Chapter 5 Manifolds, Tangent Spaces, Cotangent

Chapter 5

Manifolds, Tangent Spaces, Cotangent Spaces, Submanifolds, Manifolds With Boundary

5.1 Charts and Manifolds

In Chapter 1 we deﬁned the notion of a manifold embedded in some ambient space, RN .

In order to maximize the range of applications of the theory 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, U↵,whereeachU↵ is isomorphic to some “standard model,” e.g.,someopensubsetofEuclidean space, Rn.

293 294 CHAPTER 5. 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 vectors, di↵erential forms, etc.

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

We can still deﬁne the notion of a curve 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 derivatives. 5.1. CHARTS AND MANIFOLDS 295 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, n pri : R R,aredeﬁnedby ! 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 Hausdor↵and second-countable (which means that the topology has a countable basis). 296 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Deﬁnition 5.1. Given a topological space, M,achart (or local coordinate map)isapair,(U,'), where U is an open subset of M and ': U ⌦isahomeomorphism ! onto an open subset, ⌦= '(U), of Rn' (for some n 1). ' For any p M,achart,(U,'), is a chart at p i↵ p U. 2 2 If (U,')isachart,thenthefunctionsxi = pri ' are called local coordinates and for every p U,thetuple 2 (x1(p),...,xn(p)) is the set of coordinates of p w.r.t. the chart.

1 The inverse, (⌦,' ), of a chart is called a local parametrization.

Given any two charts, (Ui,'i)and(Uj,'j), if Ui Uj = ,wehavethetransition maps, j \ 6 ; 'i : 'i(Ui Uj) 'j(Ui Uj)and 'i : ' (U \ U ) ! ' (U \ U ), deﬁned by j j i \ j ! i i \ j j 1 i 1 ' = ' ' and ' = ' ' . i j i j i j 5.1. CHARTS AND MANIFOLDS 297

i j 1 Clearly, 'j =('i ) .

j i Observe that the transition maps 'i (resp. 'j)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 5.2. Given a topological space, M,given some integer n 1andgivensomek such that k is either an integerk 1ork = ,aCk n-atlas (or k 1 n-atlas of class C ), ,isafamilyofcharts, (Ui,'i) , such that A { } n (1) 'i(Ui) R for all i; ✓ (2) The Ui cover M,i.e.,

M = Ui; i [ j (3) Whenever Ui Uj = ,thetransitionmap'i (and i k \ 6 ; j 'j)isaC -di↵eomorphism. When k = ,the'i are smooth di↵eomorphisms. 1 298 CHAPTER 5. 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 previous charts in an existing atlas.

Technically, given a Ck n-atlas, ,onM,foranyother chart, (U,'), we say that (U,')isA compatible with the 1 1 k atlas i↵every map 'i ' and ' 'i is C (whenever U UA = ). \ i 6 ;

Two atlases and 0 on M are compatible i↵every 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. 5.1. CHARTS AND MANIFOLDS 299 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 atlases compatible with . e A

Definition 5.3. Given some integer n 1andgiven some k such that k is either an integer k 1ork = , a Ck-manifold of dimension n consists of a topological1 space, M,togetherwithanequivalenceclass, ,ofCk n-atlases, on M.Anyatlas, ,intheequivalenceclassA is called a di↵erentiable structureA of class Ck (and A dimension n) on M.WesaythatM is modeled on Rn. When k = ,wesaythatM is a smooth manifold. 1 Remark: It might have been better to use the terminology abstract manifold rather than manifold, to empha- size the fact that the space M is not a priori a subspace of RN ,forsomesuitableN. 300 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES We can allow k =0intheabovedefinitions.Condition (3) in Definition 5.2 is void, since a C0-di↵eomorphism j is just a homeomorphism, but 'i is always a homeomorphism.

In this case, M is called a topological manifold of dimension 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, n',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 connected, so is every manifold. 5.1. CHARTS AND MANIFOLDS 301 1

Figure 5.1: 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 2 2 2 3 M1 = (x, y) R y = x x . { 2 | }

This curve showed in Figure 5.1 and called a nodal cubic is also deﬁned as the parametric curve x =1 t2 y = t(1 t2). 302 CHAPTER 5. 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, ': U ⌦, where ⌦is some connected open subset of R (that is,! an open interval), since ' is a homeomorphism.

However, U O consists of four disconnected components and ⌦{'}(O)oftwodisconnectedcomponents, contradicting the fact that ' is a homeomorphism. 5.1. CHARTS AND MANIFOLDS 303 1

Figure 5.2: 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 2 2 3 M2 = (x, y) R y = x . { 2 | }

This curve showed in Figure 5.2 and called a cuspidal cubic is also deﬁned as the parametric curve x = t2 y = t3. 304 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES

Consider the map, ': M2 R,givenby ! '(x, y)=y1/3. 2/3 1 Since x = y on M2,weseethat' is given by 1 2 3 ' (t)=(t ,t ) and clearly, ' is a homeomorphism, so M2 is a topological manifold.

However, in the atlas consisting of the single chart, ': M2 R ,thespaceM2 is also a smooth manifold! { ! } Indeed, as there is a single chart, condition (3) of Deﬁni- tion 5.2 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 vector7! at t, 2 is c0(t)=(2t, 3t )). 5.1. CHARTS AND MANIFOLDS 305 However, this apparent paradox has to do with the fact 2 that, as a parametric curve, M2 is not immersed in R since c0 is not injective (see Deﬁnition 5.19 (a)), whereas as an abstract manifold, with this single chart, M2 is di↵eomorphic to R.

Now, we also have the chart, : M2 R,givenby ! (x, y)=y, 1 with given by 1 2/3 (u)=(u ,u). Then, observe that 1 1/3 ' (u)=u , amapthatisnot di↵erentiable at u =0.Therefore,the 1 atlas ': M2 R, : M2 R is not C and thus, { ! ! } 1 with respect to that atlas, M2 is not a C -manifold. 306 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES The example of the cuspidal cubic shows a peculiarity of k the deﬁnition of a C (or C1)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 function ! from some open subset, U,ofRn,toRm,thenthegraph, (f) Rn+m,off given by ✓ n+m (f)= (x, f(x)) R x U { 2 | 2 } is a smooth manifold with respect to the atlas consisting of the single chart, ':(f) U,givenby ! '(x, f(x)) = x, 1 with its inverse, ' : U (f), given by ! 1 ' (x)=(x, f(x)). 5.1. CHARTS AND MANIFOLDS 307 The notion of a submanifold using the concept of “adapted chart” (see Deﬁnition 5.18 in Section 5.6) gives a more satisfactory treatment of Ck (or smooth) submanifolds 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 obvious (or natural) way and a slight variation of Deﬁnition 5.2 is more convenient in such a situation: 308 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Deﬁnition 5.4. Given a set, M,givensomeintegern 1andgivensomek such that k is either an integer k 1 or k = ,aCk n-atlas (or n-atlas of class Ck), ,is afamilyofcharts,1 (U ,') ,suchthat A { i i } (1) Each Ui is a subset of M and 'i : Ui 'i(Ui)isa ! n bijection onto an open subset, 'i(Ui) R ,foralli; ✓ (2) The Ui cover M,i.e.,

M = Ui; i [ (3) Whenever Ui Uj = ,theset'i(Ui Uj)isopen n \ 6 ; j \ i k in R and the transition map 'i (and 'j)isaC - di↵eomorphism. 5.1. CHARTS AND MANIFOLDS 309 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 5.2.

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 dicult 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 'i : Ui 'i(Ui)are homeomorphisms, so M is a manifold according! to Deﬁ- nition 5.3.

We require M to be Hausdor↵and second-countable with this topology. 310 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Thus, we are back to the original notion of a manifold where it is assumed that M is already a topological space.

If the underlying topological space of a manifold is compact, then M has some ﬁnite atlas.

Also, if is some atlas for M and (U,')isachartin , for anyA (nonempty) open subset, V U,wegetachart,A ✓ (V,' V ), and it is obvious that this chart is compatible with . A Thus, (V,' 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. 5.1. CHARTS AND MANIFOLDS 311 Interesting manifolds often occur as the result of a quotient construction.

For example, real projective spaces and Grassmannians are obtained this way.

In this situation, the natural topology on the quotient object is the quotient topology but, unfortunately, even if the original space is Hausdor↵, the quotient topology may not be.

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

First, let us review the notion of quotient topology. 312 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Definition 5.5. 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 open set in X.

Given an equivalence relation R on a topological space X, if ⇡ : X X/R is the projection sending every x X to its equivalence! class [x]inX/R,thespaceX/R equipped2 with the quotient topology determined by ⇡ is called the quotient space of X modulo R.

Thus a set, V ,ofequivalenceclassesinX/R is open i↵ 1 ⇡ (V )isopeninX,whichisequivalenttothefactthat [x] V [x]isopeninX. 2 S It is immediately verified that Definition 5.5 defines topologies and that f : X Y and ⇡ : X X/R are continuous when Y and X/R!are given these! quotient topologies. 5.1. CHARTS AND MANIFOLDS 313 One should be careful that if X and Y are topological 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 5.6. Acontinuousmap,f : X Y ,isan open map (or simply open)iff(U)isopenin!Y 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 continuous surjective map f if either f is open or f is closed. 314 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES If : G X X is an action of a group G on a topological· space⇥ !X and if for every g G,themapfromX to itself given by x g x is continuous,2 then it can be show that the projection7! ·⇡ : X X/G is an open map. ! Furthermore, if G is a ﬁnite group, then ⇡ is a closed map.

Unfortunately, the Hausdor↵separation property is not necessarily preserved under quotient.

Nevertheless, it is preserved in some special important cases. 5.1. CHARTS AND MANIFOLDS 315 Proposition 5.1. Let X and Y be topological spaces, let f : X Y be a continuous surjective map, and assume that! X is compact and that Y has the quotient topology determined by f. Then Y is Hausdor↵i↵ f is a closed map.

Another simple criterion uses continuous open maps.

Proposition 5.2. Let f : X Y be a surjective continuous map between topological! spaces. If f is an open map then Y is Hausdor↵i↵the set (x ,x ) X X f(x )=f(x ) { 1 2 2 ⇥ | 1 2 } is closed in X X. ⇥

Note that the hypothesis of Proposition 5.2 implies that Y has the quotient topology determined by f. 316 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES AspecialcaseofProposition5.2isdiscussedinTu.

Given a topological space, X,andanequivalencerelation, R,onX,wesaythatR is open if the projection map, ⇡ : 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 Hausdor↵i↵ R is closed in X X. ⇥

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

Proposition 5.3. If X is a topological space and R is an open equivalence relation on X, then for any basis, B↵ , for the topology of X, the family ⇡(B↵) is a basis{ } for the topology of X/R, where ⇡ {: X }X/R is the projection map. Consequently, if X is! second- countable, then so is X/R. 5.1. CHARTS AND MANIFOLDS 317 Example 1. The sphere 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 N : S N R and S : S S R ,where { }! { }! N =(0, , 0, 1) Rn+1 (the north pole) and ··· 2 S =(0, , 0, 1) Rn+1 (the south pole) be the maps called··· respectively 2 stereographic projection from the north pole and stereographic projection from the south pole given by 1 (x ,...,x )= (x ,...,x ) N 1 n+1 1 x 1 n n+1 and 1 S(x1,...,xn+1)= (x1,...,xn). 1+xn+1 318 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES The inverse stereographic projections are given by

1 N (x1,...,xn)= n 1 2 2x1,...,2xn, x 1 n x2 +1 i i=1 i i=1 ⇣ ✓X ◆ ⌘ and P 1 S (x1,...,xn)= n 1 2 2x1,...,2xn, x +1 . n x2 +1 i i=1 i i=1 ⇣ ✓X ◆ ⌘ P 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. 5.1. CHARTS AND MANIFOLDS 319 Furthermore, it is easily checked that on the overlap, U U = Sn N,S ,thetransitionmaps N \ S { } 1 1 = S N N S are given by 1 (x1,...,xn) n 2 (x1,...,xn), 7! i=1 xi that is, the inversion of centerP O =(0,...,0) and power 1. Clearly, this map is smooth on Rn O ,sowecon- { } clude that (UN ,N )and(US,S)formasmoothatlasfor Sn.

Example 2. Smooth manifolds in RN .

Any m-dimensional manifold M in RN is a smooth mani- 1 fold, because by Lemma 2.22, the inverse maps ' : U ⌦oftheparametrizations':⌦ U are charts that yield! smooth transition functions. !

In particular, by Theorem 2.28, any linear Lie group is a smooth manifold. 320 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES n Example 3. The projective space RP .

n n To deﬁne an atlas on RP it is convenient to view RP as the set of equivalence classes of vectors in Rn+1 0 modulo the equivalence relation, { } u v i↵ v = u, for some =0 R. ⇠ 6 2 n Given any p =[x1,...,xn+1] RP ,wecall(x1,...,xn+1) the homogeneous coordinates2 of p.

It is customary to write (x : : x )insteadof 1 ··· n+1 [x1,...,xn+1]. (Actually, in most books, the indexing n starts with 0, i.e., homogeneous coordinates for RP are written as (x : : x ).) 0 ··· n n Now, RP can also be viewed as the quotient of the sphere, Sn,undertheequivalencerelationwhereanytwo antipodal points, x and x,areidentiﬁed. n It is not hard to show that the projection ⇡ : Sn RP is both open and closed. ! 5.1. CHARTS AND MANIFOLDS 321 Since Sn is compact and second-countable, we can ap- ply our previous results to prove that under the quotient n topology, RP is Hausdor↵, second-countable, and compact.

We define charts in the following way. For any i,with 1 i n +1,let   n Ui = (x1 : : xn+1) RP xi =0 . { ··· 2 | 6 } Observe that Ui is well defined, because if (y1 : : yn+1)=(x1 : : xn+1), then there is some =0sothat··· y = z ,for···i =1,...,n+1. 6 i i n We can define a homeomorphism, 'i,ofUi onto R ,as follows:

x1 xi 1 xi+1 xn+1 ' (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. 322 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES n n We can also deﬁne the maps, i,fromR to Ui RP , given by ✓

i(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 'i and i are mutual inverses, so the 'i are homeomorphisms. On the overlap, Ui Uj, (where i = j), as x =0,wehave \ 6 j 6 1 (' ' )(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

n Intuitively, the space RP is obtained by gluing the open subsets Ui on their overlaps. Even for n =3,thisisnot easy to visualize! 5.1. CHARTS AND MANIFOLDS 323 Example 4. The Grassmannian G(k, n).

Recall that G(k, n)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 i↵there is an invertible k k- 1 k ⇥ matrix, ⇤= (ij), such that k v = u , 1 j k. j ij i   i=1 X 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⇥ i↵ B = A⇤, for some invertible k k matrix, ⇤. ⇥ 324 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES (Note the analogy with projective spaces where two vectors u, v represent the same point i↵ v = u for some invertible R.) 2 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 = A⇤, for some invertible k k matrix, ⇤, ⇥ is open and that the graph of this equivalence relation is closed.

By Proposition 5.2, the Grassmannian G(k, n)isHaus- dor↵and second-countable. 5.1. CHARTS AND MANIFOLDS 325 We can deﬁne the domain of charts (according to Deﬁ- nition 5.2) 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 A . { 1 k} ⇥ S 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, 'S : US R ⇥ , ! where 'S(A)=the(n k) k matrix obtained by delet- ⇥ 1 ing the rows of index in S from AAS . 326 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES 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 = A⇤, with ⇤invertible, if we write A B A = 1 and B = 1 , A B ✓ 2◆ ✓ 2◆ as B = A⇤, we get B1 = A1⇤andB2 = A2⇤, from which we deduce that

B1 1 Ik B = = B 1 B B 1 ✓ 2◆ ✓ 2 1 ◆ Ik Ik A1 1 = = A . A ⇤⇤ 1A 1 A A 1 A 1 ✓ 2 1 ◆ ✓ 2 1 ◆ ✓ 2◆

Therefore, our map is indeed well-deﬁned. 5.1. CHARTS AND MANIFOLDS 327

It is clearly injective and we can deﬁne its inverse S as follows:

Let ⇡S be the permutation of 1,...,n sending 1,...,k to S deﬁned such that if S{= i <}

If PS is the permutation matrix associated with ⇡S,for any (n k) k matrix M,let ⇥ I (M)=P k . S S M ✓ ◆

The e↵ect of S is to “insert into M”therowsofthe identity matrix Ik as the rows of index from S.

At this stage, we have charts that are bijections from (n k) k subsets, US,ofG(k, n)toopensubsets,namely,R ⇥ . 328 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Then, the reader can check that the transition map 1 ' ' from ' (U U )to' (U U )isgivenby T S S S \ T T S \ T 1 M (P + P M)(P + P M) , 7! 3 4 1 2 where

P P 1 2 = P 1P P P T S ✓ 3 4◆

1 is the matrix of the permutation ⇡ ⇡ . T S This map is smooth, as it is given by determinants, and so, the charts (US,'S)formasmoothatlasforG(k, n).

Finally, it is easy to check that the conditions of Definition 5.2 are satisfied, so the atlas just defined makes G(k, n) into a topological space and a smooth manifold. 5.1. CHARTS AND MANIFOLDS 329 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,⇤,such that U = B⇤, 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 >U = Ik.

Consequently, the space of such matrices is closed and n k clearly bounded in R ⇥ and thus, compact. 330 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES 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 diculty 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} { } 5.1. CHARTS AND MANIFOLDS 331 Example 5. 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,'i)onM1 and (Vj, j)onM2, we declare that (Ui Vj,'i j)isachartonM1 M2, ⇥ ⇥n1+n2 ⇥ where 'i j : Ui Vj R is deﬁned so that ⇥ ⇥ ! ' (p, q)=(' (p), (q)), for all (p, q) U V . i ⇥ j i j 2 i ⇥ j

We deﬁne Ck-maps between manifolds as follows: 332 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Deﬁnition 5.7. Given any two Ck-manifolds, M and N,ofdimensionm and n respectively, a Ck-map if a continuous functions, h: M N,sothatforevery p M,thereissomechart,(!U,'), at p and some chart, (V,2 ), at q = h(p), with h(U) V and ✓ 1 h ' : '(U) (V ) ! a Ck-function.

It is easily shown that Deﬁnition 5.7 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 i↵for every p M,thereissomechart,(! U,'), at p so that 2

1 f ' : '(U) R ! is a Ck-function. 5.1. CHARTS AND MANIFOLDS 333 If U is an open subset of M,thesetofCk-functions on U is denoted by k(U). In particular, k(M)denotesthe set of Ck-functionsC on the manifold, MC .

Observe that k(U)isaring. C On the other hand, if M is an open interval of R,say M =]a, b[, then :]a, b[ N is called a Ck-curve in N. ! One checks immediately that a function, :]a, b[ N, is a Ck-map i↵for every q N,thereissomechart! (V, )atq and some open subinterval2 ]c, d[of]a, b[, so that (]c, d[) V and ✓ :]c, d[ (V ) ! is a Ck-function. 334 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES It is clear that the composition of Ck-maps is a Ck-map. A Ck-map, h: M N,betweentwomanifoldsisaCk- ! 1 di↵eomorphism i↵ 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. 5.2. TANGENT VECTORS, TANGENT SPACES 335 5.2 Tangent Vectors, Tangent Spaces

Let M be a Ck manifold of dimension n,withk 1. The purpose of the next three sections is to deﬁne the tangent space Tp(M), at a point p of a manifold M.

We provide three deﬁnitions of the notion of a tangent vector to a manifold and prove their equivalence.

The ﬁrst deﬁnition uses equivalence classes of curves on amanifoldandisthemostintuitive.

The second deﬁnition makes heavy use of the charts and of the transition functions.

It is also quite intuitive and it is easy to see that that it is equivalent to the first definition. 336 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES The second definition is the most convenient one to define the manifold structure of the tangent bundle T (M)(see Section 6.1).

The third deﬁnition (given in the next section) is based on the view that a tangent vector v,atp,inducesadif- ferential operator on functions f,deﬁnedlocallynearp; namely, the map f v(f)isalinearformsatisfying an additional property7! akin to the rule for taking the derivative of a product (the Leibniz property).

Such linear forms are called point-derivations.Thisthird deﬁnition is more intrinsic than the ﬁrst two but more abstract.

However, for any point p on the manifold M and for any chart whose domain contains p,thereisaconvenientbasis of the tangent space Tp(M). 5.2. TANGENT VECTORS, TANGENT SPACES 337 The third definition is also the most convenient one to define vector fields.

AfewtechnicalcomplicationsarisewhenM is not a smooth manifold (when k = )buttheseareeasily overcome using “stationary germs.”6 1

As pointed out by Serre, the relationship between the first definition and the third definition of the tangent space at p is best described by a nondegenerate pairing which shows that Tp(M)isthedualofthespaceofpoint- derivations at p that vanish on stationary germs. This pairing is presented in Section 5.4. 338 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES The most intuitive method to define tangent vectors is to use curves.

Let p M be any point on M and let :] ✏,✏[ M be a C21-curve passing through p,thatis,with (0)! = p.

Unfortunately, if M is not embedded in any RN ,the derivative 0(0) does not make sense.

However, for any chart, (U,'), at p,themap' is a 1 n C -curve in R and the tangent vector v =(' )0(0) is well deﬁned.

The trouble is that di↵erent curves may yield the same v!

To remedy this problem, we deﬁne an equivalence relation on curves through p as follows: 5.2. TANGENT VECTORS, TANGENT SPACES 339 Deﬁnition 5.8. Given a Ck manifold, M,ofdimension n,foranyp M,twoC1-curves, :] ✏ ,✏ [ M and 2 1 1 1 ! 2 :] ✏2,✏2[ M,throughp (i.e., 1(0) = 2(0) = p) are equivalent !i↵there is some chart, (U,'), at p so that

(' )0(0) = (' )0(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.

Definition 5.9. (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 class2 of C1-curves through p on M,modulotheequivalence relation defined in Definition 5.8. The set of all tangent vectors at p is denoted by Tp(M). 340 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES

It is obvious that Tp(M)isavectorspace.

Observe that the map that sends a curve, :] ✏,✏[ M,throughp (with (0) = p)toitstangent ! n vector, (' )0(0) R (for any chart (U,'), at p), 2 n induces a map, ': Tp(M) R . ! It is easy to check that ' is a linear bijection (by deﬁnition of the equivalence relation on curves through p).

This shows that Tp(M)isavectorspaceofdimension n =dimensionofM.

In particular, if M is an n-dimensional smooth manifold N in R and if is a curve in M through p,then0(0) = u is well deﬁned as a vector in RN ,andtheequivalenceclass of all curves through p such that (' )0(0) is the same vector in some chart ': U ⌦canbeidentiﬁedwith u. ! 5.2. TANGENT VECTORS, TANGENT SPACES 341

Thus, the tangent space TpM to M at p is isomorphic to

1 0(0) :( ✏,✏) M is a C -curve with (0) = p . { | ! }

In the special case of a linear Lie group G,Proposition 2.30 shows that the exponential map exp: g G is a ! di↵eomorphism from some open subset of g containing 0 to some open subset of G containing I.

For every g G,sinceL is a di↵eomorphism, the map 2 g L exp: L (g) G is a di↵eomorphism from some g g ! open subset of Lg(g)containing0tosomeopensubsetof G containing g.Furthermore,

L (g)=gg = gX X g . g { | 2 } Thus, we obtain smooth parametrizations of G whose inverses are charts on G,andsincebydefinitionofg,for every X g,thecurve(t)=getX is a curve through 2 g in G such that 0(0) = gX,weseethatthe tangent space TgG to G at g is isomorphic to gg. 342 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES One should observe that unless M = Rn,inwhichcase, n for any p, q R ,thetangentspaceTq(M)isnaturally 2 isomorphic to the tangent space Tp(M)bythetranslation q p,foranarbitrarymanifold,thereisnorelationship between T (M)andT (M)whenp = q. p q 6 The second way of defining tangent vectors has the ad- vantage that it makes it easier to define tangent bundles (see Section 6.1). 5.2. TANGENT VECTORS, TANGENT SPACES 343 Definition 5.10. (Tangent Vectors, Version 2) Given any Ck-manifold, M,ofdimensionn,withk 1, for any p M,considerthetriples,(U,', u), where (U,') 2 is any chart at p and u is any vector in Rn.

Say that two such triples (U,', u)and(V, ,v)areequiv- alent i↵

1 ( ' )0 (u)=v. '(p) A tangent vector to M at p is an equivalence class of triples, [(U,', u)], for the above equivalence relation.

The intuition behind Deﬁnition 5.10 is quite clear: The vector u is considered as a tangent vector to Rn at '(p). 344 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES If (U,')isachartonM at p,wecandeﬁneanatu- n n ral isomorphism, ✓U,',p : R Tp(M), between R and ! n Tp(M), as follows: For any u R , 2 ✓ : u [(U,', u)]. U,',p 7! One immediately check that the above map is indeed linear and a bijection.

The equivalence of this definition with the definition in terms of curves (Definition 5.9) is easy to prove. 5.2. TANGENT VECTORS, TANGENT SPACES 345 Proposition 5.4. Let M be any Ck-manifold of dimension n, with k 1. For every p M, for every chart, (U,'), at p, if x =[] is any2 tangent vector (version 1) given by some equivalence class of C1- curves :] ✏, +✏[ M through p (i.e., p = (0)), then the map !

x [(U,', (' )0(0))] 7! is an isomorphism between Tp(M)-version 1 and Tp(M)- version 2.

For simplicity of notation, we also use the notation TpM for Tp(M)(resp.Tp⇤M for Tp⇤(M)). 346 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES 5.3 Tangent Vectors as Derivations

One of the defects of the above deﬁnitions of a tangent vector is that it has no clear relation to the Ck-di↵erential structure of M.

In particular, the deﬁnition does not seem to have any- thing to do with the functions deﬁned locally at p.

There is another way to define tangent vectors that re- veals this connection more clearly. Moreover, such a definition is more intrinsic, i.e., does not refer explicitly to charts. 5.3. TANGENT VECTORS AS DERIVATIONS 347 As a first step, consider the following: Let (U,')bea chart at p M (where M is a Ck-manifold of dimen- 2 sion n,withk 1) and let xi = pri ',theith local coordinate (1 i n).   For any real-valued function f defined on U p,set 3

1 @ @(f ' ) f = , 1 i n. @x @X   ✓ i◆p i '(p) (Here, (@g/@X ) denotes the partial derivative of a func- i |y tion g : Rn R with respect to the ith coordinate, eval- uated at y.)! 348 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES We would expect that the function that maps f to the above value is a linear map on the set of functions deﬁned locally at p,butthereistechnicaldiculty:

The set of real-valued functions deﬁned locally at p is not avectorspace!

To see this, observe that if f is defined on an open U p and g is defined on a di↵erent open V p,thenwe3do not know how to define f + g. 3

The problem is that we need to identify functions that agree on a smaller open subset. This leads to the notion of germs.

Definition 5.11. Given any Ck-manifold, M,ofdimension n,withk 1, for any p M,alocally defined function at p is a pair, (U, f), where2 U is an open subset of M containing p and f is a real-valued function defined on U. 5.3. TANGENT VECTORS AS DERIVATIONS 349 Two locally defined functions, (U, f)and(V,g), at p are equivalent i↵there 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 5.11 is indeed an equivalence relation.

For example, for every a ( 1, 1), the locally deﬁned 2 n functions (R 1 , 1/(1 x)) and (( 1, 1), n1=0 x )at a are equivalent.{ } P Of course, the value at p of all the functions, f,inany germ, f,isf(p). Thus, we set f(p)=f(p).

We can define addition of germs, multiplication of a germ by a scalar, and multiplication of germs as follows. 350 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES If (U, f)and(V,g)aretwolocallydefinedfunctionsat p,wedefine(U V,f + g), (U V,fg)and(U,f)as the locally defined\ functions at p\given by (f + g)(q)= f(q)+g(q)and(fg)(q)=f(q)g(q)forallq U V , 2 \ and (f)(q)=f(q)forallq U,with R. 2 2 If f =[f]andg =[g]aretwogermsatp,andthen

[f]+[g]=[f + g] [f]=[f] [f][g]=[fg].

Therefore, the germs at p form a ring.

The ring of germs of Ck-functions at p is denoted (k) . OM,p When k = ,weusuallydropthesuperscript . 1 1 5.3. TANGENT VECTORS AS DERIVATIONS 351 Remark: Most readers will most likely be puzzled by the notation (k) . OM,p In fact, it is standard in algebraic geometry, but it is not as commonly used in di↵erential geometry.

For any open subset, U,ofamanifold,M,thering, k k (k) (U), of C -functions on U is also denoted M (U)(cer- tainlyC by people with an algebraic geometryO bent!).

Then, it turns out that the map U (k)(U)isasheaf , 7! OM denoted (k),andthering (k) is the stalk of the sheaf OM OM,p (k) at p. OM Such rings are called local rings.

Roughly speaking, all the “local” information about M at p is contained in the local ring (k) .(Thisistobetaken OM,p with a grain of salt. In the Ck-case where k< ,we also need the “stationary germs,” as we will see shortly.)1 352 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Now that we have a rigorous way of dealing with functions locally deﬁned at p,observethatthemap

@ v : f f i 7! @x ✓ i◆p yields the same value for all functions f in a germ f at p.

(k) Furthermore, the above map is linear on M,p. More is 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 ' )0('(p)) = 0, then @ f =0. @x ✓ i◆p

The ﬁrst property says that vi is a point-derivation. 5.3. TANGENT VECTORS AS DERIVATIONS 353

1 As to the second property, when (f ' )0('(p)) = 0, we say that f is stationary at p.

It is easy to check (using the chain rule) that being stationary at p does not depend on the chart, (U,'), at p or on the function chosen in a germ, f.Therefore,the notion of a stationary germ makes sense:

Deﬁnition 5.12. We say that a germ f at p M is a 1 2 stationary germ i↵(f ' )0('(p)) = 0 for some chart, (U,'), at p and some function, f,inthegerm,f.

The Ck-stationary germs form a subring of (k) (but not OM,p an ideal!) denoted (k) . SM,p

Remarkably, it turns out that the set of linear forms on (k) that vanish on (k) is isomorphic to the tangent OM,p SM,p space Tp(M).

First, we prove that this space has @ ,..., @ @x1 p @xn p as a basis. ⇣ ⌘ ⇣ ⌘ 354 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Proposition 5.5. Given any Ck-manifold, M, of dimension n, with k 1, for any p M and any chart 2 (U,') at p, the n functions, @ ,..., @ , de- @x1 p @xn p ﬁned on (k) by ⇣ ⌘ ⇣ ⌘ OM,p

1 @ @(f ' ) f = 1 i n, @x @X   ✓ i◆p i '(p) are linear forms that vanish on (k) . SM,p (k) (k) Every linear form L on M,p that vanishes on M,p can be expressed in a uniqueO way as S n @ L = , i @x i=1 i p X ✓ ◆ where i R. Therefore, the 2 @ ,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 5.3. TANGENT VECTORS AS DERIVATIONS 355 To deﬁne our third version of tangent vectors, we need to deﬁne point-derivations.

Deﬁnition 5.13. Given any Ck-manifold, M,ofdimension n,withk 1, for any p M,aderivation at p 2(k) in M or point-derivation on is a linear form 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. 2O

(k) (k) Let p (M)denotethesetofpoint-derivationson . D OM,p 356 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES As expected, point-derivations vanish on constant functions.

(k) Proposition 5.6. 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 5.7. Given any Ck-manifold, M, of dimension n, with k 1, for any p M, the linear (k) (k) 2 forms on that vanish on are exactly the OM,p SM,p point-derivations on (k) that vanish on (k) . OM,p SM,p 5.3. TANGENT VECTORS AS DERIVATIONS 357 Remarks: Proposition 5.7 says that any linear form on (k) (k) (k) that vanishes on belongs to p (M), the set OM,p SM,p D of point-derivations on (k) . OM,p However, in general, when k = ,apoint-derivationon (k) 6 1 (k) does not necessarily vanish on . OM,p SM,p We will see in Proposition 5.11 that this is true for k = . 1 358 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Here is now our third deﬁnition of a tangent vector.

Deﬁnition 5.14. (Tangent Vectors, Version 3) Given any Ck-manifold, M,ofdimensionn,withk 1, for any p M,atangent vector to M at p is any point- 2 (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- 2 dimensional vector space isomorphic to R and @ d @t x = dt x is a basis vector of Tx(R). For every C k-function, f,locallydeﬁnedatx,wehave

@ df f = = f 0(x). @t dt ✓ ◆x x @ Thus, @t x is: compute the derivative of a function at x. 5.3. TANGENT VECTORS AS DERIVATIONS 359 We now prove the equivalence of version 1 and version 3 of the deﬁnitions of a tangent vector.

Proposition 5.8. Let M be any Ck-manifold of dimension n, with k 1. For any p M, let u be any tangent vector (version 1) given by2 some equivalence class of C1-curves, :] ✏, +✏[ M, through p (i.e., ! p = (0)). Then, the map L deﬁned on (k) by u OM,p

L (f)=(f )0(0) u is a point-derivation that vanishes on (k) . SM,p Furthermore, the map u L deﬁned above is an 7! u isomorphism between Tp(M) and the space of linear forms on (k) that vanish on (k) . OM,p SM,p

We show in the next section that the the space of lin- (k) (k) ear forms on M,p that vanish on M,p is isomorphic to (k) (k) O S (k) (k) ( / )⇤ (the dual of the quotient space / ). OM,p SM,p OM,p SM,p 360 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Even though this is just a restatement of Proposition 5.5, we state the following proposition because of its practical usefulness:

Proposition 5.9. Given any Ck-manifold, M, of dimension n, with k 1, for any p M and any chart (U,') at p, the n tangent vectors,2

@ @ ,..., , @x @x ✓ 1◆p ✓ n◆p form a basis of TpM.

When M is a smooth manifold, things get a little simpler.

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 [36]): 5.3. TANGENT VECTORS AS DERIVATIONS 361

Proposition 5.10. If g : Rn R is a Ck-function ! (k 2) on a convex open U, about p Rn, then for every q U, we have 2 2 n @g g(q)=g(p)+ (q p ) @X i i i=1 ip n X 1 @2g + (q p )(q p ) (1 t) dt. i i j j @X @X i,j=1 Z0 i j (1 t)p+tq X In particular, if g C1(U), then the integral as a 2 function of q is C1.

Proposition 5.11. Let M be any C1-manifold of dimension n. For any p M, any point-derivation on ( ) ( ) 2 M,p1 vanishes on M,p1 , the ring of stationary germs. O S ( ) Consequently, T (M)= p1 (M). p D 362 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Proposition 5.11 shows that in the case of a smooth manifold, in Deﬁnition 5.14, we can omit the requirement that point-derivations vanish on stationary germs, since this is automatic.

Remark: In the case of smooth manifolds (k = ) some authors, including Morita [29] (Chapter 1, Defini-1 tion 1.32) and O’Neil [31] (Chapter 1, Definition 9), define derivations as linear derivations with domain 1(M), the set of all smooth funtions on the entire manifold,C M.

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 a point-derivation at p,thenv(f)=v(g)wheneverf,g 1(M)agreelo- cally at p. 2C 5.3. TANGENT VECTORS AS DERIVATIONS 363 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 be proved that this property is local because on smooth manifolds, “bump functions” exist (see Section 7.1, Proposition 7.2).

Unfortunately, this argument breaks down for Ck-manifolds with k< and in this case the ring of germs at p can’t be avoided.1 364 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES

5.4 Tangent and Cotangent Spaces Revisited ~

(k) (k) The space of linear forms on M,p that vanish on M,p turns out to be isomorphic toO the dual of the quotientS (k) (k) space / ,andthisfactshowsthatthedual(T M)⇤ OM,p SM,p p of the tangent space TpM,calledthecotangent space to M at p,canbeviewedasthequotientspace (k) / (k) . OM,p SM,p This provides a fairly intrinsic deﬁnition of the cotangent space to M at p.Fornotationalsimplicity,wewriteTp⇤M instead of (TpM)⇤.

(k) As the subspace of linear forms on M,p that vanish on (k) (k) O (k) is isomorphic to the dual ( / )⇤ of the space SM,p OM,p SM,p (k) / (k) ,weseethatthelinearforms OM,p SM,p @ @ ,..., @x @x ✓ 1◆p ✓ n◆p (k) (k) also form a basis of ( / )⇤. OM,p SM,p 5.4. TANGENT AND COTANGENT SPACES REVISITED ~ 365 There is a conceptually clearer way to deﬁne a canonical isomorphism 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 [34] (Chapter III, Section 8) for analytic manifolds and can be adapted to our situation.

(k) Deﬁne the map, ! : Tp(M) R,sothat ⇥OM,p !

!([], f)=(f )0(0), for all [] T (M)andallf (k) (with f f). 2 p 2OM,p 2 It is easy to check that the above expression does not depend on the representatives chosen in the equivalences classes []andf and that ! is bilinear.

However, as deﬁned, ! is degenerate because !([], f)=0 if f is a stationary germ. 366 CHAPTER 5. 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

!([], [f]) = (f )0(0),

(k) (k) where [] Tp(M)and[f] M,p/ M,p,whichwealso 2 (k) 2(kO) S denote ! : Tp(M) ( / ) R. ⇥ OM,p SM,p !

Proposition 5.12. The map (k) (k) ! : Tp(M) ( / ) R deﬁned so that ⇥ OM,p SM,p !

!([], [f]) = (f )0(0),

(k) (k) for all [] Tp(M) and all [f] M,p/ M,p, is a non- degenerate2 pairing (with f f2). O S 2 Consequently, there is a canonical isomorphism be- (k) (k) tween Tp(M) and ( M,p/ M,p)⇤ and a canonical iso- O S (k) (k) morphism between T ⇤(M) and / . p OM,p SM,p 5.4. TANGENT AND COTANGENT SPACES REVISITED ~ 367

In view of Proposition 5.12, 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, ! 2 v v, where v(f)=f(v), for all f E⇤ 7! 2 is a linear isomorphism. e e Observe that we can view !(u, f)=!([], [f]) as the result of computing the directional derivative of the locally defined function f f in the direction u (given by a curve ). 2

Proposition 5.12 also suggests the following definition: 368 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Definition 5.15. Given any Ck-manifold, M,ofdimension n,withk 1, for any p M,thetangent space 2 at p,denotedTp(M), is the space of point-derivations on (k) that vanish on (k) . OM,p SM,p (k) (k) Thus, T (M)canbeidentifiedwith( / )⇤. p OM,p SM,p The space (k) / (k) is called the cotangent space at p; OM,p SM,p it is isomorphic to the dual Tp⇤(M), of Tp(M).

Observe that if x = pr ',as i i @ x = , @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 ⇣ ⌘ ⇣ ⌘ 5.4. TANGENT AND COTANGENT SPACES REVISITED ~ 369 Given any Ck-function f,onU,wedenotetheimageof (k) (k) f in T ⇤(M)= / by df . p OM,p SM,p p This is the di↵erential 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)deﬁnedby

df (v)=v(f), for all v T (M). p 2 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 ⇣ ⌘ ⇣ ⌘ For simplicity of notation, we often omit the subscript p unless confusion arises. 370 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Remark: Strictly speaking, a tangent vector, v T (M), 2 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:2C set

v(f)=v(f).

Given any chart, (U,'), at p,sincev can be written in a unique way as n @ v = , i @x i=1 i p X ✓ ◆ we get

n @ v(f)= f. i @x i=1 i p X ✓ ◆ This shows that v(f)isthedirectional derivative of f in the direction v. 5.4. TANGENT AND COTANGENT SPACES REVISITED ~ 371 ( ) It is also possible to deﬁne T (M)justintermsof 1 . p OM,p ( ) Let m 1 be the ideal of germs that vanish at p. M,p ✓OM,p 2 Then, we also have the ideal mM,p,whichconsistsofall ﬁnite linear combinations of products of two elements in mM,p,anditturnsoutthatTp⇤(M)isisomorphicto 2 mM,p/mM,p (see Warner [36], Lemma 1.16).

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 proposition: 372 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Proposition 5.13. 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 = ,Proposition5.10showsthateverystation- 1 2 ary germ that vanishes at p belongs to mM,p.

Therefore, when k = ,wehave 1 ( ) 2 sM,p1 = mM,p and so, we obtain the result quoted above (from Warner):

( ) ( ) 2 T ⇤(M)= 1 / 1 = m /m . p OM,p SM,p ⇠ M,p M,p 5.4. TANGENT AND COTANGENT SPACES REVISITED ~ 373 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 [36] (Lemma 1.16). (k) (2) The ideal mM,p is in fact the unique maximal ideal of (k) . OM,p 374 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES

This is because if f (k) does not vanish at p,then 2OM,p 1/f belongs to (k) ,andanyproperidealcontaining OM,p m(k) and f would be equal to (k) ,whichisabsurd. M,p OM,p

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. 5.5. TANGENT MAPS 375 5.5 Tangent Maps

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. ! 2 We ﬁnd it convenient to use Version 3 of the deﬁnition of atangentvector.So,letu T (M)beapoint-derivation 2 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 2ON,h(p) 2 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 2 376 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES The germ of all locally defined functions at p of the form g h,withg g,willbedenotedg h. 2 Then, we set dh (u)(g)=u(g h). p

Moreover, if g is a stationary germ at h(p), then for some chart, (V, )onN at q = h(p), we have 1 (g )0( (q)) = 0 and, for some chart (U,')atp on M,weget

1 1 1 (g h ' )0('(p)) = (g )( (q))(( h ' )0('(p))) =0, which means that g h is stationary at p.

Therefore, dhp(u) Th(p)(M). It is also clear that dhp is alinearmap. 2 5.5. TANGENT MAPS 377 Deﬁnition 5.16. Given any two Ck-manifolds, M and N,ofdimensionm and n,respectively,foranyCk-map, h: M N and for every p M,thedi↵erential of h at p or tangent! map dh : T (M2) T (N)(alsodenoted p p ! h(p) Tph: Tp(M) Th(p)(N)), is the linear map deﬁned so that !

dh (u)(g)=T h(u)(g)=u(g h), p p for every u T (M)andeverygerm,g (k) . 2 p 2ON,h(p)

The linear map dhp (= Tph)issometimesdenotedhp0 or Dph.

The chain rule is easily generalized to manifolds.

Proposition 5.14. Given any two Ck-maps f : M N and g : N P between smooth Ck- manifolds,! for any p M!, we have 2 d(g f) = dg df . p f(p) p 378 CHAPTER 5. 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 Tpf is explicitly. Since N = R,germs(offunctionsonR)att0 = f(p)arejustgerms k of C -functions, g : R R,locallydeﬁnedatt0. ! Then, for any u T (M)andeverygermg at t , 2 p 0 T f(u)(g)=u(g f). p If we pick a chart, (U,'), 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 Tpf(u)(g)is when u = @ . @xi p ⇣ ⌘ 5.5. TANGENT MAPS 379 In this case,

1 @ @(g f ' ) T f (g)= . p @x @X ✓ i◆p! i '(p) Using the chain rule, we ﬁnd that @ @ dg Tpf (g)= f . @xi @xi dt ✓ ◆p! ✓ ◆p t0 Therefore, we have d T f(u)=u(f) . p dt t0 This shows that we can identify Tpf with the linear form in Tp⇤(M)deﬁnedby

df (u)=u(f),uT M, p 2 p by identifying Tt0R with R. 380 CHAPTER 5. 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

Proposition 5.15. Given any Ck-manifold, M, of dimension n, with k 1, for any p M and any chart (U,') at p, the nlinear maps, 2

(dx1)p,...,(dxn)p, form a basis of Tp⇤M, where (dxi)p, the di↵erential of xi at p, is identified with the linear form in Tp⇤M such that (dx ) (v)=v(x ), for every v T M i p i 2 p (by identifying TR with R). 5.5. TANGENT MAPS 381 In preparation for the definition of the flow of a vector field (which will be needed to define the exponential map in Lie group theory), we need to define the tangent vector to a curve on a manifold.

Given a Ck-curve, :]a, b[ M,onaCk-manifold, M, for any t ]a, b[, wewouldliketodeﬁnethetangent! 0 2 vector to the curve at t0 as a tangent vector to M at p = (t0).

We do this as follows: Recall that d is a basis vector dt t0 of T ( )= . t0 R R So, deﬁne the tangent vector to the curve at t0,de- d noted ˙ (t0)(or0(t0), or dt (t0)), by

d ˙ (t )=d . 0 t0 dt t0! 382 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES We ﬁnd it necessary to deﬁne curves (in a manifold) whose domain is not an open interval.

Amap, :[a, b] M,isaCk-curve in M if it is the restriction of some!Ck-curve, :]a ✏, b + ✏[ M,for some (small) ✏>0. ! e Note that for such a curve (if k 1) the tangent vector, ˙ (t), is deﬁned for all t [a, b]. 2 Acontinuouscurve, :[a, b] M,ispiecewise Ck i↵ ! there a sequence, a0 = a, a1,...,am = b,sothatthe k restriction, i,of to each [ai,ai+1]isaC -curve, for i =0,...,m 1.

This implies that i0(ai+1)andi0+1(ai+1)aredeﬁnedfor i =0,...,m 1, but there may be a jump in the tangent vector to at ai+1,thatis,wemayhave 0(a ) = 0 (a ). i i+1 6 i+1 i+1 5.5. TANGENT MAPS 383 Sometime, especially in the case of a linear Lie group, it is more convenient to deﬁne the tangent map in terms of Version 1 of a tangent vector.

Given any Ck-map h: M N,foreveryp M,itis easy to show that for every! equivalence class 2u =[]of curves through p in M,allcurvesoftheformh (with u)throughh(p)inN belong to the same equivalence class,2 and can make the following deﬁnition.

Deﬁnition 5.17. (Using Version 1 of a tangent vector) Given any two Ck-manifolds M and N,ofdimensionm and n respectively, for any Ck-map h: M N and for every p M,thedi↵erential of h at p or !tangent map dh : T (2M) T (N)(alsodenotedT h: T (M) p p ! h(p) p p ! Th(p)(N)), is the linear map deﬁned such that for every equivalence class u =[]ofcurves in M with (0) = p,

dhp(u)=Tph(u)=v, where v is the equivalence class of all curves through h(p) in N of the form h ,with u. 2 384 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES

If M is a manifold in RN1 and N is a manifold in RN2 (for N1 some N1,N2 1), then 0(0) R and (h )0(0) N2 2 2 R ,sointhiscasethedeﬁnitionofdhp = Thp is just Deﬁnition 2.18; namely, for any curve in M such that (0) = p and 0(0) = u,

dh (u)=Th (u)=(h )0(0). p p For example, consider the linear Lie group SO(3), pick any vector u R3,andletf : SO(3) R3 be given by 2 ! f(R)=Ru, R SO(3). 2

3 To compute df R : TRSO(3) TRuR ,sinceTRSO(3) = 3 3 ! Rso(3) and TRuR = R ,pickanytangentvectorRB 2 Rso(3) = TRSO(3) (where B is any 3 3skewsymmetric matrix), let (t)=RetB be the curve⇥ through R such that 0(0) = RB,andcompute

tB df R(RB)=(f((t)))0(0) = (Re u)0(0) = RBu. 5.5. TANGENT MAPS 385 Therefore, we see that

df (X)=Xu, X T SO(3) = Rso(3). R 2 R

If we express the skew symmetric matrix B so(3) as 2 B = ! for some vector ! R3,thenwehave ⇥ 2

df R(R! )=R! u = R(! u). ⇥ ⇥ ⇥

Using the isomorphism of the Lie algebras (R3, )and ⇥ so(3), the tangent map df R is given by

df (R!)=R(! u). R ⇥ 386 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Here is another example inspired by an optimization problem investigated by Taylor and Kriegman.

Pick any two vectors u, v R3,andletf : SO(3) R be the function given by 2 !

2 f(R)=(u>Rv) .

To compute df R : TRSO(3) Tf(R)R,sinceTRSO(3) = ! Rso(3) and Tf(R)R = R,againpickanytangentvector RB Rso(3) = TRSO(3) (where B is any 3 3skew symmetric2 matrix), let (t)=RetB be the curve⇥ through R such that 0(0) = RB,andcompute

df R(RB)=(f((t)))0(0) tB 2 =((u>Re v) )0(0)

= u>RBvu>Rv + u>Rvu>RBv

=2u>RBvu>Rv. 5.5. TANGENT MAPS 387 Therefore,

df (X)=2u>Xvu>Rv, X Rso(3). R 2 Unlike the case of functions defined on vector spaces, in order to define the gradient of f,afunctiondefinedon SO(3), a “nonflat” manifold, we need to pick a Rieman- nian metric on SO(3).

We will explain how to do this in Chapter 11. 388 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES 5.6 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 di↵erent 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 ,hasforitsopensetsallsubsetsoftheform Y U,whereU is an arbitary open subset of X.). \ 5.6. SUBMANIFOLDS, IMMERSIONS, EMBEDDINGS 389 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 | {z } (x ,...,x ) (x ,...,x , 0,...,0). 1 m 7! 1 m n m k Deﬁnition 5.18. Given a C -manifold,| {zM},ofdimension n,asubset,N,ofM is an m-dimensional submanifold of M (where 0 m n)i↵foreverypoint,p N, there is a chart (U,')ofM(in the maximal atlas for2M), with p U,sothat 2 m '(U N)='(U) (R 0n m ). \ \ ⇥{ }

(We write 0n m =(0,...,0).) n m | {z } The subset, U N,ofDefinition5.18issometimescalled a slice of (U,'\)andwesaythat(U,')isadapted to N (See O’Neill [31] or Warner [36]). 390 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Other authors, including Warner [36], use the term submanifold in a broader sense than us and they use the word embedded submanifold for what is defined in Defi- nition 5.18.

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

Proposition 5.16. Given a Ck-manifold, M, of dimension n, for any submanifold, N, of M of dimension m n, the family of pairs (U N,' U N), where (U,') ranges over the charts over\ any atlas\ for M, is an atlas for N, where N 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: 5.6. SUBMANIFOLDS, IMMERSIONS, EMBEDDINGS 391 Proposition 5.17. Given a Ck-manifold, M, of dimension n and a submanifold, N, of M of dimension m n, for any p N and any chart, (W,⌘), of N at p, there is some chart,2 (U,'), of M at p so that

m '(U N)='(U) (R 0n m ) \ \ ⇥{ } and

' U N = ⌘ U N, \ \ where p U N W . 2 \ ✓

It is also useful to deﬁne more general kinds of “submanifolds.” 392 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Deﬁnition 5.19. Let h: N M be a Ck-map of manifolds. !

(a) The map h is an immersion of N into M i↵ dhp is injective for all p N. 2 (b) The set h(N)isanimmersed submanifold of M i↵ h is an injective immersion. (c) The map h is an embedding of N into M i↵ h is an injective immersion such that the induced map, N h(N), is a homeomorphism, where h(N)is given! the subspace topology (equivalently, h is an open map from N into h(N)withthesubspacetopology). We say that h(N)(withthesubspacetopology) is an embedded submanifold of M.

(d) The map h is a submersion of N into M i↵ dhp is surjective for all p N. 2 5.6. SUBMANIFOLDS, IMMERSIONS, EMBEDDINGS 393 Again, we warn our readers that certain authors (such as Warner [36]) call h(N), in (b), a submanifold of M! We prefer the terminology immersed submanifold.

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 necessarily 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. 394 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES

Immersions of R into R3 are parametric curves and immersions of R2 into R3 are parametric surfaces. These have been extensively studied, for example, see DoCarmo [11], Berger and Gostiaux [3] or Gallier [16].

Immersions (i.e., subsets of the form h(N), where N is an immersion) are generally neither injective immersions (i.e., subsets of the form h(N), where N is an injective immersion) nor embeddings (or submanifolds).

For example, immersions can have self-intersections, as the plane curve (nodal cubic) shown in Figure 5.3 and given by: x = t2 1; y = t(t2 1). 1

Figure 5.3: A nodal cubic; an immersion, but not an immersed submanifold. 5.6. SUBMANIFOLDS, IMMERSIONS, EMBEDDINGS 395 Injective immersions are generally not embeddings (or submanifolds) because h(N)maynotbehomeomorphic to N.

An example is given by the Lemniscate of Bernoulli shown in Figure 5.4, an injective immersion of R into R2:

t(1 + t2) x = , 1+t4 t(1 t2) y = . 1+t4

Figure 5.4: Lemniscate of Bernoulli; an immersed submanifold, but not an embedding.

When t =0,thecurvepassesthroughtheorigin.

When t ,thecurvetendstotheoriginfromthe left and from7! 1 above, and when t + ,thecurvetends tends to the origin from the right7! and1 from above. 396 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Therefore, the inverse of the map deﬁning the Lemniscate of Bernoulli is not continuous at the origin.

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), 7! where c R. 2 One can show that the image of R under this immersion is closed in T 2 i↵ c is rational. Moreover, the image of this immersion is dense in T 2 but not closed i↵ c is irrational. 5.6. SUBMANIFOLDS, IMMERSIONS, EMBEDDINGS 397

The above example can be adapted to the torus in R3: One can show that the immersion given by

t ((2 + cos t)cos(p2 t), (2 + cos t)sin(p2 t), sin t), 7! is dense but not closed in the torus (in R3)givenby

(s, t) ((2 + cos s)cost, (2 + cos s)sint, sin s), 7! where s, t R. 2 There is, however, a close relationship between submanifolds and embeddings. 398 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Proposition 5.18. If N is a submanifold of M, then the inclusion map, j : N M, is an embedding. Con- versely, if h: N M !is an embedding, then h(N) with the subspace! topology is a submanifold of M and h is a di↵eomorphism between N and h(N).

In summary, embedded submanifolds and (our) submanifolds coincide.

Some authors refer to spaces of the form h(N), where h 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. 5.7. MANIFOLDS WITH BOUNDARY ~ 399 5.7 Manifolds With Boundary ~

Up to now, we have deﬁned manifolds locally di↵eomorphic 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 di↵eomorphic to open sets in Rm.

Perhaps the simplest example is the (closed) upper half space,

m m H = (x1,...,xm) R xm 0 . { 2 | } 400 CHAPTER 5. 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 ⇥{ } !

m m @H = x H xm =0 { 2 | }

m 1 m is isomorphic to R and is called the boundary of H . We also deﬁne the interior of Hm as

m m m Int(H )=H @H .

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. e e e e e e e e 5.7. MANIFOLDS WITH BOUNDARY ~ 401 1 We say that f is a (smooth) di↵eomorphism i↵ 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 5.1 and Definition 5.2.

So, for instance, given a topological space, M,achart is now pair, (U,'), where U is an open subset of M and ': U ⌦isahomeomorphismontoanopensubset, ! n' ⌦='(U), of H (for some n' 1), etc. 402 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Deﬁnition 5.20. Given some integer n 1andgiven some k such that k is either an integer k 1ork = , a Ck-manifold of dimension n with boundary consists1 of a topological space, M,togetherwithanequivalence class, ,ofCk n-atlases, on M (where the charts are now A deﬁned in terms of open subsets of Hn).

Any atlas, ,intheequivalenceclass is called a di↵erentiable structureA of class Ck (andA dimension n) on M.WesaythatM is modeled on Hn.

When k = ,wesaythatM is a smooth manifold with boundary. 1

It remains to deﬁne what is the boundary of a manifold with boundary! 5.7. MANIFOLDS WITH BOUNDARY ~ 403 By deﬁnition, the boundary, @M,ofamanifold(with boundary), M,isthesetofallpoints,p M,such 2 that there is some chart, (U↵,'↵), with p U↵ and n 2 '↵(p) @H . 2 We also let Int(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 di↵erent.

Note that manifolds as deﬁned earlier (In Deﬁnition 5.3) are also manifolds with boundary: their boundary is just empty. 404 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES 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, p2 @M. 2 The only notion that requires more care is that of a submanifold. For more on this, see Hirsch [19], Chapter 1, Section 4.

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). ⇥ 5.7. MANIFOLDS WITH BOUNDARY ~ 405 There is a generalization of the notion of a manifold with boundary called manifold with corners and such manifolds are closed under products (see Hirsch [19], 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 5.19. If M is a manifold with boundary of dimension n, for any p @M on the boundary on M, for any chart, (U,')2, with p M, we have 2 '(p) @Hn. 2

Using Proposition 5.19, we immediately derive the fact that @M is a manifold of dimension n 1. 406 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES