Since x = y on M2,weseethatϕ− is given by 1 2 3 ϕ− (t)=(t ,t ) and clearly, ϕ is a homeomorphism, so M2 is a topological manifold.

However, in the atlas consisting of the single chart, ϕ: M R ,thespaceM is also a smooth manifold! { 2 → } 2 Indeed, as there is a single chart, condition (3) of Defini- tion 4.1.2 holds vacuously.

This fact is somewhat unexpected because the cuspidal cubic is usually not considered smooth at the origin, since the tangent vector of the parametric curve, c: t (t2,t3), at the origin is the zero vector (the velocity vector-→ at t, 2 is c,(t)=(2t, 3t )). 4.1. MANIFOLDS 245 However, this apparent paradox has to do with the fact 2 that, as a parametric curve, M2 is not immersed in R since c, is not injective (see Definition 4.4.4 (a)), whereas as an abstract manifold, with this single chart, M2 is diffeomorphic to R.

Now, we also have the chart, ψ: M R,givenby 2 → ψ(x, y)=y, 1 with ψ− given by 1 2/3 ψ− (u)=(u ,u). Then, observe that 1 1/3 ϕ ψ− (u)=u , ◦ amapthatisnot differentiable at u =0.Therefore,the 1 atlas ϕ: M2 R,ψ: M2 R is not C and thus, with { → → } 1 respect to that atlas, M2 is not a C -manifold. 246 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES The example of the cuspidal cubic shows a peculiarity of k the definition of a C (or C∞)manifold:

If a space, M,happenstobeatopologicalmanifoldbe- cause it has an atlas consisting of a single chart, then it is automatically a smooth manifold!

In particular, if f: U Rm is any continuous from some open subset,→U,ofRn,toRm,thenthegraph, Γ(f) Rn+m,off given by ⊆ Γ(f)= (x, f(x)) Rn+m x U { ∈ | ∈ } is a smooth manifold with respect to the atlas consisting of the single chart, ϕ:Γ(f) U,givenby → ϕ(x, f(x)) = x, 1 with its inverse, ϕ− : U Γ(f), given by → 1 ϕ− (x)=(x, f(x)). 4.1. MANIFOLDS 247 The notion of a using the concept of “adapted chart” (see Definition 4.4.1 in Section 4.4) gives a more satisfactory treatment of Ck (or smooth) of Rn.

The example of the cuspidal cubic also shows clearly that whether a topological space is a Ck-manifold or a smooth manifold depends on the choice of atlas.

In some cases, M does not come with a topology in an ob- vious (or natural) way and a slight variation of Definition 4.1.2 is more convenient in such a situation: 248 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Definition 4.1.4 Given a set, M,givensomeinteger n 1andgivensomek such that k is either an integer k ≥ 1ork = ,aCk n-atlas (or n-atlas of class Ck), ,isafamilyofcharts,≥ ∞ (U ,ϕ) ,suchthat A { i i } (1) Each Ui is a subset of M and ϕi: Ui ϕi(Ui)isa onto an open subset, ϕ (U ) →Rn,foralli; i i ⊆ (2) The Ui cover M,i.e.,

M = Ui; i ! (3) Whenever Ui Uj = ,thesetϕi(Ui Uj)isopen n ∩ ( ∅ j ∩ i k in R and the transition map ϕi (and ϕj)isaC - diffeomorphism. 4.1. MANIFOLDS 249 Then, the notion of a chart being compatible with an atlas and of two atlases being compatible is just as before and we get a new definition of a manifold, analogous to Definition 4.1.2.

But, this time, we give M the topology in which the open sets are arbitrary unions of domains of charts (the Ui’s in amaximalatlas).

It is not difficult to verify that the axioms of a topology are verified and M is indeed a topological space with this topology.

It can also be shown that when M is equipped with the above topology, then the maps ϕi: Ui ϕi(Ui)arehome- omorphisms, so M is a manifold according→ to Definition 4.1.3.

We require M to be Hausdorffand second-countable with this topology.

Thus, we are back to the original notion of a manifold where it is assumed that M is already a topological space. 250 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES If the underlying topological space of a manifold is com- pact, then M has some finite atlas.

Also, if is some atlas for M and (U,ϕ)isachartin , for anyA (nonempty) open subset, V U,wegetachart,A (V,ϕ ! V ), and it is obvious that this⊆ chart is compatible with . A Thus, (V,ϕ ! V )isalsoachartforM.Thisobservation shows that if U is any open subset of a Ck-manifold, M,thenU is also a Ck-manifold whose charts are the restrictions of charts on M to U.

Interesting manifolds often occur as the result of a quo- tient construction.

For example, real projective spaces and Grassmannians are obtained this way. 4.1. MANIFOLDS 251 In this situation, the natural topology on the quotient object is the quotient topology but, unfortunately, even if the original space is Hausdorff, the quotient topology may not be.

Therefore, it is useful to have criteria that insure that a quotient topology is Hausdorff(or second-countable). We will present two criteria.

First, let us review the notion of quotient topology. 252 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Definition 4.1.5 Given any topological space, X,and any set, Y ,foranysurjectivefunction,f: X Y ,we define the quotient topology on Y determined→ by f (also called the identification topology on Y determined by 1 f), by requiring a subset, V ,ofY to be open if f − (V ) is an in X.

Given an equivalence relation R on a topological space X,ifπ: X X/R is the projection sending every x X to its equivalence→ class [x]inX/R,thespaceX/R∈ equipped with the quotient topology determined by π is called the quotient space of X modulo R.Thusaset, 1 V ,ofequivalenceclassesinX/R is open iff π− (V )is open in X,whichisequivalenttothefactthat [x] V [x] is open in X. ∈ #

It is immediately verified that Definition 4.1.5 defines topologies and that f: X Y and π: X X/R are continuous when Y and X/R→ are given these→ quotient topologies. 4.1. MANIFOLDS 253 One should be careful that if X and Y are topolog- ! ical spaces and f: X Y is a continuous surjective map, Y does not necessarily→ have the quotient topology determined by f.

Indeed, it may not be true that a subset V of Y is open 1 when f − (V )isopen.However,thiswillbetrueintwo important cases.

Definition 4.1.6 Acontinuousmap,f: X Y ,isan open map (or simply open)iff(U)isopeninY→whenever U is open in X,andsimilarly,f: X Y ,isaclosed map (or simply closed)iff(F )isclosedin→ Y whenever F is closed in X.

Then, Y has the quotient topology induced by the con- tinuous surjective map f if either f is open or f is closed. 254 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Unfortunately, the Hausdorffseparation property is not necessarily preserved under quotient. Nevertheless, it is preserved in some special important cases.

Proposition 4.1.7 Let X and Y be topological spaces, let f: X Y be a continuous surjective map, and as- sume that→X is compact and that Y has the quotient topology determined by f.ThenY is Hausdorffiff f is a closed map.

Another simple criterion uses continuous open maps.

Proposition 4.1.8 Let f: X Y be a surjective con- tinuous map between topological→ spaces. If f is an open map then Y is Hausdorffiffthe set (x ,x ) X X f(x )=f(x ) { 1 2 ∈ × | 1 2 } is closed in X X. ×

Note that the hypothesis of Proposition 4.1.8 implies that Y has the quotient topology determined by f. 4.1. MANIFOLDS 255 AspecialcaseofProposition4.1.8isdiscussedinTu.

Given a topological space, X,andanequivalencerelation, R,onX,wesaythatR is open if the projection map, π: X X/R,isanopenmap,whereX/R is equipped with the→ quotient topology.

Then, if R is an open equivalence relation on X,the topological quotient space X/R is Hausdorffiff R is closed in X X. ×

The following proposition yields a sufficient condition for second-countability:

Proposition 4.1.9 If X is a topological space and R is an open equivalence relation on X,thenforany basis, Bα ,forthetopologyofX,thefamily π(Bα) is a basis{ for} the topology of X/R,whereπ: X { X/R} is the projection map. Consequently, if X is→ second- countable, then so is X/R. 256 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Example 1. The Sn. Using the stereographic projections (from the north pole and the south pole), we can define two charts on Sn and show that Sn is a smooth manifold. Let n n n n σN : S N R and σS: S S R ,where N =(0−, {, 0}, 1)→ Rn+1 (the north−{ pole)}→ and S =(0, ···, 0, 1)∈ Rn+1 (the south pole) be the maps called respectively··· − stereographic∈ projection from the north pole and stereographic projection from the south pole given by 1 σ (x ,...,x )= (x ,...,x ) N 1 n+1 1 x 1 n − n+1 and 1 σS(x1,...,xn+1)= (x1,...,xn). 1+xn+1 4.1. MANIFOLDS 257 The inverse stereographic projections are given by 1 σN− (x1,...,xn)= 1 n (2x ,...,2x , ( x2) 1) ( n x2)+1 1 n i − i=1 i i=1 % and $ 1 σS− (x1,...,xn)= 1 n (2x ,...,2x , ( x2)+1). ( n x2)+1 1 n − i i=1 i i=1 % $ n n Thus, if we let UN = S N and US = S S , −{ } −{ }n we see that UN and US are two open subsets covering S , both homeomorphic to Rn. 258 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Furthermore, it is easily checked that on the overlap, U U = Sn N,S ,thetransitionmaps N ∩ S −{ } 1 1 σ σ− = σ σ− S ◦ N N ◦ S are given by 1 (x1,...,xn) n 2 (x1,...,xn), -→ i=1 xi that is, the inversion of center$ O =(0,...,0) and power 1. Clearly, this map is smooth on Rn O ,sowecon- −{ } clude that (UN ,σN )and(US,σS)formasmoothatlasfor Sn.

Example 2. The projective space RPn.

To define an atlas on RPn it is convenient to view RPn as the set of equivalence classes of vectors in Rn+1 0 modulo the equivalence relation, −{ } u v iff v = λu, for some λ =0 R. ∼ ( ∈ n Given any p =[x1,...,xn+1] RP ,wecall(x1,...,xn+1) the homogeneous coordinates∈ of p. 4.1. MANIFOLDS 259 It is customary to write (x : : x )insteadof 1 ··· n+1 [x1,...,xn+1]. (Actually, in most books, the indexing starts with 0, i.e., homogeneous coordinates for RPn are written as (x : : x ).) 0 ··· n Now, RPn can also be viewed as the quotient of the sphere, Sn,undertheequivalencerelationwhereanytwo antipodal points, x and x,areidentified. − It is not hard to show that the projection π: Sn RPn is both open and closed. →

Since Sn is compact and second-countable, we can ap- ply our previous results to prove that under the quotient topology, RPn is Hausdorff, second-countable, and com- pact. 260 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES We define charts in the following way. For any i,with 1 i n +1,let ≤ ≤ U = (x : : x ) RPn x =0 . i { 1 ··· n+1 ∈ | i ( } Observe that Ui is well defined, because if (y : : y )=(x : : x ), then there is some λ =0 1 ··· n+1 1 ··· n+1 ( so that yi = λzi,fori =1,...,n+1.

n We can define a homeomorphism, ϕi,ofUi onto R ,as follows:

x1 xi 1 xi+1 xn+1 ϕ (x : : x )= ,..., − , ,..., , i 1 ··· n+1 x x x x & i i i i ' where the ith component is omitted. Again, it is clear that this map is well defined since it only involves ratios. 4.1. MANIFOLDS 261 n n We can also define the maps, ψi,fromR to Ui RP , given by ⊆

ψi(x1,...,xn)=(x1: : xi 1:1:xi: : xn), ··· − ··· where the 1 goes in the ith slot, for i =1,...,n+1.

One easily checks that ϕi and ψi are mutual inverses, so the ϕi are homeomorphisms. On the overlap, Ui Uj, (where i = j), as x =0,wehave ∩ ( j ( 1 (ϕ ϕ− )(x ,...,x )= j ◦ i 1 n x1 xi 1 1 xi xj 1 xj+1 xn ,..., − , , ,..., − , ,..., . x x x x x x x & j j j j j j j ' (We assumed that i

Intuitively, the space RPn is obtained by gluing the open subsets Ui on their overlaps. Even for n =3,thisisnot easy to visualize! 262 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Example 3. The Grassmannian G(k,n).

Recall that G(n, k)isthesetofallk-dimensional linear subspaces of Rn,alsocalledk-planes.

Every k-plane, W ,isthelinearspanofk linearly indepen- n dent vectors, u1,...,uk,inR ;furthermore,u1,...,uk and v ,...,v both span W iffthere is an invertible k k- 1 k × matrix, Λ= (λij), such that k v = λ u , 1 j k. j ij i ≤ ≤ i=1 % Obviously, there is a bijection between the collection of n k linearly independent vectors, u1,...,uk,inR and the collection of n k matrices of rank k. × Furthermore, two n k matrices A and B of rank k represent the same k-plane× iff B = AΛ, for some invertible k k matrix, Λ. × 4.1. MANIFOLDS 263 (Note the analogy with projective spaces where two vec- tors u, v represent the same point iff v = λu for some invertible λ R.) ∈ n k The set of n k matrices of rank k is a subset of R × , in fact, an open× subset.

One can show that the equivalence relation on n k matrices of rank k given by × B = AΛ, for some invertible k k matrix, Λ, × is open and that the graph of this equivalence relation is closed.

By Proposition 4.1.8, the Grassmannian G(k,n)isHaus- dorffand second-countable. 264 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES We can define the domain of charts (according to Def- inition 4.1.2) on G(k,n)asfollows:Foreverysubset, S = i1,...,ik of 1,...,n ,letUS be the subset of n k{ matrices,} A,ofrank{ k} whose rows of index in S ×= i ,...,i form an invertible k k matrix denoted { 1 k} × AS.

Observe that the k k matrix consisting of the rows of 1 × the matrix AAS− whose index belong to S is the identity matrix, Ik.

(n k) k Therefore, we can define a map, ϕ : U R − × , S S → where ϕS(A)=the(n k) k matrix obtained by delet- − × 1 ing the rows of index in S from AAS− . 4.1. MANIFOLDS 265 We need to check that this map is well defined, i.e., that it does not depend on the matrix, A,representingW .

Let us do this in the case where S = 1,...,k ,which is notationally simpler. The general case{ can be} reduced to this one using a suitable permutation.

If B = AΛ, with Λinvertible, if we write A B A = 1 and B = 1 , A B & 2 ' & 2 ' as B = AΛ, we get B1 = A1ΛandB2 = A2Λ, from which we deduce that

B1 1 Ik B1− = 1 = B B B− & 2 ' & 2 1 ' Ik Ik A1 1 1 1 = 1 = A1− . A ΛΛ A− A A− A & 2 − 1 ' & 2 1 ' & 2 '

Therefore, our map is indeed well-defined. 266 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES

It is clearly injective and we can define its inverse, ψS,as follows: Let πS be the permutation of 1,...,n swaping 1,...,k and S and leaving every other{ element} fixed { } (i.e., if S = i1,...,ik ,thenπS(j)=ij and πS(ij)=j, for j =1,...,k{ ). }

If PS is the permutation matrix associated with πS,for any (n k) k matrix, M,let − × I ψ (M)=P k . S S M & ' The effect of ψS is to “insert into M”therowsofthe identity matrix Ik as the rows of index from S.

At this stage, we have charts that are from (n k) k subsets, US,ofG(k,n)toopensubsets,namely,R − × . 4.1. MANIFOLDS 267 Then, the reader can check that the transition map 1 ϕ ϕ− from ϕ (U U )toϕ (U U )isgivenby T ◦ S S S ∩ T T S ∩ T 1 M (C + DM)(A + BM)− , -→ where AB = P P , CD T S & ' is the matrix of the permutation πT πS (this permutation “shuffles” S and T ). ◦

This map is smooth, as it is given by determinants, and so, the charts (US,ϕS)formasmoothatlasforG(k,n).

Finally, it is easy to check that the conditions of Definition 4.1.2 are satisfied, so the atlas just defined makes G(k,n) into a topological space and a smooth manifold. 268 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES The Grassmannian G(k,n)isactuallycompact.Tosee this, observe that if W is any k-plane, then using the Gram-Schmidt orthonormalization procedure, every basis B =(b1,...,bk)forW yields an orthonormal basis U = (u1,...,uk)andthereisaninvertiblematrix,Λ,such that U = BΛ, where the the columns of B are the bjsandthecolumns of U are the ujs.

The matrices U have orthonormal columns and are char- acterized by the equation

U 0U = Ik.

Consequently, the space of such matrices is closed an n k clearly bounded in R × and thus, compact. 4.1. MANIFOLDS 269 The Grassmannian G(k,n)isthequotientofthisspace under our usual equivalence relation and G(k,n)isthe image of a compact set under the projection map, which is clearly continuous, so G(k,n)iscompact.

Remark: The reader should have no difficulty proving that the collection of k-planes represented by matrices in US is precisely the set of k-planes, W ,supplementaryto the (n k)-plane spanned by the n k canonical basis − − vectors ejk+1,...,ejn (i.e., span(W ejk+1,...,ejn )= n ∪{ } R ,whereS = i1,...,ik and j ,...,j ={ 1,...,n} S). { k+1 n} { }− 270 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Example 4. Product Manifolds.

k Let M1 and M2 be two C -manifolds of dimension n1 and n2,respectively.

The topological space, M1 M2,withtheproducttopol- ogy (the opens of M M ×are arbitrary unions of sets of 1 × 2 the form U V ,whereU is open in M1 and V is open × k in M2)canbegiventhestructureofaC -manifold of dimension n1 + n2 by defining charts as follows:

For any two charts, (Ui,ϕi)onM1 and (Vj,ψj)onM2, we declare that (U V ,ϕ ψ )isachartonM M , i × j i × j 1 × 2 where ϕ ψ : U V Rn1+n2 is defined so that i × j i × j → ϕ ψ (p, q)=(ϕ (p),ψ (q)), for all (p, q) U V . i × j i j ∈ i × j

We define Ck-maps between manifolds as follows: 4.1. MANIFOLDS 271 Definition 4.1.10 Given any two Ck-manifolds, M and N,ofdimensionm and n respectively, a Ck-map if a con- tinuous functions, h: M N,sothatforevery p M,thereissomechart,(→ U,ϕ), at p and some chart, (V,∈ψ), at q = h(p), with f(U) V and ⊆ 1 ψ h ϕ− : ϕ(U) ψ(V ) ◦ ◦ −→ a Ck-function.

It is easily shown that Definition 4.1.10 does not depend on the choice of charts. In particular, if N = R,weobtain a Ck-function on M.

One checks immediately that a function, f: M R,isa Ck-map ifffor every p M,thereissomechart,(→ U,ϕ), at p so that ∈ 1 f ϕ− : ϕ(U) R ◦ −→ is a Ck-function. 272 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES If U is an open subset of M,setofCk-functions on U is denoted by k(U). In particular, k(M)denotestheset of Ck-functionsC on the manifold, MC.Observethat k(U) is a ring. C

On the other hand, if M is an open interval of R,say M =]a, b[,thenγ:]a, b[ N is called a Ck-curve in N. One checks immediately→ that a function, γ:]a, b[ N,is a Ck-map ifffor every q N,thereissomechart,(→V,ψ), at q so that ∈ ψ γ:]a, b[ ψ(V ) ◦ −→ is a Ck-function.

It is clear that the composition of Ck-maps is a Ck-map. A Ck-map, h: M N,betweentwomanifoldsisaCk- → 1 diffeomorphism iff h has an inverse, h− : N M (i.e., 1 1 → 1 h− h =idM and h h− =idN ), and both h and h− are k ◦ ◦ 1 C -maps (in particular, h and h− are homeomorphisms). Next, we define tangent vectors. 4.2. TANGENT VECTORS, TANGENT SPACES, COTANGENT SPACES 273 4.2 Tangent Vectors, Tangent Spaces, Cotangent Spaces

Let M be a Ck manifold of dimension n,withk 1. The most intuitive method to define tangent vectors≥ is to use curves. Let p M be any point on M and let γ:] (,([ M be∈ a C1-curve passing through p,that is, with− γ(0)→ = p.Unfortunately,ifM is not embed- N ded in any R ,thederivativeγ,(0) does not make sense. However, for any chart, (U,ϕ), at p,themapϕ γ is a 1 n ◦ C -curve in R and the tangent vector v =(ϕ γ),(0) is well defined. The trouble is that different curves◦ may yield the same v!

To remedy this problem, we define an equivalence relation on curves through p as follows: 274 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Definition 4.2.1 Given a Ck manifold, M,ofdimen- sion n,foranyp M,twoC1-curves, γ :] ( ,( [ M ∈ 1 − 1 1 → and γ2:] (2,(2[ M,throughp (i.e., γ1(0) = γ2(0) = p) are equivalent− iff→there is some chart, (U,ϕ), at p so that

(ϕ γ ),(0) = (ϕ γ ),(0). ◦ 1 ◦ 2

Now, the problem is that this definition seems to depend on the choice of the chart. Fortunately, this is not the case.

This leads us to the first definition of a tangent vector.

Definition 4.2.2 (Tangent Vectors, Version 1) Given any Ck-manifold, M,ofdimensionn,withk 1, for any p M,atangent vector to M at p is any≥ equivalence class∈ of C1-curves through p on M,modulotheequiva- lence relation defined in Definition 4.2.1. The set of all tangent vectors at p is denoted by Tp(M). 4.2. TANGENT VECTORS, TANGENT SPACES, COTANGENT SPACES 275

It is obvious that Tp(M)isavectorspace.

Observe that the map that sends a curve, γ:] (,([ M,throughp (with γ(0) = p)toitstangent − → n vector, (ϕ γ),(0) R (for any chart (U,ϕ), at p), induces a map,◦ ϕ: T ∈(M) Rn. p → It is easy to check that ϕ is a linear bijection (by definition of the equivalence relation on curves through p).

This shows that Tp(M)isavectorspaceofdimension n =dimensionofM. 276 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES One should observe that unless M = Rn,inwhichcase, for any p, q Rn,thetangentspaceT (M)isnaturally ∈ q isomorphic to the Tp(M)bythetranslation q p,foranarbitrarymanifold,thereisnorelationship between− T (M)andT (M)whenp = q. p q ( One of the defects of the above definition of a tangent vector is that it has no clear relation to the Ck-differential structure of M.

In particular, the definition does not seem to have any- thing to do with the functions defined locally at p. 4.2. TANGENT VECTORS, TANGENT SPACES, COTANGENT SPACES 277 There is another way to define tangent vectors that re- veals this more clearly. Moreover, such a def- inition is more intrinsic, i.e., does not refer explicitly to charts.

As a first step, consider the following: Let (U,ϕ)bea chart at p M (where M is a Ck-manifold of dimen- ∈ sion n,withk 1) and let xi = pri ϕ,theith local coordinate (1 ≥i n). ◦ ≤ ≤ For any function, f,definedonU p,set 2 1 ∂ ∂(f ϕ− ) f = ◦ , 1 i n. ∂x ∂X ≤ ≤ & i'p i (ϕ(p) ( (Here, (∂g/∂X ) denotes the( partial of a func- i y ( tion g: Rn R with| respect to the ith coordinate, eval- uated at y.)→ 278 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES We would expect that the function that maps f to the above value is a on the set of functions defined locally at p,butthereistechnicaldifficulty:

The set of functions defined locally at p is not avector space!

To see this, observe that if f is defined on an open U p and g is defined on a different open V p,thenwe2do not know how to define f + g. 2

The problem is that we need to identify functions that agree on a smaller open subset. This leads to the notion of germs. 4.2. TANGENT VECTORS, TANGENT SPACES, COTANGENT SPACES 279 Definition 4.2.3 Given any Ck-manifold, M,ofdimen- sion n,withk 1, for any p M,alocally defined function at p is≥ a pair, (U, f), where∈ U is an open sub- set of M containing p and f is a function defined on U. Two locally defined functions, (U, f)and(V,g), at p are equivalent iffthere is some open subset, W U V , containing p so that ⊆ ∩ f ! W = g ! W. The equivalence class of a locally defined function at p, denoted [f]orf,iscalledagerm at p.

One should check that the relation of Definition 4.2.3 is indeed an equivalence relation.

Of course, the value at p of all the functions, f,inany , f,isf(p). Thus, we set f(p)=f(p). 280 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES One should also check that we can define addition of germs, multiplication of a germ by a scalar and multi- plication of germs, in the obvious way:

If f and g are two germs at p,andλ R,then ∈ [f]+[g]=[f + g] λ[f]=[λf] [f][g]=[fg]. (Of course, f + g is the function locally defined so that (f + g)(x)=f(x)+g(x)andsimilarly,(λf)(x)=λf(x) and (fg)(x)=f(x)g(x).)

Therefore, the germs at p form a ring.

The ring of germs of Ck-functions at p is denoted (k) M,p.Whenk = ,weusuallydropthesuperscript O . ∞ ∞ Remark: Most readers will most likely be puzzled by the notation (k) . OM,p In fact, it is standard in , but it is not as commonly used in differential geometry. 4.2. TANGENT VECTORS, TANGENT SPACES, COTANGENT SPACES 281 For any open subset, U,ofamanifold,M,thering, k k (k) (U), of C -functions on U is also denoted M (U)(cer- Ctainly by people with an algebraic geometryO bent!).

Then, it turns out that the map U (k)(U)isasheaf , -→ OM denoted (k),andthering (k) is the of the sheaf OM OM,p (k) at p. OM Such rings are called local rings.Roughlyspeaking,all the “local” information about M at p is contained in the local ring (k) .(Thisistobetakenwithagrainof OM,p salt. In the Ck-case where k< ,wealsoneedthe “stationary germs,” as we will see shortly.)∞

Now that we have a rigorous way of dealing with functions locally defined at p,observethatthemap ∂ v : f f i -→ ∂x & i'p yields the same value for all functions f in a germ f at p. 282 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES

(k) Furthermore, the above map is linear on M,p.Moreis true. O

(1) For any two functions f,g locally defined at p,we have ∂ ∂ ∂ (fg)=f(p) g + g(p) f. ∂x ∂x ∂x & i'p & i'p & i'p 1 (2) If (f ϕ− ),(ϕ(p)) = 0, then ◦ ∂ f =0. ∂x & i'p

The first property says that vi is a point-derivation .

1 As to the second property, when (f ϕ− ),(ϕ(p)) = 0, we say that f is stationary at p. ◦ 4.2. TANGENT VECTORS, TANGENT SPACES, COTANGENT SPACES 283 It is easy to check (using the chain rule) that being sta- tionary at p does not depend on the chart, (U,ϕ), at p or on the function chosen in a germ, f.Therefore,the notion of a stationary germ makes sense:

Definition 4.2.4 We say that a germ f at p M is a 1 ∈ stationary germ iff(f ϕ− ),(ϕ(p)) = 0 for some chart, (U,ϕ), at p and some function,◦ f,inthegerm,f.The Ck-stationary germs form a subring of (k) (but not an OM,p ideal!) denoted (k) . SM,p

Remarkably, it turns out that the dual of the , (k) / (k) ,isisomorphictothetangentspace,T (M). OM,p SM,p p First, we prove that the subspace of linear forms on (k) OM,p (k) ∂ ∂ that vanish on M,p has ,..., as a basis. S ∂x1 p ∂xn p ) * ) * 284 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Proposition 4.2.5 Given any Ck-manifold, M,ofdi- mension n,withk 1,foranyp M,then func- ∂ ≥∂ ∈ (k) tions, ,..., ,definedon M,p by ∂x1 p ∂xn p O ) * ) * 1 ∂ ∂(f ϕ− ) f = ◦ , 1 i n ∂x ∂X ≤ ≤ & i'p i (ϕ(p) ( ( (k) are linear forms that vanish( on M,p.Everylinear (k) S (k) form, L,on M,p that vanishes on M,p can be ex- pressed in a uniqueO way as S n ∂ L = λ , i ∂x i=1 i p % & ' where λ R.Therefore,the i ∈ ∂ ,i=1,...,n ∂x & i'p form a basis of the vector space of linear forms on (k) that vanish on (k) . OM,p SM,p 4.2. TANGENT VECTORS, TANGENT SPACES, COTANGENT SPACES 285

(k) As the subspace of linear forms on M,p that vanish on (k) (k) O (k) is isomorphic to the dual, ( / )∗,ofthespace SM,p OM,p SM,p (k) / (k) ,weseethatthe OM,p SM,p ∂ ,i=1,...,n ∂x & i'p (k) (k) also form a basis of ( / )∗. OM,p SM,p To define our second version of tangent vectors, we need to define point-derivations.

Definition 4.2.6 Given any Ck-manifold, M,ofdimen- sion n,withk 1, for any p M,aderivation at p in ≥ ∈(k) M or point-derivation on is a , v,on OM,p (k) ,suchthat OM,p v(fg)=v(f)g(p)+f(p)v(g), (k) for all germs f, g M,p.TheaboveiscalledtheLeibniz property. ∈O 286 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES As expected, point-derivations vanish on constant func- tions.

(k) Proposition 4.2.7 Every point-derivation, v,on M,p, vanishes on germs of constant functions. O

Recall that we observed earlier that the ∂ are point- ∂xi p derivations at p.Therefore,wehave ) *

Proposition 4.2.8 Given any Ck-manifold, M,ofdi- mension n,withk 1,foranyp M,thelinear (k) ≥ (k) ∈ forms on that vanish on are exactly the OM,p SM,p point-derivations on (k) that vanish on (k) . OM,p SM,p 4.2. TANGENT VECTORS, TANGENT SPACES, COTANGENT SPACES 287 Remarks:

(k) (1) If we let p (M)denotethesetofpoint-derivations (k) D on M,p,thenProposition4.2.8saysthatanylin- O (k) (k) ear form on M,p that vanishes on M,p belongs to (k) O S p (M), so we have the inclusion D (k) (k) (k) ( / )∗ (M). OM,p SM,p ⊆Dp

However, in general, when k = ,apoint-derivation (k) ( ∞ (k) on does not necessarily vanish on . OM,p SM,p We will see in Proposition 4.2.15 that this is true for k = . ∞ (2) In the case of smooth manifolds (k = )some authors, including Morita and O’Neil,∞ define point- derivations as linear derivations with domain ∞(M), the set of all smooth funtions on the entire manifold,C M. 288 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES This definition is simpler in the sense that it does not require the definition of the notion of germ but it is not local, because it is not obvious that if v is apoint-derivationatp,thenv(f)=v(g)whenever f,g ∞(M)agreelocallyatp. ∈C In fact, if two smooth locally defined functions agree near p it may not be possible to extend both of them to the whole of M.

However, it can proved that this property is local be- cause on smooth manifolds, “bump functions” exist (see Section 4.6, Proposition 4.6.2).

Unfortunately, this argument breaks down for Ck- manifolds with k< and in this case the ring of germs at p can’t be avoided.∞ 4.2. TANGENT VECTORS, TANGENT SPACES, COTANGENT SPACES 289 Here is now our second definition of a tangent vector.

Definition 4.2.9 (Tangent Vectors, Version 2) Given any Ck-manifold, M,ofdimensionn,withk 1, for any p M,atangent vector to M at p is any≥ point- ∈ (k) (k) derivation on M,p that vanishes on M,p,thesubspace of stationary germs.O S

Let us consider the simple case where M = R.Inthis case, for every x R,thetangentspace,Tx(R), is a one- dimensional vector∈ space isomorphic to R and ∂ d R ∂t x = dt x is a basis vector of Tx( ). + , ( For every C( k-function, f,locallydefinedatx,wehave ∂ df f = = f ,(x). ∂t dt & 'x (x ( Thus, ∂ is: compute the derivative( of a function at x. ∂t x ( + , 290 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES We now prove the equivalence of the two Definitions of a tangent vector.

Proposition 4.2.10 Let M be any Ck-manifold of dimension n,withk 1.Foranyp M,letu be any tangent vector (version≥ 1) given by∈ some equivalence class of C1-curves, γ:] (, +([ M,throughp (i.e., − → p = γ(0)). Then, the map L defined on (k) by u OM,p L (f)=(f γ),(0) u ◦ is a point-derivation that vanishes on (k) .Further- SM,p more, the map u Lu defined above is an isomor- -→ (k) (k) phism between T (M) and ( / )∗,thespaceof p OM,p SM,p linear forms on (k) that vanish on (k) . OM,p SM,p 4.2. TANGENT VECTORS, TANGENT SPACES, COTANGENT SPACES 291 There is a conceptually clearer way to define a canonical between T (M)andthedualof (k) / (k) p OM,p SM,p in terms of a nondegenerate pairing between Tp(M)and (k) / (k) OM,p SM,p This pairing is described by Serre in [?](ChapterIII, Section 8) for analytic manifolds and can be adapted to our situation.

Define the map, ω: T (M) (k) R,sothat p ×OM,p → ω([γ], f)=(f γ),(0), ◦ for all [γ] T (M)andallf (k) (with f f). ∈ p ∈OM,p ∈ It is easy to check that the above expression does not depend on the representatives chosen in the equivalences classes [γ]andf and that ω is bilinear.

However, as defined, ω is degenerate because ω([γ], f)=0 if f is a stationary germ. 292 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Thus, we are led to consider the pairing with domain T (M) ( (k) / (k) )givenby p × OM,p SM,p ω([γ], [f]) = (f γ),(0), ◦ (k) (k) where [γ] Tp(M)and[f] M,p/ M,p,whichwealso ∈ (k) ∈(k)O S denote ω: Tp(M) ( M,p/ M,p) R.Then,thefollow- ing result holds: × O S →

Proposition 4.2.11 The map ω: T (M) ( (k) / (k) ) R defined so that p × OM,p SM,p → ω([γ], [f]) = (f γ),(0), ◦ (k) (k) for all [γ] Tp(M) and all [f] M,p/ M,p,isa nondegenerate∈ pairing (with f ∈f).O Consequently,S ∈ there is a canonical isomorphism between Tp(M) and (k) (k) ( M,p/ M,p)∗ and a canonical isomorphism between O S (k) (k) T ∗(M) and / . p OM,p SM,p 4.2. TANGENT VECTORS, TANGENT SPACES, COTANGENT SPACES 293

In view of Proposition 4.2.11, we can identify Tp(M)with (k) (k) (k) (k) ( / )∗ and T ∗(M)with / . OM,p SM,p p OM,p SM,p Remark: Also recall that if E is a finite dimensional space, the map iE: E E∗∗ defined so that, for any v E, → ∈ v v, where v(f)=f(v), for all f E∗ -→ ∈ is a linear isomorphism. " " Observe that we can view ω(u, f)=ω([γ], [f]) as the re- sult of computing the of the locally defined function f f in the direction u (given by a curve γ). Proposition∈ 4.2.11 also suggests the following definition: 294 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Definition 4.2.12 Given any Ck-manifold, M,ofdi- mension n,withk 1, for any p M,thetangent space ≥ ∈ at p,denotedTp(M), is the space of point-derivations on (k) (k) M,p that vanish on M,p.Thus,Tp(M)canbeidenti- O (k) (k) S (k) (k) fied with ( M,p/ M,p)∗.Thespace M,p/ M,p is called the cotangentO spaceS at p;itisisomorphictothedual,O S Tp∗(M), of Tp(M).

Even though this is just a restatement of Proposition 4.2.5, we state the following proposition because of its practical usefulness: 4.2. TANGENT VECTORS, TANGENT SPACES, COTANGENT SPACES 295 Proposition 4.2.13 Given any Ck-manifold, M,of dimension n,withk 1,foranyp M and any chart (U,ϕ) at p,then≥tangent vectors,∈ ∂ ∂ ,..., , ∂x ∂x & 1'p & n'p form a basis of TpM.

Observe that if x = pr ϕ,as i i ◦ ∂ x = δ , ∂x j i,j & i'p the images of x ,...,x in (k) / (k) constitute the dual 1 n OM,p SM,p ∂ ∂ basis of the basis ,..., of Tp(M). ∂x1 p ∂xn p ) * ) * 296 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Given any Ck-function, f,onM,wedenotetheimageof (k) (k) f in T ∗(M)= / by df . p OM,p SM,p p This is the differential of f at p.

(k) (k) Using the isomorphism between M,p/ M,p and (k) (k) O S ( / )∗∗ described above, df corresponds to the OM,p SM,p p linear map in Tp∗(M)definedbydf p(v)=v(f), for all v T (M). ∈ p

With this notation, we see that (dx1)p,...,(dxn)p is a basis of Tp∗(M), and this basis is dual to the basis ∂ ∂ ,..., of Tp(M). ∂x1 p ∂xn p ) * ) * 4.2. TANGENT VECTORS, TANGENT SPACES, COTANGENT SPACES 297 For simplicity of notation, we often omit the subscript p unless confusion arises. Remark: Strictly speaking, a tangent vector, v T (M), ∈ p is defined on the space of germs, (k) ,atp.However,itis OM,p often convenient to define v on Ck-functions, f k(U), where U is some open subset containing p.Thisiseasy:∈C Set v(f)=v(f). Given any chart, (U,ϕ), at p,sincev can be written in a unique way as n ∂ v = λ , i ∂x i=1 i p % & ' we get n ∂ v(f)= λ f. i ∂x i=1 i p % & ' This shows that v(f)isthedirectional derivative of f in the direction v. 298 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES When M is a smooth manifold, things get a little sim- pler. Indeed, it turns out that in this case, every point- derivation vanishes on stationary germs.

To prove this, we recall the following result from calculus (see Warner [?]):

Proposition 4.2.14 If g: Rn R is a Ck-function (k 2)onaconvexopen,U,about→ p Rn,thenfor every≥ q U,wehave ∈ ∈ n ∂g g(q)=g(p)+ (q p ) ∂X i − i i=1 i(p n % ( ( 1 ∂2g + (q p )(q p() (1 t) dt. i − i j − j − ∂X ∂X i,j=1 -0 i j ((1 t)p+tq % ( − ( In particular, if g C∞(U),thentheintegralasa ∈ ( function of q is C∞. 4.2. TANGENT VECTORS, TANGENT SPACES, COTANGENT SPACES 299

Proposition 4.2.15 Let M be any C∞-manifold of dimension n.Foranyp M,anypoint-derivationon ( ) ( ) ∈ M,p∞ vanishes on M,p∞ ,theringofstationarygerms. O S ( ) Consequently, T (M)= p∞ (M). p D

Proposition 4.2.15 shows that in the case of a smooth manifold, in Definition 4.2.9, we can omit the requirement that point-derivations vanish on stationary germs, since this is automatic.

( ) It is also possible to define T (M)justintermsof ∞ . p OM,p ( ) Let mM,p M,p∞ be the ideal of germs that vanish at ⊆O 2 p.Then,wealsohavetheidealmM,p,whichconsistsof all finite linear combinations of products of two elements in mM,p,anditturnsoutthatTp∗(M)isisomorphicto 2 mM,p/mM,p (see Warner [?], Lemma 1.16). 300 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES

Actually, if we let m(k) (k) denote the ideal of M,p ⊆OM,p Ck-germs that vanish at p and s(k) (k) denote the M,p ⊆SM,p ideal of stationary Ck-germs that vanish at p,adapting Warner’s argument, we can prove the following proposi- tion:

Proposition 4.2.16 We have the inclusion, (m(k) )2 s(k) and the isomorphism M,p ⊆ M,p (k) (k) (k) (k) ( / )∗ = (m /s )∗. OM,p SM,p ∼ M,p M,p (k) (k) As a consequence, Tp(M) ∼= (mM,p/sM,p)∗ and (k) (k) Tp∗(M) ∼= mM,p/sM,p.

When k = ,Proposition4.2.14showsthateverysta- ∞ 2 tionary germ that vanishes at p belongs to mM,p.

( ) 2 Therefore, when k = ,wehavesM,p∞ = mM,p and so, we obtain the result quoted∞ above (from Warner):

( ) ( ) 2 T ∗(M)= ∞ / ∞ = m /m . p OM,p SM,p ∼ M,p M,p 4.2. TANGENT VECTORS, TANGENT SPACES, COTANGENT SPACES 301 Remarks:

(1) The isomorphism (k) (k) (k) (k) ( / )∗ = (m /s )∗ OM,p SM,p ∼ M,p M,p yields another proof that the linear forms in (k) (k) ( M,p/ M,p)∗ are point-derivations, using the argu- mentO fromS Warner [?] (Lemma 1.16). (k) (2) The ideal mM,p is in fact the unique maximal ideal (k) (k) of M,p.Thisisbecauseiff M,p does not vanish O ∈O (k) at p,thenitisaninvertibleelementof M,p and any (k) O (k) ideal containing mM,p and f would be equal to M,p, which it absurd. O

Thus, (k) is a local ring (in the sense of commuta- OM,p tive algebra) called the local ring of germs of Ck- functions at p.Theseringsplayacrucialrolein algebraic geometry. 302 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Yet one more way of defining tangent vectors will make it a little easier to define tangent bundles.

Definition 4.2.17 (Tangent Vectors, Version 3) Given any Ck-manifold, M,ofdimensionn,withk 1, for any p M,considerthetriples,(U,ϕ, u), where (≥U,ϕ)isany chart∈ at p and u is any vector in Rn.Saythattwosuch triples (U,ϕ, u)and(V,ψ,v)areequivalent iff 1 (ψ ϕ− ), (u)=v. ◦ ϕ(p) A tangent vector to M at p is an equivalence class of triples, [(U,ϕ, u)], for the above equivalence relation.

The intuition behind Definition 4.2.17 is quite clear: The vector u is considered as a tangent vector to Rn at ϕ(p). 4.2. TANGENT VECTORS, TANGENT SPACES, COTANGENT SPACES 303 If (U,ϕ)isachartonM at p,wecandefineanaturaliso- n n morphism, θU,ϕ,p: R Tp(M), between R and Tp(M), as follows: For any u→ Rn, ∈ θ : u [(U,ϕ, u)]. U,ϕ,p -→ One immediately check that the above map is indeed lin- ear and a bijection.

The equivalence of this definition with the definition in terms of curves (Definition 4.2.2) is easy to prove. 304 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Proposition 4.2.18 Let M be any Ck-manifold of dimension n,withk 1.Foreveryp M,forevery chart, (U,ϕ),atp,if≥ x is any tangent∈ vector (ver- sion 1) given by some equivalence class of C1-curves, γ:] (, +([ M,throughp (i.e., p = γ(0)), then the map− → x [(U,ϕ, (ϕ γ),(0))] -→ ◦ is an isomorphism between Tp(M)-version 1 and Tp(M)- version 3.

For simplicity of notation, we also use the notation TpM for Tp(M)(resp.Tp∗M for Tp∗(M)).

After having explored thorougly the notion of tangent vector, we show how a Ck-map, h: M N,betweenCk → manifolds, induces a linear map, dhp: Tp(M) Th(p)(N), for every p M. → ∈ 4.2. TANGENT VECTORS, TANGENT SPACES, COTANGENT SPACES 305 We find it convenient to use Version 2 of the definition of atangentvector.So,letu T (M)beapoint-derivation ∈ p on (k) that vanishes on (k) . OM,p SM,p We would like dh (u)tobeapoint-derivationon (k) p ON,h(p) that vanishes on (k) . SN,h(p)

Now, for every germ, g (k) ,ifg g is any locally ∈ON,h(p) ∈ defined function at h(p), it is clear that g h is locally k ◦ defined at p and is C and that if g1,g2 g then g1 h and g h are equivalent. ∈ ◦ 2 ◦ The germ of all locally defined functions at p of the form g h,withg g,willbedenotedg h. ◦ ∈ ◦ Then, we set dh (u)(g)=u(g h). p ◦ 306 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Moreover, if g is a stationary germ at h(p), then for some chart, (V,ψ)onN at q = h(p), we have 1 (g ψ− ),(ψ(q)) = 0 and, for some chart (U,ϕ)atp on M,weget◦ 1 1 1 (g h ϕ− ),(ϕ(p)) = (g ψ− )(ψ(q))((ψ h ϕ− ),(ϕ(p))) ◦ ◦ ◦ ◦ ◦ =0, which means that g h is stationary at p. ◦

Therefore, dhp(u) Th(p)(M). It is also clear that dhp is a linear map. We∈ summarize all this in the following definition: 4.2. TANGENT VECTORS, TANGENT SPACES, COTANGENT SPACES 307 Definition 4.2.19 Given any two Ck-manifolds, M and N,ofdimensionm and n,respectively,foranyCk-map, h: M N,andforeveryp M,thedifferential of → ∈ h at p or tangent map, dhp: Tp(M) Th(p)(N), is the linear map defined so that → dh (u)(g)=u(g h), p ◦ for every u T (M)andeverygerm,g (k) .The ∈ p ∈ON,h(p) linear map dhp is also denoted Tph (and sometimes, hp, or Dph).

The chain rule is easily generalized to manifolds.

Proposition 4.2.20 Given any two Ck-maps f: M N and g: N P between smooth Ck-manifolds, for any→p M,wehave→ ∈ d(g f) = dg df . ◦ p f(p) ◦ p 308 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES In the special case where N = R,aCk-map between the manifolds M and R is just a Ck-function on M.

It is interesting to see what df p is explicitly. Since N = R, germs (of functions on R)att0 = f(p)arejustgermsof Ck-functions, g: R R,locallydefinedatt . → 0 Then, for any u T (M)andeverygermg at t , ∈ p 0 df (u)(g)=u(g f). p ◦ If we pick a chart, (U,ϕ), on M at p,weknowthatthe ∂ form a basis of Tp(M), with 1 i n. ∂xi p ≤ ≤ ) * Therefore, it is enough to figure out what df p(u)(g)is when u = ∂ . ∂xi p ) * 4.2. TANGENT VECTORS, TANGENT SPACES, COTANGENT SPACES 309 In this case,

1 ∂ ∂(g f ϕ− ) df (g)= ◦ ◦ . p ∂x ∂X .& i'p/ i (ϕ(p) ( ( Using the chain rule, we find that ( ∂ ∂ dg df p (g)= f . ∂xi ∂xi dt .& 'p/ & 'p (t0 ( ( Therefore, we have ( d df (u)=u(f) . p dt (t0 ( This shows that we can identify df ( with the linear form p( in Tp∗(M)definedby df (u)=u(f),uT M, p ∈ p by identifying Tt0R with R. 310 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES

This is consistent with our previous definition of df p as (k) (k) the image of f in Tp∗(M)= M,p/ M,p (as Tp(M)is (k) (k) O S isomorphic to ( / )∗). OM,p SM,p Again, even though this is just a restatement of facts we already showed, we state the following proposition be- cause of its practical usefulness: Proposition 4.2.21 Given any Ck-manifold, M,of dimension n,withk 1,foranyp M and any chart (U,ϕ) at p,then≥linear maps, ∈

(dx1)p,...,(dxn)p, form a basis of Tp∗M,where(dxi)p,thedifferential of xi at p,isidentifiedwiththelinearforminTp∗M such that (dx ) (v)=v(xi),foreveryv T M (by i p ∈ p identifying TλR with R). 4.2. TANGENT VECTORS, TANGENT SPACES, COTANGENT SPACES 311 In preparation for the definition of the flow of a vector field (which will be needed to define the exponential map in theory), we need to define the tangent vector to a curve on a manifold.

Given a Ck-curve, γ:]a, b[ M,onaCk-manifold, M, for any t ]a, b[, wewouldliketodefinethetangent→ 0 ∈ vector to the curve γ at t0 as a tangent vector to M at p = γ(t0).

We do this as follows: Recall that d is a basis vector dt t0 of T (R)=R. t0 ( ( So, define the tangent vector to the curve γ at t0,de- dγ notedγ ˙ (t0)(orγ,(t0), or dt (t0)) by d γ˙ (t )=dγ . 0 t0 dt . (t0/ ( ( ( 312 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Sometime, it is necessary to define curves (in a manifold) whose domain is not an open interval.

Amap,γ:[a, b] M,isaCk-curve in M if it is the restriction of some→ Ck-curve, γ:]a (, b + ([ M,for some (small) (>0. − → " Note that for such a curve (if k 1) the tangent vector, γ˙ (t), is defined for all t [a, b]. ≥ ∈ Acontinuouscurve,γ:[a, b] M,ispiecewise Ck iff → there a sequence, a0 = a, a1,...,am = b,sothatthe k restriction, γi,ofγ to each [ai,ai+1]isaC -curve, for i =0,...,m 1. −

This implies that γi,(ai+1)andγi,+1(ai+1)aredefinedfor i =0,...,m 1, but there may be a jump in the tangent − vector to γ at ai,thatis,wemayhave γ,(a ) = γ, (a ). i i+1 ( i+1 i+1 4.3. TANGENT AND COTANGENT BUNDLES, VECTOR FIELDS 313 4.3 Tangent and Cotangent Bundles, Vector Fields

Let M be a Ck-manifold (with k 2). Roughly speaking, avectorfieldonM is the assignment,≥ p X(p), of a tangent vector, X(p) T (M), to a point-→p M. ∈ p ∈ Generally, we would like such assignments to have some properties when p varies in M,forexample, to be Cl,forsomel related to k.

Now, if the collection, T (M), of all tangent spaces, Tp(M), was a Cl-manifold, then it would be very easy to define what we mean by a Cl-vector field: We would simply require the map, X: M T (M), to be Cl. → If M is a Ck-manifold of dimension n,thenwecanindeed k 1 make T (M)intoaC − -manifold of dimension 2n and we now sketch this construction. 314 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES We find it most convenient to use Version 3 of the def- inition of tangent vectors, i.e., as equivalence classes of triples (U,ϕ, x), with x Rn. ∈ First, we let T (M)bethedisjointunionofthetangent spaces T (M), for all p M.Formally, p ∈ T (M)= (p, v) p M,v T (M) . { | ∈ ∈ p } There is a natural projection, π: T (M) M, with π(p, v)=p. →

We still have to give T (M)atopologyandtodefinea k 1 C − -atlas.

For every chart, (U,ϕ), of M (with U open in M)we 1 2n define the function, ϕ: π− (U) R ,by → 1 ϕ(p, v)=(ϕ(p),θU,−ϕ,p(v)), " 1 where (p, v) π− (U)andθU,ϕ,p is the isomorphism n ∈" between R and Tp(M)describedjustafterDefinition 4.2.17.

1 It is obvious that ϕ is a bijection between π− (U)and ϕ(U) Rn,anopensubsetofR2n. × " 4.3. TANGENT AND COTANGENT BUNDLES, VECTOR FIELDS 315 We give T (M)theweakesttopologythatmakesallthe ϕ continuous, i.e., we take the collection of subsets of the 1 n form ϕ− (W ), where W is any open subset of ϕ(U) R , × as" a basis of the topology of T (M). " One easily checks that T (M)isHausdorffandsecond- countable in this topology. If (U,ϕ)and(V,ψ)areover- lapping charts, then the transition function 1 n n ψ ϕ− : ϕ(U V ) R ψ(U V ) R ◦ ∩ × −→ ∩ × is given by " " 1 1 1 ψ ϕ− (z,x)=(ψ ϕ− (z), (ψ ϕ− ), (x)), ◦ ◦ ◦ z with (z,x) ϕ(U V ) Rn. " "∈ ∩ 1 × k 1 It is clear that ψ ϕ− is a C − -map. Therefore, T (M) k 1◦ is indeed a C − -manifold of dimension 2n,calledthe ". "

Remark: Even if the manifold M is naturally embed- ded in RN (for some N n =dim(M)), it is not at all obvious how to view the≥ tangent bundle, T (M), as em- N bedded in R ,,forsomesuitableN ,.Hence,weseethat the definition of an abtract manifold is unavoidable. 316 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Asimilarconstructioncanbecarriedoutforthecotan- gent bundle.

In this case, we let T ∗(M)bethedisjointunionofthe cotangent spaces Tp∗(M),

T ∗(M)= (p, ω) p M,ω T ∗(M) . { | ∈ ∈ p }

We also have a natural projection, π: T ∗(M) M,and we can define charts as follows: For any chart,→ (U,ϕ), on 1 2n M,wedefinethefunctionϕ: π− (U) R by → ∂ ∂ ϕ(p, ω)= ϕ(p),ω " ,...,ω , ∂x ∂x . .& 1'p/ .& n'p// " 1 ∂ where (p, ω) π− (U)andthe are the basis of ∈ ∂xi p Tp(M)associatedwiththechart()U,ϕ*).

k 1 Again, one can make T ∗(M)intoaC − -manifold of di- mension 2n,calledthecotangent bundle. 4.3. TANGENT AND COTANGENT BUNDLES, VECTOR FIELDS 317 Another method using Version 3 of the definition of tan- gent vectors is presented in Section ??.

For each chart (U,ϕ)onM,weobtainachart 1 n 2n ϕ∗: π− (U) ϕ(U) R R → × ⊆ on T ∗(M)givenby " ϕ∗(p, ω)=(ϕ(p),θU,∗ ϕ,π(ω)(ω)) 1 for all (p, ω) π− (U), where ∈" n θ∗ = ι θ0 : T ∗(M) R . U,ϕ,p ◦ U,ϕ,p p →

n Here, θU,0ϕ,p: Tp∗(M) (R )∗ is obtained by dualizing n → n n the map, θ : R T (M)andι:(R )∗ R is the U,ϕ,p → p → isomorphism induced by the canonical basis (e1,...,en) of Rn and its dual basis.

For simplicity of notation, we also use the notation TM for T (M)(resp.T ∗M for T ∗(M)). 318 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Observe that for every chart, (U,ϕ), on M,thereisa bijection 1 n τ : π− (U) U R , U → × given by 1 τU (p, v)=(p, θU,−ϕ,p(v)). 1 Clearly, pr1 τU = π,onπ− (U)asillustratedbythe following commutative◦ diagram:

1 τU n π− (U) / U R II uu II uu × II uu II uu π II uu pr1 $ U zu

Thus, locally, that is, over U,thebundleT (M)lookslike the product manifold U Rn. × We say that T (M)islocally trivial (over U)andwecall τU a trivializing map.

For any p M,thevectorspace 1 ∈ π− (p)= p Tp(M) ∼= Tp(M)iscalledthe fibre above{ p}.× 4.3. TANGENT AND COTANGENT BUNDLES, VECTOR FIELDS 319

1 Observe that the restriction of τU to π− (p)isanisomor- phism between p T (M) = T (M)and { }× p ∼ p p Rn = Rn,foranyp M. { }× ∼ ∈ Furthermore, for any two overlapping charts (U,ϕ)and (V,ψ), there is a function gUV : U V GL(n, R)such that ∩ → 1 (τ τ − )(p, x)=(p, g (p)(x)) U ◦ V UV for all p U V and all x Rn,withg (p)givenby ∈ ∩ ∈ UV 1 g (p)=(ϕ ψ− ), . UV ◦ ϕ(p)

n Obviously, gUV (p)isalinearisomorphismofR for all p U V .ThemapsgUV (p)arecalledthetransition functions∈ ∩ of the tangent bundle. 320 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES All these ingredients are part of being a . For more on bundles, see Lang [?], Gallot, Hulin and Lafontaine [?], Lafontaine [?]orBottandTu[?].

When M = Rn,observethat T (M)=M Rn = Rn Rn,i.e.,thebundleT (M)is (globally) trivial.× ×

Given a Ck-map, h: M N,betweentwoCk-manifolds, we can define the function,→ dh: T (M) T (N), (also denoted Th,orh ,orDh)bysetting → ∗ dh(u)=dh (u), iff u T (M). p ∈ p

We leave the next proposition as an exercise to the reader (A proof can be found in Berger and Gostiaux [?]). 4.3. TANGENT AND COTANGENT BUNDLES, VECTOR FIELDS 321 Proposition 4.3.1 Given a Ck-map, h: M N,be- tween two Ck-manifolds M and N (with k → 1), the k 1 ≥ map dh: T (M) T (N) is a C − map. →

We are now ready to define vector fields.

Definition 4.3.2 Let M be a Ck+1 manifold, with k 1. For any open subset, U of M,avector field on U is≥ any section, X,ofT (M) over U,i.e.,anyfunction, X: U T (M), such that π X =idU .Wealsosaythat X is a→lifting of U into T (M◦ ).

We say that X is a Ck-vector field on U iff X is a section over U and a Ck-map.

The set of Ck-vector fields over U is denoted Γ(k)(U, T (M)). Given a curve, γ:[a, b] M,avector field, X,along γ is any section of T (M→)overγ,i.e.,aCk-function, X:[a, b] T (M), such that π X = γ.Wealsosay that X lifts→ γ into T (M). ◦ 322 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Clearly, Γ(k)(U, T (M)) is a real vector space. For short, the space Γ(k)(M,T(M)) is also denoted by Γ(k)(T (M)) (or X(k)(M) or even Γ(T (M)) or X(M)).

Remark: We can also define a Cj-vector field on U as asection,X,overU which is a Cj-map, where 0 j k. Then, we have the vector space, Γ(j)(U, T (M)),≤etc ≤.

If M = Rn and U is an open subset of M,then T (M)=Rn Rn and a section of T (M)overU is simply afunction,X×,suchthat X(p)=(p, u), with u Rn, ∈ for all p U.Inotherwords,X is defined by a function, f: U R∈n (namely, f(p)=u). → This corresponds to the “old” definition of a vector field in the more basic case where the manifold, M,isjustRn. 4.3. TANGENT AND COTANGENT BUNDLES, VECTOR FIELDS 323 For any vector field X Γ(k)(U, T (M)) and for any p U,wehaveX(p)=(∈p, v)forsomev T (M), and∈ ∈ p it is convenient to denote the vector v by Xp so that X(p)=(p, Xp).

In fact, in most situations it is convenient to identify X(p) with X T (M), and we will do so from now on. p ∈ p This amounts to identifying the isomorphic vector spaces p T (M)andT (M). { }× p p Let us illustrate the advantage of this convention with the next definition.

Given any Ck-function, f k(U), and a vector field, X Γ(k)(U, T (M)), we define∈C the vector field, fX,by ∈ (fX) = f(p)X ,p U. p p ∈ 324 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Obviously, fX Γ(k)(U, T (M)), which shows that Γ(k)(U, T (M)) is∈ also a k(U)-module. C For any chart, (U,ϕ), on M it is easy to check that the map ∂ p ,p U, -→ ∂x ∈ & i'p is a Ck-vector field on U (with 1 i n). This vector ≤ ≤ field is denoted ∂ or ∂ . ∂xi ∂xi ) * 4.3. TANGENT AND COTANGENT BUNDLES, VECTOR FIELDS 325 Definition 4.3.3 Let M be a Ck+1 manifold and let X be a Ck vector field on M.IfU is any open subset of M and f is any function in k(U), then the C of f with respect to X,denotedX(f)orLXf,isthe function on U given by X(f)(p)=X (f)=X (f),pU. p p ∈ Observe that

X(f)(p)=df p(Xp), where df p is identified with the linear form in Tp∗(M) defined by df (v)=v(f),vT M, p ∈ p by identifying Tt0R with R (see the discussion following Proposition 4.2.20). The Lie derivative, LXf,isalsode- noted X[f]. 326 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES As a special case, when (U,ϕ)isachartonM,thevector field, ∂ ,justdefinedaboveinducesthefunction ∂xi ∂ p f, f U, -→ ∂x ∈ & i'p denoted ∂ (f)or ∂ f. ∂xi ∂xi ) * k 1 It is easy to check that X(f) − (U). As a conse- quence, every vector field X ∈Γ(kC)(U, T (M)) induces a linear map, ∈ k k 1 L : (U) − (U), X C −→ C given by f X(f). -→

It is immediate to check that LX has the Leibniz property, i.e., LX(fg)=LX(f)g + fLX(g).

Linear maps with this property are called derivations. 4.3. TANGENT AND COTANGENT BUNDLES, VECTOR FIELDS 327 Thus, we see that every vector field induces some kind of differential operator, namely, a derivation.

Unfortunately, not every derivation of the above type arises from a vector field, although this turns out to be true in the smooth case i.e., when k = (for a proof, see Gallot, Hulin and Lafontaine [?] or Lafontaine∞ [?]).

In the rest of this section, unless stated otherwise, we assume that k 1. The following easy proposition holds (c.f. Warner [?≥]): 328 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Proposition 4.3.4 Let X be a vector field on the Ck+1- manifold, M,ofdimensionn.Then,thefollowingare equivalent: (a) X is Ck.

(b) If (U,ϕ) is a chart on M and if f1,...,fn are the functions on U uniquely defined by n ∂ X ! U = f , i ∂x i=1 i % k then each fi is a C -map. (c) Whenever U is open in M and f k(U),then k 1 ∈C X(f) − (U). ∈C

Given any two Ck-vector field, X, Y ,onM,foranyfunc- tion, f k(M), we defined above the function X(f) and Y (f∈). C

Thus, we can form X(Y (f)) (resp. Y (X(f))), which are k 2 in − (M). C 4.3. TANGENT AND COTANGENT BUNDLES, VECTOR FIELDS 329 Unfortunately, even in the smooth case, there is generally no vector field, Z,suchthat Z(f)=X(Y (f)), for all f k(M). ∈C This is because X(Y (f)) (and Y (X(f))) involve second- order derivatives.

However, if we consider X(Y (f)) Y (X(f)), then second- order derivatives cancel out and− there is a unique vector field inducing the above differential operator.

Intuitively, XY YX measures the “failure of X and Y to commute.” −

Proposition 4.3.5 Given any Ck+1-manifold, M,of dimension n,foranytwoCk-vector fields, X, Y ,on k 1 M,thereisauniqueC − -vector field, [X, Y ],such that k 1 [X, Y ](f)=X(Y (f)) Y (X(f)), for all f − (M). − ∈C 330 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Definition 4.3.6 Given any Ck+1-manifold, M,ofdi- mension n,foranytwoCk-vector fields, X, Y ,onM,the k 1 Lie bracket,[X, Y ], of X and Y ,istheC − vector field defined so that k 1 [X, Y ](f)=X(Y (f)) Y (X(f)), for all f − (M). − ∈C

We also have the following simple proposition whose proof is left as an exercise (or, see Do Carmo [?]):

Proposition 4.3.7 Given any Ck+1-manifold, M,of dimension n,foranyCk-vector fields, X, Y, Z,onM, for all f,g k(M),wehave: ∈C (a) [[X, Y ],Z]+[[Y,Z],X]+[Z, X],Y]=0 (Jacobi identity). (b) [X, X]=0. (c) [fX,gY]=fg[X, Y ]+fX(g)Y gY (f)X. − (d) [ , ] is bilinear. − − 4.3. TANGENT AND COTANGENT BUNDLES, VECTOR FIELDS 331 Consequently, for smooth manifolds (k = ), the space ( ) ∞ of vector fields, Γ ∞ (T (M)), is a vector space equipped with a bilinear operation, [ , ], that satisfies the Jacobi ( ) − − identity. This makes Γ ∞ (T (M)) a .

Let ϕ: M N be a diffeomorphism between two mani- folds. Then,→ vector fields can be transported from N to M and conversely.

Definition 4.3.8 Let ϕ: M N be a diffeomorphism between two Ck+1 manifolds.→ For every Ck vector field, Y ,onN,thepull-back of Y along ϕ is the vector field, ϕ∗Y ,onM,givenby 1 (ϕ∗Y ) = dϕ− (Y ),pM. p ϕ(p) ϕ(p) ∈ For every Ck vector field, X,onM,thepush-forward of X along ϕ is the vector field, ϕ X,onN,givenby ∗ 1 ϕ X =(ϕ− )∗X, ∗ that is, for every p M, ∈ (ϕ X)ϕ(p) = dϕp(Xp), ∗ or equivalently,

(ϕ X)q = dϕϕ 1(q)(Xϕ 1(q)),qN. ∗ − − ∈ 332 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES It is not hard to check that 1 Lϕ Xf = LX(f ϕ) ϕ− , ∗ ◦ ◦ for any function f Ck(N). ∈ One more notion will be needed to when we deal with Lie algebras.

Definition 4.3.9 Let ϕ: M N be a Ck+1-map of manifolds. If X is a Ck vector→ field on M and Y is a Ck vector field on N,wesaythatX and Y are ϕ-related iff dϕ X = Y ϕ. ◦ ◦

Proposition 4.3.10 Let ϕ: M N be a Ck+1-map of manifolds, let X and Y be C→k vector fields on M k and let X1,Y1 be C vector fields on N.IfX is ϕ- related to X1 and Y is ϕ-related to Y1,then[X, Y ] is ϕ-related to [X1,Y1]. 4.4. SUBMANIFOLDS, IMMERSIONS, 333 4.4 Submanifolds, Immersions, Embeddings

Although the notion of submanifold is intuitively rather clear, technically, it is a bit tricky.

In fact, the reader may have noticed that many different definitions appear in books and that it is not obvious at first glance that these definitions are equivalent.

What is important is that a submanifold, N,ofagiven manifold, M,notonlyhavethetopologyinducedM but also that the charts of N be somewhow induced by those of M.

(Recall that if X is a topological space and Y is a subset of X,thenthesubspace topology on Y or topology induced by X on Y has for its open sets all subsets of the form Y U,whereU is an arbitary open subset of X.). ∩ 334 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Given m, n,with0 m n,wecanviewRm as a subspace of Rn using≤ the inclusion≤ m m m n m n R = R (0,...,0) / R R − = R , ∼ ×{ } → × n m − given by 0 12 3 (x ,...,x ) (x ,...,x , 0,...,0). 1 m -→ 1 m n m − Definition 4.4.1 Given a Ck-manifold,0 12M3,ofdimen- sion n,asubset,N,ofM is an m-dimensional subman- ifold of M (where 0 m n)iffforeverypoint,p N, there is a chart, (U,ϕ≤), of≤M,withp U,sothat∈ ∈ m ϕ(U N)=ϕ(U) (R 0n m ). ∩ ∩ ×{ − } (We write 0n m =(0,...,0).) − n m − 0 12 3 The subset, U N,ofDefinition4.4.1issometimescalled a slice of (U,ϕ∩)andwesaythat(U,ϕ)isadapted to N (See O’Neill [?]orWarner[?]). 4.4. SUBMANIFOLDS, IMMERSIONS, EMBEDDINGS 335 Other authors, including Warner [?], use the term sub- ! manifold in a broader sense than us and they use the word embedded submanifold for what is defined in Defi- nition 4.4.1.

The following proposition has an almost trivial proof but it justifies the use of the word submanifold:

Proposition 4.4.2 Given a Ck-manifold, M,ofdi- mension n,foranysubmanifold,N,ofM of dimen- sion m n,thefamilyofpairs(U N,ϕ ! U N), where (U,≤ϕ) ranges over the charts over∩ any atlas∩ for M,isanatlasforN,whereN is given the subspace topology. Therefore, N inherits the structure of a Ck- manifold.

In fact, every chart on N arises from a chart on M in the following precise sense: 336 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Proposition 4.4.3 Given a Ck-manifold, M,ofdi- mension n and a submanifold, N,ofM of dimension m n,foranyp N and any chart, (W,η),ofN at p,thereissomechart,≤ ∈ (U,ϕ),ofM at p so that m ϕ(U N)=ϕ(U) (R 0n m ) ∩ ∩ ×{ − } and ϕ ! U N = η ! U N, ∩ ∩ where p U N W . ∈ ∩ ⊆

It is also useful to define more general kinds of “subman- ifolds.”

Definition 4.4.4 Let ϕ: N M be a Ck-map of man- ifolds. →

(a) The map ϕ is an of N into M iff dϕp is injective for all p N. ∈ (b) The set ϕ(N)isanimmersed submanifold of M iff ϕ is an injective immersion. 4.4. SUBMANIFOLDS, IMMERSIONS, EMBEDDINGS 337 (c) The map ϕ is an of N into M iff ϕ is an injective immersion such that the induced map, N ϕ(N), is a homeomorphism, where ϕ(N) is given−→ the subspace topology (equivalently, ϕ is an open map from N into ϕ(N)withthesubspacetopol- ogy). We say that ϕ(N)(withthesubspacetopology) is an embedded submanifold of M.

(d) The map ϕ is a of N into M iff dϕp is surjective for all p N. ∈

Again, we warn our readers that certain authors (such ! as Warner [?]) call ϕ(N), in (b), a submanifold of M! We prefer the terminology immersed submanifold. 338 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES The notion of immersed submanifold arises naturally in the framework of Lie groups.

Indeed, the fundamental correspondence between Lie groups and Lie algebras involves Lie subgroups that are not nec- essarily closed.

But, as we will see later, subgroups of Lie groups that are also submanifolds are always closed.

It is thus necessary to have a more inclusive notion of submanifold for Lie groups and the concept of immersed submanifold is just what’s needed.

Immersions of R into R3 are parametric curves and im- mersions of R2 into R3 are parametric surfaces. These have been extensively studied, for example, see DoCarmo [?], Berger and Gostiaux [?] or Gallier [?]. 4.4. SUBMANIFOLDS, IMMERSIONS, EMBEDDINGS 339 Immersions (i.e., subsets of the form ϕ(N), where N is an immersion) are generally neither injective immersions (i.e., subsets of the form ϕ(N), where N is an injective immersion) nor embeddings (or submanifolds).

For example, immersions can have self-intersections, as the plane curve (nodal cubic): x = t2 1; y = t(t2 1). − − Injective immersions are generally not embeddings (or submanifolds) because ϕ(N)maynotbehomeomorphic to N.

An example is given by the Lemniscate of Benoulli, an injective immersion of R into R2: t(1 + t2) x = , 1+t4 t(1 t2) y = − . 1+t4 340 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Another interesting example is the immersion of R into the 2-torus, T 2 = S1 S1 R4,givenby × ⊆ t (cos t, sin t, cos ct, sin ct), -→ where c R. ∈ One can show that the image of R under this immersion is closed in T 2 iff c is rational. Moreover, the image of this immersion is dense in T 2 but not closed iff c is irrational.

The above example can be adapted to the torus in R3: One can show that the immersion given by t ((2 + cos t)cos(√2 t), (2 + cos t)sin(√2 t), sin t), -→ is dense but not closed in the torus (in R3)givenby

(s, t) ((2 + cos s)cost, (2 + cos s)sint, sin s), -→ where s, t R. ∈ There is, however, a close relationship between submani- folds and embeddings. 4.4. SUBMANIFOLDS, IMMERSIONS, EMBEDDINGS 341 Proposition 4.4.5 If N is a submanifold of M,then the inclusion map, j: N M,isanembedding.Con- versely, if ϕ: N M →is an embedding, then ϕ(N) with the subspace→ topology is a submanifold of M and ϕ is a diffeomorphism between N and ϕ(N).

In summary, embedded submanifolds and (our) subman- ifolds coincide.

Some authors refer to spaces of the form ϕ(N), where ϕ is an injective immersion, as immersed submanifolds.

However, in general, an immersed submanifold is not a submanifold.

One case where this holds is when N is compact, since then, a bijective continuous map is a homeomorphism.

Our next goal is to review and promote to manifolds some standard results about ordinary differential equations. 342 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES 4.5 Integral Curves, Flow of a Vector , One-Parameter Groups of Diffeomorphisms

We begin with integral curves and (local) flows of vector fields on a manifold.

k 1 Definition 4.5.1 Let X be a C − vector field on a k C -manifold, M,(k 2) and let p0 be a point on M. An integral curve (or≥ trajectory) for X with initial k 1 condition p is a C − -curve, γ: I M,sothat 0 → γ˙ (t)=X , for all t I and γ(0) = p , γ(t) ∈ 0 where I =]a, b[ R is an open interval containing 0. ⊆

What definition 4.5.1 says is that an integral curve, γ, with initial condition p0 is a curve on the manifold M passing through p0 and such that, for every point p = γ(t) on this curve, the tangent vector to this curve at p,i.e., γ˙ (t), coincides with the value, Xp,ofthevectorfieldX at p. 4.5. INTEGRAL CURVES, FLOW, ONE-PARAMETER GROUPS 343 Given a vector field, X,asabove,andapointp M, 0 ∈ is there an integral curve through p0?Issuchacurve unique? If so, how large is the open interval I?

We provide some answers to the above questions below.

k 1 k Definition 4.5.2 Let X be a C − vector field on a C - manifold, M,(k 2) and let p be a point on M.Alocal ≥ 0 flow for X at p0 is a map, ϕ: J U M, × → where J R is an open interval containing 0 and U is an ⊆ open subset of M containing p0,sothatforeveryp U, the curve t ϕ(t, p)isanintegralcurveofX with initial∈ condition p-→.

Thus, a local flow for X is a family of integral curves for all points in some small open set around p0 such that these curves all have the same domain, J,independently of the initial condition, p U. ∈ 344 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES The following theorem is the main existence theorem of local flows.

This is a promoted version of a similar theorem in the classical theory of ODE’s in the case where M is an open subset of Rn.

Theorem 4.5.3 (Existence of a local flow) Let X be k 1 k a C − vector field on a C -manifold, M,(k 2)and ≥ let p0 be a point on M.ThereisanopenintervalJ R containing 0 and an open subset U M containing⊆ ⊆ p0,sothatthereisaunique local flow ϕ: J U M for X at p .Whatthismeansisthatifϕ : J× U → M 0 1 × → and ϕ2: J U M are both local flows with domain × → k 1 J U,thenϕ = ϕ .Furthermore,ϕ is C − . × 1 2

Theorem 4.5.3 holds under more general hypotheses, namely, when the vector field satisfies some Lipschitz condition, see Lang [?] or Berger and Gostiaux [?].

Now, we know that for any initial condition, p0,thereis some integral curve through p0. 4.5. INTEGRAL CURVES, FLOW, ONE-PARAMETER GROUPS 345 However, there could be two (or more) integral curves γ : I M and γ : I M with initial condition p . 1 1 → 2 2 → 0

This leads to the natural question: How do γ1 and γ2 differ on I I ?Thenextpropositionshowstheydon’t! 1 ∩ 2

k 1 Proposition 4.5.4 Let X be a C − vector field on a k C -manifold, M,(k 2)andletp0 be a point on M. If γ : I M and γ≥: I M are any two integral 1 1 → 2 2 → curves both with initial condition p0,thenγ1 = γ2 on I I . 1 ∩ 2

Proposition 4.5.4 implies the important fact that there is a unique maximal integral curve with initial condition p.

Indeed, if γj: Ij M j K is the family of all integral curves with{ initial→ condition} ∈ p (for some big index set,

K), if we let I(p)= j K Ij,wecandefineacurve, ∈ γp: I(p) M,sothat → # γ (t)=γ (t), if t I . p j ∈ j 346 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Since γ and γ agree on I I for all j, l K,thecurve j l j ∩ l ∈ γp is indeed well defined and it is clearly an integral curve with initial condition p with the largest possible domain (the open interval, I(p)).

The curve γp is called the maximal integral curve with initial condition p and it is also denoted by γ(p, t).

Note that Proposition 4.5.4 implies that any two distinct integral curves are disjoint, i.e., do not intersect each other. 4.5. INTEGRAL CURVES, FLOW, ONE-PARAMETER GROUPS 347 Consider the vector field in R2 given by ∂ ∂ X = y + x . − ∂x ∂y

If we write γ(t)=(x(t),y(t)), the differential equation, γ˙ (t)=X(γ(t)), is expressed by

x,(t)= y(t) − y,(t)=x(t), or, in matrix form, x 0 1 x , = . y 10− y & , ' & '& ' 348 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES x 0 1 If we write X = and A = ,thenthe y 10− above equation is written& ' as & '

X, = AX.

Now, as A A2 An etA = I + t + t2 + + tn + , 1! 2! ··· n! ··· we get 2 3 n d tA A A 2 A n 1 tA (e )=A+ t+ t + + t − + = Ae , dt 1! 2! ··· (n 1)! ··· −

tA so we see that e p is a solution of the ODE X, = AX with initial condition X = p,andbyuniqueness, X = etAp is the solution of our ODE starting at X = p. 4.5. INTEGRAL CURVES, FLOW, ONE-PARAMETER GROUPS 349

x0 Thus, our integral curve, γp,throughp = is the y0 circle given by & ' x cos t sin t x = 0 . y sin t −cos t y & ' & '& 0 '

Observe that I(p)=R,foreveryp R2. ∈ The following interesting question now arises: Given any p0 M,ifγp0: I(p0) M is the maximal integral curve with∈ initial condition→p and, for any t I(p ), if p = 0 1 ∈ 0 1 γp0(t1) M,thenthereisamaximalintegralcurve, γ : I(p ∈) M,withinitialconditionp ; p1 1 → 1

What is the relationship between γp0 and γp1,ifany?The answer is given by 350 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES

k 1 Proposition 4.5.5 Let X be a C − vector field on k a C -manifold, M,(k 2)andletp0 be a point on M.Ifγ : I(p ) ≥M is the maximal integral p0 0 → curve with initial condition p0,foranyt1 I(p0),if p = γ (t ) M and γ : I(p ) M is the∈ maximal 1 p0 1 ∈ p1 1 → integral curve with initial condition p1,then

I(p1)=I(p0) t1 and γp (t)=γγ (t )(t)=γp (t+t1), − 1 p0 1 0 for all t I(p ) t ∈ 0 − 1

Proposition 4.5.5 says that the traces γp0(I(p0)) and

γp1(I(p1)) in M of the maximal integral curves γp0 and γp1 are identical; they only differ by a simple reparametriza- tion (u = t + t1).

It is useful to restate Proposition 4.5.5 by changing point of view.

So far, we have been focusing on integral curves, i.e., given any p M,welett vary in I(p )andgetanintegral 0 ∈ 0 curve, γp0,withdomainI(p0). 4.5. INTEGRAL CURVES, FLOW, ONE-PARAMETER GROUPS 351

Instead of holding p0 M fixed, we can hold t R fixed and consider the set ∈ ∈ (X)= p M t I(p) , Dt { ∈ | ∈ } i.e., the set of points such that it is possible to “travel for t units of time from p”alongthemaximalintegralcurve, γp,withinitialconditionp (It is possible that (X)= ). Dt ∅

By definition, if t(X) = ,thepointγp(t)iswellde- fined, and so, weD obtain a( map,∅ ΦX: (X) M,withdomain (X), given by t Dt → Dt X Φt (p)=γp(t).

The above suggests the following definition: 352 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES

k 1 Definition 4.5.6 Let X be a C − vector field on a Ck-manifold, M,(k 2). For any t R,let ≥ ∈ (X)= p M t I(p) Dt { ∈ | ∈ } and (X)= (t, p) R M t I(p) D { ∈ × | ∈ } and let ΦX: (X) M be the map given by D → X Φ (t, p)=γp(t). The map ΦX is called the (global) flow of X and (X)is called its domain of definition.Foranyt R suchD that X ∈ t(X) = ,themap,p t(X) Φ (t, p)=γp(t), is D ( ∅ X X∈D X -→ denoted by Φt (i.e., Φt (p)=Φ (t, p)=γp(t)).

Observe that (X)= (I(p) p ). D ×{ } p M !∈ 4.5. INTEGRAL CURVES, FLOW, ONE-PARAMETER GROUPS 353

X Also, using the Φt notation, the property of Proposition 4.5.5 reads ΦX ΦX =ΦX , ( ) s ◦ t s+t ∗ whenever both sides of the equation make sense.

Indeed, the above says X X X X Φs (Φt (p)) = Φs (γp(t)) = γγp(t)(s)=γp(s+t)=Φs+t(p).

Using the above property, we can easily show that the X X X Φt are invertible. In fact, the inverse of Φt is Φ t. − We summarize in the following proposition some addi- tional properties of the domains (X), t(X)andthe X D D maps Φt : 354 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES

k 1 Theorem 4.5.7 Let X be a C − vector field on a Ck-manifold, M,(k 2). The following properties hold: ≥

(a) For every t R,if t(X) = ,then t(X) is open (this is trivially∈ trueD if ( X∅)= ). D Dt ∅ (b) The domain, (X), of the flow, ΦX,isopenand Dk 1 X the flow is a C − map, Φ : (X) M. D → X k 1 (c) Each Φt : t(X) t(X) is a C − -diffeomorphism D X →D− with inverse Φ t. − (d) For all s, t R,thedomainofdefinitionof X X ∈ Φs Φt is contained but generally not equal to ◦ X X s+t(X). However, dom(Φs Φt )= s+t(X) if s Dand t have the same sign. Moreover,◦ D on dom(ΦX ΦX),wehave s ◦ t ΦX ΦX =ΦX . s ◦ t s+t

We may omit the superscript, X,andwriteΦinsteadof ΦX if no confusion arises.

The reason for using the terminology flow in referring to the map ΦX can be clarified as follows: 4.5. INTEGRAL CURVES, FLOW, ONE-PARAMETER GROUPS 355 For any t such that (X) = ,everyintegralcurve,γ , Dt ( ∅ p with initial condition p t(X), is defined on some open interval containing [0,t],∈ andD we can picture these curves as “flow lines” along which the points p flow (travel) for atimeintervalt.

Then, ΦX(t, p)isthepointreachedby“flowing”forthe amount of time t on the integral curve γp (through p) starting from p.

Intuitively, we can imagine the flow of a fluid through M,andthevectorfieldX is the field of velocities of the flowing particles.

Given a vector field, X,asabove,itmayhappenthat (X)=M,forallt R. Dt ∈ In this case, namely, when (X)=R M,wesaythat the vector field X is completeD . × 356 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES

X Then, the Φt are diffeomorphisms of M and they form agroup.

X The family Φt t R acalleda1-parameter group of X. { } ∈ In this case, ΦX induces a group homomorphism, (R, +) Diff(M), from the additive group R to the −→k 1 group of C − -diffeomorphisms of M.

By abuse of language, even when it is not the case that X t(X)=M for all t,thefamily Φt t R is called a local 1D-parameter group of X,eventhoughitis{ } ∈ not agroup, because the composition ΦX ΦX may not be defined. s ◦ t 4.5. INTEGRAL CURVES, FLOW, ONE-PARAMETER GROUPS 357 If we go back to the vector field in R2 given by ∂ ∂ X = y + x , − ∂x ∂y

x0 since the integral curve, γp(t), through p = is x0 given by & ' x cos t sin t x = 0 , y sin t −cos t y & ' & '& 0 ' the global flow associated with X is given by cos t sin t ΦX(t, p)= p, sin t −cos t & '

X and each diffeomorphism, Φt ,istherotation, cos t sin t ΦX = . t sin t −cos t & ' 358 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES

X The 1-parameter group, Φt t R,generatedbyX is the group of rotations in the{ plane,} ∈SO(2).

More generally, if B is an n n invertible matrix that has areallogarithmA (that is,× if eA = B), then the matrix A defines a vector field, X,inRn,with n ∂ X = (a x ) , ij j ∂x i,j=1 i % whose integral curves are of the form, tA γp(t)=e p, and we have γp(1) = Bp.

X The one-parameter group, Φt t R,generatedbyX is tA { } ∈ given by e t R. { } ∈ When M is compact, it turns out that every vector field is complete, a nice and useful fact. 4.5. INTEGRAL CURVES, FLOW, ONE-PARAMETER GROUPS 359

k 1 Proposition 4.5.8 Let X be a C − vector field on a Ck-manifold, M,(k 2). If M is compact, then X is complete, i.e., (X≥)=R M.Moreover,themap X D × t Φt is a homomorphism from the additive group -→ k 1 R to the group, Diff(M),of(C − )diffeomorphisms of M.

Remark: The proof of Proposition 4.5.8 also applies when X is a vector field with compact support (this means that the closure of the set p M X(p) =0 is compact). { ∈ | ( }

If ϕ: M N is a diffeomorphism and X is a vector field on M,itcanbeshownthatthelocal1-parametergroup→ associated with the vector field, ϕ X,is ∗ X 1 ϕ Φt ϕ− t R. { ◦ ◦ } ∈

Apointp M where a vector field vanishes, i.e., X(p)=0,iscalleda∈ critical point of X. 360 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Critical points play a major role in the study of vec- tor fields, in differential topology (e.g., the celebrated Poincar´e–Hopf index theorem) and especially in , but we won’t go into this here.

Another famous theorem about vector fields says that every smooth vector field on a sphere of even dimension (S2n)mustvanishinatleastonepoint(theso-called “hairy-ball theorem.”

On S2,itsaysthatyoucan’tcombyourhairwithout having a singularity somewhere. Try it, it’s true!).

Let us just observe that if an integral curve, γ,passes through a critical point, p,thenγ is reduced to the point p,i.e.,γ(t)=p,forallt.

Then, we see that if a maximal integral curve is defined on the whole of R,eitheritisinjective(ithasnoself- intersection), or it is simply periodic (i.e., there is some T>0sothatγ(t + T )=γ(t), for all t R and γ is injective on [0,T[),oritisreducedtoasinglepoint.∈ 4.5. INTEGRAL CURVES, FLOW, ONE-PARAMETER GROUPS 361 We conclude this section with the definition of the Lie derivative of a vector field with respect to another vector field.

Say we have two vector fields X and Y on M.Forany p M,wecanflowalongtheintegralcurveofX with ∈ initial condition p to Φt(p)(fort small enough) and then evaluate Y there, getting Y (Φt(p)).

Now, this vector belongs to the tangent space TΦt(p)(M), but Y (p) T (M). ∈ p

So to “compare” Y (Φt(p)) and Y (p), we bring back Y (Φt(p)) to Tp(M)byapplyingthetangentmap,dΦ t,atΦt(p), − to Y (Φt(p)) (Note that to alleviate the notation, we use the slight abuse of notation dΦ t instead of d(Φ t)Φ (p).) − − t 362 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES

Then, we can form the difference dΦ t(Y (Φt(p))) Y (p), divide by t and consider the limit as−t goes to 0. − Definition 4.5.9 Let M be a Ck+1 manifold. Given any two Ck vector fields, X and Y on M,foreveryp M,theLie derivative of Y with respect to X at p∈, denoted (LX Y )p,isgivenby

dΦ t(Y (Φt(p))) Y (p) (LX Y )p =lim − − t 0 t −→d = (dΦ t(Y (Φt(p)))) . dt − (t=0 ( ( ( It can be shown that (LX Y )p is our old friend, the Lie bracket, i.e., (LX Y )p =[X, Y ]p. (For a proof, see Warner [?]orO’Neill[?]). 4.5. INTEGRAL CURVES, FLOW, ONE-PARAMETER GROUPS 363 In terms of Definition 4.3.8, observe that

((Φ t) Y )(p) Y (p) (LX Y )p =lim − ∗ − t 0 −→ t (Φ∗Y )(p) Y (p) =lim t − t 0 t −→d = (Φ∗Y )(p) , dt t (t=0 1 ( since (Φ t)− =Φt. ( − ( 364 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES 4.6 Partitions of Unity

To study manifolds, it is often necessary to construct var- ious objects such as functions, vector fields, Riemannian metrics, volume forms, etc., by gluing together items con- structed on the domains of charts.

Partitions of unity are a crucial technical tool in this glu- ing process.

The first step is to define “bump functions”(alsocalled plateau functions). For any r>0, we denote by B(r) the open ball B(r)= (x ,...,x ) Rn x2 + + x2

Given a topological space, X,foranyfunction, f: X R,thesupport of f,denotedsuppf,isthe closed→ set supp f = x X f(x) =0 . { ∈ | ( } 4.6. PARTITIONS OF UNITY 365 Proposition 4.6.1 There is a smooth function, b: Rn R,sothat → 1 if x B(1) b(x)= 0 if x ∈ Rn B(2). 4 ∈ −

Proposition 4.6.1 yields the following useful technical re- sult:

Proposition 4.6.2 Let M be a smooth manifold. For any open subset, U M,anyp U and any smooth function, f: U R⊆,thereexistanopensubset,∈ V , with p V and→ a smooth function, f: M R,defined on the∈ whole of M,sothatV is compact,→ " V U, supp f U ⊆ ⊆ and " f(q)=f(q), for all q V. ∈ " 366 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES

If X is a (Hausdorff) topological space, a family, Uα α I, { } ∈ of subsets Uα of X is a cover (or covering )ofX iff X = α I Uα.Acover, Uα α I,suchthateachUα is open is an∈ open cover. { } ∈ #

If Uα α I is a cover of X,foranysubset,J I,the { } ∈ ⊆ subfamily Uα α J is a subcover of Uα α I if X = { } ∈ { } ∈ α J Uα,i.e., Uα α J is still a cover of X. ∈ { } ∈ # Given a cover Vβ β J,wesaythatafamily Uα α I is { } ∈ { } ∈ a refinement of Vβ β J if it is a cover and if there is a function, h: I {J,sothat} ∈ U V ,forallα I. → α ⊆ h(α) ∈

Afamily Uα α I of subsets of X is locally finite ifffor every point,{ p} ∈ X,thereissomeopensubset,U,with p U,sothat∈U U = for only finitely many α I. ∈ ∩ α ( ∅ ∈ 4.6. PARTITIONS OF UNITY 367 Aspace,X,isparacompact iffevery open cover has an open locally finite refinement. Remark: Recall that a space, X,iscompact iffit is Hausdorffand if every open cover has a finite subcover. Thus, the notion of paracompactess (due to Jean Dieudonn´e) is a generalization of the notion of compactness.

Recall that a topological space, X,issecond-countable if it has a countable basis, i.e., if there is a countable family of open subsets, Ui i 1,sothateveryopensubsetofX { } ≥ is the union of some of the Ui’s.

Atopologicalspace,X,iflocally compact iffit is Haus- dorffand for every a X,thereissomecompactsubset, K,andsomeopensubset,∈ U,witha U and U K. ∈ ⊆ As we will see shortly, every locally compact and second- countable topological space is paracompact. 368 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES It is important to observe that every manifold (even not second-countable) is locally compact.

Finally, we define partitions of unity.

Definition 4.6.3 Let M be a (smooth) manifold. A partition of unity on M is a family, fi i I,ofsmooth functions on M (the index set I may{ be} ∈ uncountable) such that

(a) The family of supports, supp fi i I,islocallyfinite. { } ∈ (b) For all i I and all p M,wehave0 fi(p) 1, and ∈ ∈ ≤ ≤ f (p)=1, for every p M. i ∈ i I %∈ Note that condition (b) implies that supp fi i I is a { } ∈ cover of M.If Uα α J is a cover of M,wesaythat { } ∈ the partition of unity fi i I is subordinate to the cover { } ∈ Uα α J if supp fi i I is a refinement of Uα α J.When { } ∈ { } ∈ { } ∈ I = J and supp fi Ui,wesaythat fi i I is sub- ⊆ { } ∈ ordinate to Uα α I with the same index set as the partition of unity{ }. ∈ 4.6. PARTITIONS OF UNITY 369 In Definition 4.6.3, by (a), for every p M,thereis some open set, U,withp U and U meets∈ only finitely many of the supports, supp∈ f .So,f (p) =0foronly i i ( finitely many i I and the infinite sum i I fi(p)is well defined. ∈ ∈ $

Proposition 4.6.4 Let X be a topological space which is second-countable and locally compact (thus, also Hausdorff). Then, X is paracompact. Moreover, ev- ery open cover has a countable, locally finite refine- ment consisting of open sets with compact closures.

Remarks: 1. Proposition 4.6.4 implies that a second-countable, lo- cally compact (Hausdorff) topological space is the union of countably many compact subsets. Thus, X is count- able at infinity,anotionthatwealreadyencountered in Proposition 3.4.10 and Theorem 3.4.11. 2. A manifold that is countable at infinity has a count- able open cover by domains of charts. It follows that M is second-countable. Thus, for manifolds, second- countable is equivalent to countable at infinity. 370 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Recall that we are assuming that our manifolds are Haus- dorffand second-countable.

Theorem 4.6.5 Let M be a smooth manifold and let Uα α I be an open cover for M.Then,thereisa { } ∈ countable partition of unity, fi i 1,subordinateto { } ≥ the cover Uα α I and the support, supp fi,ofeach { } ∈ fi is compact. If one does not require compact sup- ports, then there is a partition of unity, fα α I,sub- { } ∈ ordinate to the cover Uα α I with at most countably { } ∈ many of the fα not identically zero. (In the second case, supp f U .) α ⊆ α

We close this section by stating a famous theorem of Whitney whose proof uses partitions of unity.

Theorem 4.6.6 (Whitney, 1935) Any smooth mani- fold (Hausdorffand second-countable), M,ofdimen- sion n is diffeomorphic to a closed submanifold of R2n+1.

For a proof, see Hirsch [?], Chapter 2, Section 2, Theorem 2.14. 4.7. MANIFOLDS WITH BOUNDARY 371 4.7 Manifolds With Boundary

Up to now, we have defined manifolds locally diffeomor- phic to an open subset of Rm.

This excludes many natural spaces such as a closed disk, whose boundary is a circle, a closed ball, B(1), whose m 1 boundary is the sphere, S − ,acompactcylinder, S1 [0, 1], whose boundary consist of two circles, a M¨obius strip,× etc.

These spaces fail to be manifolds because they have a boundary, that is, neighborhoods of points on their bound- aries are not diffeomorphic to open sets in Rm.

Perhaps the simplest example is the (closed) upper half space, Hm = (x ,...,x ) Rm x 0 . { 1 m ∈ | m ≥ } 372 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Under the natural embedding m 1 m 1 m m m R − ∼= R − 0 / R ,thesubset∂H of H defined by ×{ } → ∂Hm = x Hm x =0 { ∈ | m } m 1 m is isomorphic to R − and is called the boundary of H . We also define the interior of Hm as Int(Hm)=Hm ∂Hm. −

Now, if U and V are open subsets of Hm,whereHm Rm has the subset topology, and if f: U V is a continuous⊆ function, we need to explain what we→ mean by f being smooth.

We say that f: U V ,asabove,issmooth if it has an extension, f: U →V ,whereU and V are open subsets of Rm with U →U and V V and with f asmooth ⊆ ⊆ function. " " " " " " " " 4.7. MANIFOLDS WITH BOUNDARY 373

1 We say that f is a (smooth) diffeomorphism iff f − exists 1 and if both f and f − are smooth, as just defined.

To define a manifold with boundary,wereplaceevery- where R by H in Definition 4.1.1 and Definition 4.1.2.

So, for instance, given a topological space, M,achart is now pair, (U,ϕ), where U is an open subset of M and ϕ: U Ωisahomeomorphismontoanopensubset, → Ω=ϕ(U), of Hnϕ (for some n 1), etc. ϕ ≥

Definition 4.7.1 Given some integer n 1andgiven some k such that k is either an integer k ≥1ork = , a Ck-manifold of dimension n with boundary≥ consists∞ of a topological space, M,togetherwithanequivalence class, ,ofCk n-atlases, on M (where the charts are now definedA in terms of open subsets of Hn). Any atlas, ,intheequivalenceclass is called a differentiable structureA of class Ck (andA dimension n)onM.We say that M is modeled on Hn.Whenk = ,wesay that M is a smooth manifold with boundary∞. 374 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES It remains to define what is the boundary of a manifold with boundary!

By definition, the boundary , ∂M,ofamanifold(with boundary), M,isthesetofallpoints,p M,such ∈ that there is some chart, (Uα,ϕα), with p Uα and n ∈ ϕα(p) ∂H .WealsoletInt(M)=M ∂M and call it the interior∈ of M. −

Do not confuse the boundary ∂M and the interior ! Int(M)ofamanifoldwithboundaryembeddedinRN with the topological notions of boundary and interior of M as a topological space. In general, they are different.

Note that manifolds as defined earlier (In Definition 4.1.3) are also manifolds with boundary: their boundary is just empty. 4.7. MANIFOLDS WITH BOUNDARY 375 We shall still reserve the word “manifold” for these, but for emphasis, we will sometimes call them “boundary- less.”

The definition of tangent spaces, tangent maps, etc., are easily extended to manifolds with boundary.

The reader should note that if M is a manifold with boundary of dimension n,thetangentspace,TpM,is defined for all p M and has dimension n, even for boundary points, p∈ ∂M. ∈ The only notion that requires more care is that of a sub- manifold. For more on this, see Hirsch [?], Chapter 1, Section 4. 376 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES One should also beware that the product of two manifolds with boundary is generally not amanifoldwithboundary (consider the product [0, 1] [0, 1] of two line segments). There is a generalization of× the notion of a manifold with boundary called manifold with corners and such mani- folds are closed under products (see Hirsch [?], Chapter 1, Section 4, Exercise 12).

If M is a manifold with boundary, we see that Int(M)is amanifoldwithoutboundary.Whatabout∂M?

Interestingly, the boundary, ∂M,ofamanifoldwithbound- ary, M,ofdimensionn,isamanifoldofdimensionn 1. −

Proposition 4.7.2 If M is a manifold with boundary of dimension n,foranyp ∂M on the boundary on M,foranychart,(U,ϕ)∈,withp M,wehave ϕ(p) ∂Hn. ∈ ∈

Using Proposition 4.7.2, we immediately derive the fact that ∂M is a manifold of dimension n 1. − 4.8. ORIENTATION OF MANIFOLDS 377 4.8 Orientation of Manifolds

Although the notion of orientation of a manifold is quite intuitive it is technically rather subtle.

We restrict our discussion to smooth manifolds (although the notion of orientation can also be defined for topolog- ical manifolds but more work is involved).

Intuitively, a manifold, M,isorientableifitispossibleto give a consistent orientation to its tangent space, TpM, at every point, p M. ∈ So, if we go around a closed curve starting at p M, ∈ when we come back to p,theorientationofTpM should be the same as when we started. For exampe, if we travel on a M¨obiusstrip (a manifold with boundary) dragging a coin with us, we will come back to our point of departure with the coin flipped. Try it! 378 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES To be rigorous, we have to say what it means to orient TpM (a vector space) and what consistency of orientation means.

We begin by quickly reviewing the notion of orientation of a vector space.

Let E be a vector space of dimension n.Ifu1,...,un and v1,...,vn are two bases of E,abasicandcrucialfactof linear algebra says that there is a unique linear map, g, mapping each ui to the corresponding vi (i.e., g(ui)=vi, i =1,...,n).

Then, look at the determinant, det(g), of this map. We know that det(g)=det(P ), where P is the matrix whose j-th columns consist of the coordinates of vj over the basis u1,...,un. 4.8. ORIENTATION OF MANIFOLDS 379 Either det(g)isnegativeoritispositive.Thus,wede- fine an equivalence relation on bases by saying that two bases have the same orientation iffthe determinant of the linear map sending the first basis to the second has positive determinant.

An orientation of E is the choice of one of the two equiv- alence classes, which amounts to picking some basis as an orientation frame.

The above definition is perfectly fine but it turns out that it is more convenient, in the long term, to use a definition of orientation in terms of alternate multi-linear maps (in particular, to define the notion of integration on amanifold). 380 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Recall that a function, h: Ek R,isalternate multi- linear (or alternate k-linear)i→ffitislinearineachofits arguments (holding the others fixed) and if h(...,x,...,x,...)=0, that is, h vanishes whenever two of its arguments are identical.

Using multi-linearity, we immediately deduce that h van- ishes for all k-tuples of arguments, u1,...,uk,thatare linearly dependent and that h is skew-symmetric,i.e., h(...,y,...,x,...)= h(...,x,...,y,...). − In particular, for k = n,itiseasytoseethatifu1,...,un and v1,...,vn are two bases, then

h(v1,...,vn)=det(g)h(u1,...,un), where g is the unique linear map sending each ui to vi. 4.8. ORIENTATION OF MANIFOLDS 381 This shows that any alternating n-linear function is a multiple of the determinant function and that the space of alternating n-linear maps is a one-dimensional vector n space that we will denote E∗.

We also call an alternating5n-linear map an n-form.But then, observe that two bases u1,...,un and v1,...,vn have the same orientation iff

ω(u1,...,un)andω(v1,...,vn)havethesamesignfor n all ω E∗ 0 ∈ −{ } (where 05 denotes the zero n-form).

n As E∗ is one-dimensional, picking an orientation of E is equivalent to picking a generator (a one-element 5 n basis), ω,of E∗,andtosaythatu1,...,un has positive orientation iff ω(u1,...,un) > 0. 5 382 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES n Given an orientation (say, given by ω E∗)ofE, alinearmap,f: E E,isorientation∈ preserving iff → 5 ω(f(u1),...,f(un)) > 0wheneverω(u1,...,un) > 0 (or equivalently, iffdet(f) > 0).

Now, to define the orientation of an n-dimensional man- ifold, M,weusecharts.

Given any p M,foranychart,(U,ϕ), at p,thetangent 1 ∈ n map, dϕ− : R T M makes sense. ϕ(p) → p n If (e1,...,en)isthestandardbasisofR ,asitgivesan n orientation to R ,wecanorientTpM by giving it the ori- 1 1 entation induced by the basis dϕϕ−(p)(e1),...,dϕϕ−(p)(en).

Then, the consistency of orientations of the TpM’s is given by the overlapping of charts. We require that the 1 Jacobian determinants of all ϕ ϕ− have the same sign, j ◦ i whenever (Ui,ϕi)and(Uj,ϕj)areanytwooverlapping charts. 4.8. ORIENTATION OF MANIFOLDS 383 Thus, we are led to the definition below. All definitions and results stated in the rest of this section apply to man- ifolds with or without boundary.

Definition 4.8.1 Given a smooth manifold, M,ofdi- mension n,anorientation atlas of M is any atlas so that j 1 the transition maps, ϕi = ϕj ϕi− ,(fromϕi(Ui Uj)to ϕ (U U )) all have a positive◦ Jacobian determinant∩ for j i ∩ j every point in ϕi(Ui Uj). A manifold is orientable iff its has some orientation∩ atlas.

Definition 4.8.1 can be hard to check in practice and there is an equivalent criterion is terms of n-forms which is often more convenient.

The idea is that a manifold of dimension n is orientable iff there is a map, p ωp,assigningtoeverypoint,p M, -→ n ∈ anonzeron-form, ωp Tp∗M,sothatthismapis smooth. ∈ 5 384 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES In order to explain rigorously what it means for such a map to be smooth, we can define the exterior n-bundle, n T ∗M (also denoted n∗ M)inmuchthesameway that we defined the bundles TM and T ∗M. 5 5 There is an obvious smooth projection n map, π: T ∗M M. → 5 n Then, leaving the details of the fact that T ∗M can be made into a smooth manifold (of dimension n)asan 5 exercise, a smooth map, p ωp,issimplyasmooth n -→ section of the bundle T ∗M,i.e.,asmoothmap, n ω: M T ∗M,sothatπ ω =id. → 5 ◦ 5 Definition 4.8.2 If M is an n-dimensional manifold, a n smooth section, ω Γ(M, T ∗M), is called a (smooth) ∈ n n-form.Thesetofn-forms, Γ(M, T ∗M), is also de- noted n(M). An n-form,5 ω,isanowhere-vanishing A 5 n-form on M or volume form on M iff ωp is a nonzero form for every p M.Thisisequivalenttosaying that ω (u ,...,u )∈=0,forallp M and all bases, p 1 n ( ∈ u1,...,un,ofTpM. 4.8. ORIENTATION OF MANIFOLDS 385 The determinant function, (u ,...,u ) det(u ,...,u ), 1 n -→ 1 n where the ui are expressed over the canonical basis n n (e1,...,en)ofR ,isavolumeformonR .Wewillde- note this volume form by ω0.

Another standard notation is dx1 dxn,butthis notation may be very puzzling for∧ readers···∧ not familiar with .

Observe the justification for the term volume form: the quantity det(u1,...,un)isindeedthe(signed)volumeof the parallelepiped λ u + + λ u 0 λ 1, 1 i n . { 1 1 ··· n n | ≤ i ≤ ≤ ≤ } 386 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES AvolumeformonthesphereSn Rn+1 is obtained as follows: ⊆

ωp(u1,...un)=det(p, u1,...un), n n where p S and u1,...un TpS .Astheui are orthogonal∈ to p,thisisindeedavolumeform.∈

Observe that if f is a smooth function on M and ω is any n-form, then fω is also an n-form.

Definition 4.8.3 Let ϕ: M N be a smooth map of manifolds of the same dimension,→ n,andletω n(N) ∈A be an n-form on N.Thepull-back, ϕ∗ω,ofω to M is the n-form on M given by

ϕ∗ωp(u1,...,un)=ωϕ(p)(dϕp(u1),...,dϕp(un)), for all p M and all u ,...,u T M. ∈ 1 n ∈ p

One checks immediately that ϕ∗ω is indeed an n-form on M.MoreinterestingisthefollowingProposition: 4.8. ORIENTATION OF MANIFOLDS 387 Proposition 4.8.4 (a) If ϕ: M N is a local diffeo- morphism of manifolds, where dim→ M =dimN = n, n and ω (N) is a volume form on N,thenϕ∗ω is a volume∈A form on M. (b) Assume M has a vol- ume form, ω.Then,foreveryn-form, η n(M), ∈A there is a unique smooth function, f C∞(M),so that η = fω.Ifη is a volume form, then∈ f(p) =0for all p M. ( ∈

Remark: If ϕ and ψ are smooth maps of manifolds, it is easy to prove that

(ϕ ψ)∗ = ψ∗ ϕ∗ ◦ ◦ and that ϕ∗(fω)=(f ϕ)ϕ∗ω, ◦ where f is any smooth function on M and ω is any n- form.

The connection between Definition 4.8.1 and volume forms is given by the following important theorem whose proof contains a wonderful use of partitions of unity. 388 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Theorem 4.8.5 A smooth manifold (Hausdorffand second-countable) is orientable iffit possesses a vol- ume form.

Since we showed that there is a volume form on the sphere, Sn,byTheorem4.8.5,thesphereSn is orientable.

It can be shown that the projective spaces, RPn,arenon- orientable iff n is even an thus, orientable iff n is odd. In particular, RP2 is not orientable.

Also, even though M may not be orientable, its tangent bundle, T (M), is always orientable! (Prove it).

It is also easy to show that if f: Rn+1 R is a smooth 1 → submersion, then M = f − (0) is a smooth orientable manifold. Another nice fact is that every Lie group is orientable. 4.8. ORIENTATION OF MANIFOLDS 389

By Proposition 4.8.4 (b), given any two volume forms, ω1 and ω on a manifold, M,thereisafunction,f: M R, 2 → never 0 on M such that ω2 = fω1.Thisfactsuggeststhe following definition:

Definition 4.8.6 Given an orientable manifold, M,two volume forms, ω1 and ω2,onM are equivalent iff ω2 = fω1 for some smooth function, f: M R,suchthat f(p) > 0forallp M.Anorientation→ of M is the choice of some equivalence∈ class of volume forms on M and an oriented manifold is a manifold together with a choice of orientation. If M is a manifold oriented by the volume form, ω,foreveryp M,abasis,(b ,...,b )of ∈ 1 n TpM is posively oriented iff ωp(b1,...,bn) > 0, else it is negatively oriented (where n =dim(M)).

Aconnectedorientablemanifoldhastwoorientations. 390 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES We will also need the notion of orientation-preserving dif- feomorphism.

Definition 4.8.7 Let ϕ: M N be a diffeomorphism of oriented manifolds, M and →N,ofdimensionn and say the orientation on M is given by the volume form ω1 while the orientation on N is given by the volume form ω2.We say that ϕ is orientation preserving iff ϕ∗ω2 determines the same orientation of M as ω1. 4.8. ORIENTATION OF MANIFOLDS 391 Using Definition 4.8.7 we can define the notion of a posi- tive atlas.

Definition 4.8.8 If M is a manifold oriented by the volume form, ω,anatlasforM is positive ifffor every chart, (U,ϕ), the diffeomorphism, ϕ: U ϕ(U), is ori- entation preserving, where U has the orientation→ induced by M and ϕ(U) Rn has the orientation induced by the standard orientation⊆ on Rn (with dim(M)=n).

The proof of Theorem 4.8.5 shows

Proposition 4.8.9 If a manifold, M,hasanorien- tation atlas, then there is a uniquely determined ori- entation on M such that this atlas is positive. 392 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES 4.9 Covering Maps and Universal Covering Manifolds

Covering maps are an important technical tool in alge- braic topology and more generally in geometry.

We begin with covering maps.

Definition 4.9.1 Amap,π: M N,betweentwo smooth manifolds is a covering map→(or cover )iff (1) The map π is smooth and surjective. (2) For any q N,thereissomeopensubset,V N, so that q ∈V and ⊆ ∈ 1 π− (V )= Ui, i I !∈ where the U are pairwise disjoint open subsets, U i i ⊆ M,andπ: Ui V is a diffeomorphism for every i I.Wesaythat→ V is evenly covered. ∈ The manifold, M,iscalledacovering manifold of N. 4.9. COVERING MAPS AND UNIVERSAL COVERING MANIFOLDS 393

A homomorphism of coverings, π1: M1 N and π : M N,isasmoothmap,ϕ: M →M ,sothat 2 2 → 1 → 2 π = π ϕ, 1 2 ◦ that is, the following diagram commutes: ϕ / M1C M2 . CC {{ CC {{ CC {{ π1 C! }{{ π2 N

We say that the coverings π1: M1 N and π2: M2 N are equivalent iffthere is a homomorphism,→ → ϕ: M1 M2,betweenthetwocoveringsandϕ is a dif- feomorphism.→

1 As usual, the inverse image, π− (q), of any element q N is called the fibre over q,thespaceN is called the ∈base and M is called the .

As π is a covering map, each fibre is a discrete space. 394 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES

1 Note that a homomorphism maps each fibre π1− (q)inM1 1 to the fibre π− (ϕ(q)) in M ,foreveryq M . 2 2 ∈ 1

Proposition 4.9.2 Let π: M N be a covering map. → 1 If N is connected, then all fibres, π− (q),havethe 1 same cardinality for all q N.Furthermore,ifπ− (q) is not finite then it is countably∈ infinite.

When the common cardinality of fibres is finite it is called the multiplicity of the covering (or the number of sheets).

For any integer, n>0, the map, z zn,fromtheunit circle S1 = U(1) to itself is a covering-→ with n sheets. The map, t: (cos(2πt), sin(2πt)), -→ is a covering, R S1,withinfinitelymanysheets. → 4.9. COVERING MAPS AND UNIVERSAL COVERING MANIFOLDS 395 It is also useful to note that a covering map, π: M N, is a local diffeomorphism (which means that → dπp: TpM Tπ(p)N is a bijective linear map for every p M). → ∈ The crucial property of covering manifolds is that curves in N can be lifted to M,inauniqueway.Foranymap, ϕ: P N,alift of ϕ through π is a map, ϕ: P M, so that→ → ϕ = π ϕ, ◦ " as in the following commutative diagram:

ϕ " / P C M . CC CC CC π ϕ" C!  N

We state without proof the following results:

Proposition 4.9.3 If π: M N is a covering map, then for every smooth curve,→α: I N,inN (with 0 I)andforanypoint,q M,suchthat→ π(q)= α(0)∈ ,thereisauniquesmoothcurve,∈ α: I M,lift- ing α through π such that α(0) = q. → " " 396 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Proposition 4.9.4 Let π: M N be a covering map and let ϕ: P N be a smooth→ map. For any p P → 0 ∈ and any q0 M with π(q0)=ϕ(p0),thefollowing properties hold:∈ (1) If P is connected then there is at most one lift, ϕ: P M,ofϕ through π such that ϕ(p )=q . → 0 0 (2) If P is simply connected, such a lift exists. " "

Theorem 4.9.5 Every connected manifold, M,pos- sesses a simply connected covering map, π: M M, that is, with M simply connected. Any two simply→ connected coverings of N are equivalent. 6 6 In view of Theorem 4.9.5, it is legitimate to speak of the simply connected cover, M,ofM,alsocalleduniversal covering (or cover )ofM. 6 4.9. COVERING MAPS AND UNIVERSAL COVERING MANIFOLDS 397

Given any point, p M,letπ1(M,p)denotethefunda- mental group of M ∈with basepoint p.

If ϕ: M N is a smooth map, for any p M,ifwewrite q = ϕ(p→), then we have an induced group∈ homomorphism

ϕ : π1(M,p) π1(N,q). ∗ →

Proposition 4.9.6 If π: M N is a covering map, for every p M,ifq = π(p),thentheinducedhomo-→ ∈ morphism, π : π1(M,p) π1(N,q),isinjective. ∗ →

Basic Assumption:Foranycovering,π: M N,if N is connected then we also assume that M is connected.→ 398 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Using Proposition 4.9.6, we get

Proposition 4.9.7 If π: M N is a covering map and N is simply connected, then→ π is a diffeomorphism (recall that M is connected); thus, M is diffeomorphic to the universal cover, N,ofN.

The following proposition" shows that the universal cover- ing of a space covers every other covering of that space. This justifies the terminology “universal covering.”

Proposition 4.9.8 Say π1: M1 N and π : M N are two coverings→ of N,withN con- 2 2 → nected. Every homomorphism, ϕ: M1 M2,between these two coverings is a covering map.→ As a conse- quence, if π: N N is a universal covering of N, → then for every covering, π,: M N,ofN,thereisa → covering, ϕ: N" M,ofM. → " 4.9. COVERING MAPS AND UNIVERSAL COVERING MANIFOLDS 399 The notion of deck-transformation group of a covering is also useful because it yields a way to compute the funda- mental group of the base space.

Definition 4.9.9 If π: M N is a covering map, a deck-transformation is any→ diffeomorphism, ϕ: M M,suchthatπ = π ϕ,thatis,thefollowing diagram→ commutes: ◦ ϕ / MC M . CC {{ CC {{ CC {{ π C! }{{ π N

Note that deck-transformations are just automorphisms of the covering map.

The commutative diagram of Definition 4.9.9 means that adecktransformationpermuteseveryfibre.Itisimme- diately verified that the set of deck transformations of a covering map is a group denoted Γπ (or simply, Γ), called the deck-transformation group of the covering. 400 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Observe that any deck transformation, ϕ,isaliftofπ through π.Consequently,ifM is connected, by Propo- sition 4.9.4 (1), every deck-transformation is determined by its value at a single point.

So, the deck-transformations are determined by their ac- 1 tion on each point of any fixed fibre, π− (q), with q N. ∈ 1 Since the fibre π− (q)iscountable,Γisalsocountable, that is, a discrete Lie group.

1 Moreover, if M is compact, as each fibre, π− (q), is com- pact and discrete, it must be finite and so, the deck- transformation group is also finite.

The following proposition gives a useful method for de- termining the fundamental group of a manifold. 4.9. COVERING MAPS AND UNIVERSAL COVERING MANIFOLDS 401

Proposition 4.9.10 If π: M M is the universal covering of a connected manifold,→ M,thenthedeck- transformation group, Γ,isisomorphictothefunda-6 mental group, π1(M),ofM. "

Remark: When π: M M is the universal covering of M,itcanbeshownthatthegroup→ Γactssimplyand 1 transitively on every fibre,6 π− (q). " 1 This means that for any two elements, x, y π− (q), ∈ there is a unique deck-transformation, ϕ Γsuchthat ϕ(x)=y. ∈ " So, there is a bijection between π1(M) ∼= Γandthefibre 1 π− (q). " Proposition 4.9.5 together with previous observations im- plies that if the universal cover of a connected (compact) manifold is compact, then M has a finite fundamental group. We will use this fact later, in particular, in the proof of Myers’ Theorem. 402 CHAPTER 4. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES