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Chapter 6

Manifolds, Tangent Spaces, Cotangent

Spaces, Vector Fields, Flow, Integral

Curves

6.1 Manifolds

In a previous Chapter we defined the notion of a manifoldembedded 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 conceptof a manifold to spaces that are not a priori embedded insome RN.

The basic idea is still that, whatever a manifold is, it is

a topological space that can be covered by a collection ofopen subsets, U, where each U is isomorphic to somestandard model, e.g., some open subset of Euclideanspace, Rn.

305

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306 CHAPTER 6. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES

Of course, manifolds would be very dull without functionsdefined on them and between them.

This is a general fact learned from experience: Geometryarises not just from spaces but from spaces and interestingclasses of functions between them.

In particular, we still would like to do calculus on ourmanifold and have good notions of curves, tangent vec-

tors, differential forms, etc.

The small drawback with the more general approach isthat 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

, so it itnot possible to define tangent vectors in a simple-mindedway using derivatives.

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6.1. MANIFOLDS 307

Instead, we have to resort to the notion of chart. This isnot such a strange idea.

For example, a geography atlas gives a set of maps of var-ious portions of the earh and this provides a very gooddescription of what the earth is, without actually imag-ining the earth embedded in 3-space.

Given Rn, recall that the projection functions,

pri:Rn R, are defined by

pri(x1, . . . , xn) =xi, 1in.

For technical reasons, from now on, all topological spacesunder consideration will be assumed to be Hausdorff and

second-countable (which means that the topology has acountable basis).

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Definition 6.1.1Given a topological space,M, achart(or local coordinate map) is a pair, (U, ), where U isan open subset ofMand: U is a homeomorphism

onto an open subset, =(U), ofRn (for somen1).

For anyp M, a chart, (U, ), is achart atpiffpU.If (U, ) is a chart, then the functions xi = pri arecalled local coordinatesand for every p U, the tuple

(x1(p), . . . , xn(p)) is the set ofcoordinates ofpw.r.t. thechart.

The inverse, (, 1), of a chart is called alocal parametrization.

Given any two charts, (U1, 1) and (U2, 2), ifU1 U2=, we have thetransition maps,ji : i(Ui Uj)j(Ui Uj) andij: j(Ui Uj)i(Ui Uj), defined by

ji =j 1i and

ij =i

1j .

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6.1. MANIFOLDS 309

Clearly,ij = (ji )

1.

Observe that the transition mapsj

i (resp. i

j) are maps

betweenopen subsets ofRn.

This is good news! Indeed, the whole arsenal of calculusis available for functions on Rn, and we will be able topromote many of these results to manifolds by imposingsuitable conditions on transition functions.

Definition 6.1.2 Given a topological space, M, andany two integers, n 1 and k 1, a Ck n-atlas (orn-atlas of classCk),A, is a family of charts,{(Ui, i)},such that

(1)i(Ui) Rn for alli;(2) TheUi coverM, i.e.,

M=

i

Ui;

(3) Whenever Ui Uj = , the transition map ji (and

ij) is aCk-diffeomorphism.

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We must insure that we have enough charts in order tocarry out our program of generalizing calculus on Rn tomanifolds.

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

Technically, given a Ck n-atlas,A, on M, for any other

chart, (U, ), we say that (U, ) is compatiblewith thealtas A iff every mapi1 and

1i isC

k (wheneverU Ui=).

Two atlases A and A on M are compatible iff everychart of one is compatible with the other atlas.

This is equivalent to saying that the union of the twoatlases is still an atlas.

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6.1. MANIFOLDS 311

It is immediately verified that compatibility induces anequivalence relation onCk n-atlases onM.

In fact, given an atlas, A, for M, the collection,A, ofall charts compatible with A is a maximal atlas in theequivalence class of charts compatible withA.

Definition 6.1.3Given any two integers, n 1 andk1, aCk-manifold of dimensionnconsists of a topo-logical space, M, together with an equivalence class, A,ofCk n-atlases, onM. Any atlas,A, in the equivalenceclass A is called a differentiable structure of class Ck

(and dimensionn) onM. We say that Mismodeled onR

n. Whenk =, we say thatM is asmooth manifold.

Remark: It might have been better to use the terminol-ogyabstract manifoldrather than manifold, to empha-size the fact that the space Mis not a priori a subspaceofRN, for some suitableN.

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We can allow k = 0 in the above definitions. Condition(3) in Definition 6.1.2 is void, since aC0-diffeomorphismis just a homeomorphism, but ji is always a homeomor-

phism.

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

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 ofM, it can beshown that the dimension,n, of the range of every chartis the same. This is quite easy to show ifk 1 but for

k= 0, this requires a deep theorem of Brouwer.

What happens ifn = 0? In this case, every one-pointsubset ofMis open, so every subset ofMis open, i.e.,Mis any (countable if we assumeMto be second-countable)set with the discrete topology!

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6.1. MANIFOLDS 313

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

Remark: In some cases,Mdoes not come with a topol-ogy in an obvious (or natural) way and a slight variationof Definition 6.1.2 is more convenient in such a situation:

Definition 6.1.4Given a set,M, and any two integers,n 1 andk1, aCk n-atlas(orn-atlas of classCk),

A, is a family of charts, {(Ui, i)}, such that(1) Each Ui is a subset ofM and i: Ui i(Ui) is a

bijection onto an open subset, i(Ui) Rn, for alli;

(2) TheUi coverM, i.e.,

M= i Ui;(3) Whenever Ui Uj = , the set i(Ui Uj) is open

in Rn and the transition map ji (and ij) is a C

k-diffeomorphism.

Then, the notion of a chart being compatible with an

atlas and of two atlases being compatible is just as beforeand we get a new definition of a manifold, analogous toDefinition 6.1.3.

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But, this time, we give M the topology in which theopen sets are arbitrary unions of domains of charts, Ui,more precisely, theUis of the maximal atlas defining the

differentiable structure onM.

It is not difficult to verify that the axioms of a topologyare verified andMis indeed a topological space with thistopology.

It can also be shown that when M is equipped with theabove topology, then the mapsi: Uii(Ui) are home-omorphisms, soM is a manifold according to Definition6.1.3. Thus, we are back to the original notion of a man-ifold where it is assumed that Mis already a topologicalspace.

One can also define the topology on M in terms of anythe atlases, A, definingM(not only the maximal one) byrequiringUMto be open iffi(U Ui) is open in Rn,for every chart, (Ui, i), in the altasA. This topology isthe same as the topology induced by the maximal atlas.

We also requireMto be Hausdorff and second-countablewith this topology. IfMhas a countable atlas, then it issecond-countable

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N(x1, . . . , xn+1) = 1

1 xn+1(x1, . . . , xn)

and

S(x1, . . . , xn+1) = 1

1 +xn+1(x1, . . . , xn).

The inverse stereographic projections are given by

1N (x1, . . . , xn) =

1

(ni=1 x2i ) + 1(2x1, . . . , 2xn, (n

i=1x2i ) 1)

and

1S (x1, . . . , xn) =

1

(ni=1 x2i ) + 1(2x1, . . . , 2xn, (n

i=1 x2i ) + 1).Thus, if we let UN = S

n {N} and US = Sn {S},we see thatUNandUSare two open subsets coveringS

n,both homeomorphic to Rn.

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6.1. MANIFOLDS 317

Furthermore, it is easily checked that on the overlap,UN US=Sn {N, S}, the transition maps

S 1

N =N 1

S

are given by

(x1, . . . , xn) 1n

i=1 x2i

(x1, . . . , xn),

that is, the inversion of centerO= (0, . . . , 0) and power

1. Clearly, this map is smooth on Rn

{O}, so we con-clude that (UN, N) and (US, S) form a smooth atlas forSn.

Example 2. The projective space RPn.

To define an atlas onRP

n

it is convenient to viewRP

n

as the set of equivalence classes of vectors in Rn+1 {0}modulo the equivalence relation,

u v iff v=u, for some = 0 R.

Given anyp= [x1, . . . , xn+1] RPn, we call (x1, . . . , xn+1)

thehomogeneous coordinatesofp.

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It is customary to write (x1: : xn+1) instead of[x1, . . . , xn+1]. (Actually, in most books, the indexingstarts with 0, i.e., homogeneous coordinates for RPn are

written as (x0: : xn).)

For anyi, with 1 in+ 1, let

Ui={(x1: : xn+1) RPn |xi= 0}.

Observe thatUi is well defined, because if

(y1: : yn+1) = (x1: : xn+1), then there is some= 0so thatyi=xi, fori= 1, . . . , n+ 1.

We can define a homeomorphism, i, ofUi onto Rn, as

follows:

i(x1: : xn+1) = x1xi , . . . ,xi1xi ,xi+1xi , . . . ,xn+1xi ,where the ith component is omitted. Again, it is clearthat this map is well defined since it only involves ratios.

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6.1. MANIFOLDS 319

We can also define the maps, i, from Rn toUi RP

n,given by

i(x1, . . . , xn) = (x1: : xi1: 1: xi: : xn),where the 1 goes in the ith slot, fori= 1, . . . , n+ 1.

One easily checks thati andi are mutual inverses, sothe i are homeomorphisms. On the overlap, UiUj,(wherei=j), asxj = 0, we have

(j 1i )(x1, . . . , xn) =

x1xj

, . . . ,xi1

xj,

1

xj,xixj

, . . . ,xj1

xj,xj+1

xj, . . . ,

xnxj

.

(We assumed that i < j; the casej < i is similar.) This isclearly a smooth function fromi(Ui Uj) toj(Ui Uj).

As the Ui cover RPn, see conclude that the (Ui, i) are

n+ 1 charts making a smooth atlas for RPn.

Intuitively, the space RPn is obtained by glueing the opensubsetsUi on their overlaps. Even for n= 3, this is not

easy to visualize!

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Example 3. The GrassmannianG(k, n).

Recall that G(k, n) is the set of all k-dimensional linear

subspaces ofRn, also calledk-planes.

Every k-plane,W, is the linear span ofklinearly indepen-dent vectors, u1, . . . , uk, in R

n; furthermore, u1, . . . , ukand v1, . . . , vkboth span Wiff there is an invertible kk-matrix, = (ij), such that

vi=k

j=1

ijuj, 1ik.

Obviously, there is a bijection between the collection ofklinearly independent vectors,u1, . . . , uk, in R

n and thecollection ofn k matrices of rankk.

Furthermore, two nk matrices A and B of rank krepresent the samek-plane iff

B =A, for some invertiblek k matrix, .

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6.1. MANIFOLDS 321

(Note the analogy with projective spaces where two vec-tors u, v represent the same point iffv = u for someinvertible R.)

We can define the domain of charts (according to Def-inition 6.1.4) on G(k, n) as follows: For every subset,S = {i1, . . . , ik} of{1, . . . , n}, let USbe the subset ofn k matrices, A, of rank k whose rows of index inS = {i1, . . . , ik} forms an invertible kk matrix de-

notedAS.

Observe that the k k matrix consisting of the rows ofthe matrixAA1S whose index belong toSis the identitymatrix,Ik.

Therefore, we can define a map, S: US R(nk)k

,whereS(A) = the (n k) kmatrix obtained by delet-ing the rows of index in S fromAA1S .

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We need to check that this map is well defined, i.e., thatit does not depend on the matrix,A, representingW.

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

IfB =A, with invertible, if we write

A= A1A2 and B = B1B2 ,as B = A, we get B1 = A1 and B2 = A2, fromwhich we deduce that

B1B2

B11 =

Ik

B2B11

=

IkA2

1A11= Ik

A2A11

= A1A2A11 .

Therefore, our map is indeed well-defined.

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6.1. MANIFOLDS 323

It is clearly injective and we can define its inverse,S, asfollows: LetSbe the permutation of{1, . . . , n} swaping{1, . . . , k} and Sand leaving every other element fixed

(i.e., ifS={i1, . . . , ik}, thenS(j) =ij andS(ij) =j,forj = 1, . . . , k).

IfPS is the permutation matrix associated with S, forany (n k) k matrix,M, let

S(M) =PSIkM .The effect of S is to insert into M the rows of theidentity matrixIk as the rows of index from S.

At this stage, we have charts that are bijections from

subsets,US, ofG(k, n) to open subsets, namely,R

(nk)k

.

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Then, the reader can check that the transition mapT

1S fromS(US UU) toT(US UU) is given by

M (C+DM)(A+BM)1

,where

A BC D

=PTPS,

is the matrix of the permutation TS(this permutationshufflesSandT).

This map is smooth, as it is given by determinants, andso, the charts (US, S) form a smooth atlas for G(k, n).

Finally, one can check that the conditions of Definition6.1.4 are satisfied, so the atlas just defined makesG(k, n)

into a topological space and a smooth manifold.Remark: The reader should have no difficulty provingthat the collection of k-planes represented by matricesin US is precisely set ofk-planes, W, supplementary tothe (n k)-plane spanned by the n k canonical basisvectors ejk+1, . . . , ejn (i.e., span(W {ejk+1, . . . , ejn}) =

Rn, whereS={i1, . . . , ik}and{jk+1, . . . , jn}={1, . . . , n} S).

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6.1. MANIFOLDS 325

Example 4. Product Manifolds.

LetM1andM2be twoCk-manifolds of dimensionn1and

n2, respectively.

The topological space,M1 M2, with the product topol-ogy (the opens ofM1 M2are arbitrary unions of sets ofthe formU V, where U is open in M1 andV is openin M2) can be given the structure of a C

k-manifold of

dimensionn1+n2 by defining charts as follows:

For any two charts, (Ui, i) on M1 and (Vj, j) on M2,we declare that (Ui Vj, i j) is a chart onM1 M2,wherei j: Ui Vj R

n1+n2 is defined so that

i j(p, q) = (i(p), j(q)), for all (p, q)Ui Vj.

We defineCk-maps between manifolds as follows:

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Definition 6.1.5Given any twoCk-manifolds,M andN, of dimension m and n respectively, a Ck-map if acontinuous functions,h: MN, so that for everypM, there is some chart, (U, ), atpand some chart,(V, ), atq=h(p), withf(U)V and

h 1: (U)(V)

aCk-function.

It is easily shown that Definition 6.1.5 does not depend onthe choice of charts. In particular, ifN = R, we obtainaCk-function onM.

One checks immediately that a function, f: M R, is aCk-map iff for every pM, there is some chart, (U, ),atpso that

f 1: (U) R

is aCk-function.

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6.1. MANIFOLDS 327

IfUis an open subset ofM, set ofCk-functions onU isdenoted byCk(U). In particular, Ck(M) denotes the setofCk-functions on the manifold,M. Observe that Ck(U)is a ring.

On the other hand, if M is an open interval of R, sayM=]a, b[, then: ]a, b[ Nis called aCk-curve inN.One checks immediately that a function,: ]a, b[ N, isaCk-map iff for everyqN, there is some chart, (V, ),

atqso that : ]a, b[ (V)

is aCk-function.

It is clear that the composition ofCk-maps is aCk-map.ACk-map,h: M N, between two manifolds is a Ck-

diffeomorphism iffhhas an inverse, h1: N M (i.e.,h1h= idMandhh1 = idN), and bothhandh1 areCk-maps (in particular,h and h1 are homeomorphisms).Next, we define tangent vectors.

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6.2 Tangent Vectors, Tangent Spaces,

Cotangent Spaces

Let M be a Ck

manifold of dimension n, with k 1.The most intuitive method to define tangent vectors is touse curves. LetpMbe any point onMand let: ] , [ M be a C1-curve passing through p, thatis, with (0) = p. Unfortunately, if M is not embed-ded in any RN, the derivative(0) does not make sense.

However, for any chart, (U, ), at p, the map is aC1-curve in Rn and the tangent vector v = ()(0)is well defined. The trouble is that different curves mayyield the samev!

To remedy this problem, we define an equivalence relationon curves throughpas follows:

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6.2. TANGENT VECTORS, TANGENT SPACES, COTANGENT SPACES 329

Definition 6.2.1Given a Ck manifold, M, of dimen-sionn, for anyp M, twoC1-curves,1: ] 1, 1[ Mand 2: ]2, 2[M, throughp (i.e.,1(0) =2(0) =p)areequivalentiff there is some chart, (U, ), atpso that

( 1)(0) = ( 2)

(0).

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

case.

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

Definition 6.2.2(Tangent Vectors, Version 1)GivenanyCk-manifold,M, of dimensionn, withk 1, for anyp M, a tangent vector to M atp is any equivalenceclass ofC1-curves through p on M, modulo the equiva-lence relation defined in Definition 6.2.1. The set of alltangent vectors atpis denoted byTp(M).

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It is obvious thatTp(M) is a vector space.

We will show that Tp(M) is a vector space of dimension

n= dimension ofM.

One should observe that unless M = Rn, in which case,for anyp, q Rn, the tangent space Tq(M) is naturallyisomorphic to the tangent spaceTp(M) by the translationq p, for an arbitrary manifold, there is no relationship

betweenTp(M) andTq(M) whenp=q.

One of the defects of the above definition of a tangentvector is that it has no clear relation to theCk-differentialstructure ofM.

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

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There is another way to define tangent vectors that re-veals this connection 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, ) be achart at p M (where M is a Ck-manifold of dimen-sion n, with k 1) and let xi = pri , the ith localcoordinate (1in).

For any function,f, defined onUp, set

xi

p

f=(f 1)

Xi

(p)

, 1in.

(Here, (g/Xi)|ydenotes the partial derivative of a func-

tiong:Rn

Rwith respect to the ith coordinate, eval-uated aty.)

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We would expect that the function that maps f to theabove value is a linear map on the set of functions definedlocally atp, but there is technical difficulty:

The set of functions defined locally at p is nota vectorspace!

To see this, observe that iffis defined on an openUpand g is defined on a different open V p, then we do

know how to definef+g.

The problem is that we need to identify functions thatagree on a smaller open. This leads to the notion ofgerms.

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Definition 6.2.3Given any Ck-manifold,M, of dimen-sion n, with k 1, for any p M, a locally definedfunction atpis a pair, (U, f), whereU is an open sub-set ofM containing pandf is a function defined onU.Two locally defined functions, (U, f) and (V, g), atpareequivalent iff there is some open subset, W UV,containing pso that

f W =g W.

The equivalence class of a locally defined function at p,denoted [f] or f, is called a germ atp.

One should check that the relation of Definition 6.2.3 isindeed an equivalence relation.

Of course, the value at p of all the functions, f, in anygerm,f, isf(p). Thus, we set f(p) =f(p).

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One should also check that we can define addition ofgerms, multiplication of a germ by a scalar and multi-plication of germs, in the obvious way:

Iff andgare two germs at p, and R, then

[f] + [g] = [f+g]

[f] = [f]

[f][g] = [fg].

(Of course, f+g is the function locally defined so that(f+ g)(x) =f(x) + g(x) and similarly, (f)(x) =f(x)and (f g)(x) =f(x)g(x).)

Therefore, the germs atpform a ring. The ring of germs

ofCk-functions atpis denotedO(k)M,p. Whenk =, we

usually drop the superscript.

Remark: Most readers will most likely be puzzled by

the notationO(k)M,p.

In fact, it is standard in algebraic geometry, but it is notas commonly used in differential geometry.

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For any open subset, U, of a manifold, M, the ring,

Ck(U), ofCk-functions onUis also denoted O(k)M(U) (cer-

tainly by people with an algebraic geometry bent!).

Then, it turns out that the mapU O(k)M(U) is asheaf,

denotedO(k)

M, and the ringO(k)M,p is thestalkof the sheaf

O(k)M atp.

Such rings are called local rings. Roughly speaking, allthe local information aboutMatpis contained in the

local ring O(k)M,p. (This is to be taken with a grain of

salt. In the Ck-case where k < , we also need thestationary germs, as we will see shortly.)

Now that we have a rigorous way of dealing with functionslocally defined atp, observe that the map

vi: f

xi

p

f

yields the same value for all functions fin a germfatp.

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

true.

Firstly for any two functionsf, g locally defined atp, wehave

xi

p

(f g) =f(p)

xi

p

g+g(p)

xi

p

f.

Secondly, if (f 1)((p)) = 0, then

xi

p

f= 0.

The first property says thatviis aderivation. As to the

second property, when (f 1)((p)) = 0, we say thatf is stationary atp.

It is easy to check (using the chain rule) that being sta-tionary at p does not depend on the chart, (U, ), at por on the function chosen in a germ, f. Therefore, the

notion of a stationary germ makes sense:

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We say that f is a stationary germ iff(f 1)((p)) = 0 for some chart, (U, ), at p andsome function,f, in the germ, f.

TheCk-stationary germs form a subring ofO(k)M,p(but not

an ideal!) denotedS(k)M,p.

Remarkably, it turns out that the dual of the vector space,

O(k)

M,p

/S(k)

M,p

, is isomorphic to the tangent space, Tp(M).

First, we prove that the subspace of linear forms on O(k)

M,p

that vanish onS(k)

M,phas

x1

p

, . . . ,

xn

p

as a basis.

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Proposition 6.2.4Given anyCk-manifold,M, of di-mensionn, withk1, for anypMand any chart

(U, ) at p, the n functions, x1p , . . . , xnp, de-fined onO

(k)M,p by

xi

p

f=(f 1)

Xi

(p)

, 1in

are linear forms that vanish on S(k)

M,p

. Every linear

form, L, on O(k)M,p that vanishes on S

(k)M,p can be ex-

pressed in a unique way as

L=n

i=1

i

xi

p

,

whereiR

. Therefore, the

xi

p

, i= 1, . . . , n

form a basis of the vector space of linear forms on

O(k)M,p that vanish onS

(k)M,p.

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As the subspace of linear forms on O(k)M,p that vanish on

S(k)M,pis isomorphic to the dual, (O

(k)M,p/S

(k)M,p)

, of the space

O(k)M,p/S

(k)M,p, we see that the

xi

p

, i= 1, . . . , n

also form a basis of (O(k)

M,p/S(k)M,p)

.

To define our second version of tangent vectors, we needto define linear derivations.

Definition 6.2.5Given any Ck-manifold,M, of dimen-sionn, with k 1, for any p M, a linear derivationatpis a linear form,v, onO

(k)

M,p, such that

v(fg) =f(p)v(g) +g(p)v(f),

for all germs f, g O(k)M,p. The above is called the Leib-

nitz property.

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Recall that we observed earlier that the

xi

p

are linear

derivations atp. Therefore, we have

Proposition 6.2.6Given anyCk-manifold,M, of di-mension n, with k 1, for any p M, the linearforms on O

(k)M,p that vanish on S

(k)M,p are exactly the

linear derivations onO(k)M,p that vanish onS

(k)M,p.

Here is now our second definition of a tangent vector.

Definition 6.2.7(Tangent Vectors, Version 2)Givenany Ck-manifold, M, of dimension n, with k 1, foranyp M, a tangent vector to M at p is any linear

derivation on O(k)M,p that vanishes on S(k)M,p, the subspaceof stationary germs.

Let us consider the simple case where M = R. In thiscase, for everyx R, the tangent space,Tx(R), is a one-dimensional vector space isomorphic to Rand tx= ddtx is a basis vector ofTx(R).

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For everyCk-function,f, locally defined at x, we have

tx f=df

dtx =f(x).Thus,

t

x

is: compute the derivative of a function at x.

We now prove the equivalence of the two Definitions of atangent vector.

Proposition 6.2.8LetMbe anyCk-manifold of di-mensionn, withk1. For anypM, letu be anytangent vector (version 1) given by some equivalenceclass ofC1-curves, : ] , +[ M, throughp (i.e.,

p=(0)). Then, the map Lu defined onO(k)M,p by

Lu(f) = (f )(0)

is a linear derivation that vanishes onS(k)M,p. Further-

more, the map u Lu defined above is an isomor-phism betweenTp(M) and (O

(k)M,p/S

(k)M,p)

, the space of

linear forms onO(k)M,p that vanish onS

(k)M,p.

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In view of Proposition 6.2.8, we can identify Tp(M) with

(O(k)M,p/S

(k)M,p)

.

As the spaceO(k)M,p/S

(k)M,p is finite dimensional,

(O(k)

M,p/S(k)M,p)

is canonically isomorphic to O(k)

M,p/S(k)M,p,

so we can identifyTp (M) withO(k)M,p/S

(k)M,p.

(Recall that ifE is a finite dimensional space, the map

iE: EE defined so that, for any vE,v v, wherev(f) =f(v), for allfE

is a linear isomorphism.) This also suggests the followingdefinition:

Definition 6.2.9Given any Ck-manifold,M, of dimen-sion n, with k 1, for any p M, the tangent spaceatp, denotedTp(M), is the space of linear derivations on

O(k)M,p that vanish on S

(k)M,p. Thus,Tp(M) can be identi-

fied with (O(k)M,p/S

(k)M,p)

. The space O(k)M,p/S

(k)M,p is called

the cotangent space at p; it is isomorphic to the dual,Tp (M), ofTp(M).

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Observe that ifxi=pri , as

xip xj =i,j,the images ofx1, . . . , xnin O

(k)M,p/S

(k)M,pare the dual of the

basis

x1

p

, . . . ,

xn

p

ofTp(M).

Given anyCk-function,f, onM, we denote the image of

f inTp (M) =O(k)M,p/S(k)M,pbydfp.

This is thedifferential off atp.

Using the isomorphism betweenO(k)M,p/S

(k)M,p and

(O(k)M,p/S

(k)M,p)

described above, dfp corresponds to the

linear map in Tp (M) defined by dfp(v) = v(f), for allvTp(M).

With this notation, we see that (dx1)p, . . . , (dxn)p is abasis ofTp (M), and this basis is dual to the basis

x1p , . . . , xnp ofTp(M).

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For simplicity of notation, we often omit the subscript punless confusion arises.

Remark: Strictly speaking, a tangent vector,v Tp(M),is defined on the space of germs, O(

k)M,p atp. However, it is

often convenient to define v onCk-functionsf Ck(U),whereU is some open subset containingp. This is easy:Set

v(f) =v(f).

Given any chart, (U, ), atp, sincev can be written in aunique way as

v=n

i=1

i

xi

p

,

we get

v(f) =n

i=1

i xi

p

f.

This shows that v(f) is the directional derivative offin the directionv.

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When M is a smooth manifold, things get a little sim-pler. Indeed, it turns out that in this case, every linearderivation vanishes on stationary germs.

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

Proposition 6.2.10If g:Rn R is a Ck-function(k 2) on a convex open, U, aboutp Rn, then foreveryqU, we have

g(q) =g(p) +n

i=1

g

Xi

p

(qi pi)

+n

i,j=1(qi pi)(qjpj) 1

0

(1 t) 2g

XiXj (1t)p+tqdt.In particular, if g C(U), then the integral as afunction ofq isC.

Proposition 6.2.11Let M be any C-manifold of

dimensionn. For anyp M, any linear derivationonO

()M,p vanishes on stationary germs.

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Proposition 6.2.11 shows that in the case of a smoothmanifold, in Definition 6.2.7, we can omit the requirementthat linear derivations vanish on stationary germs, since

this is automatic.

It is also possible to defineTp(M) just in terms ofO()M,p .

Let mM,p O()M,p be the ideal of germs that vanish at

p. Then, we also have the ideal m2M,p, which consists of

all finite sums of products of two elements in mM,p, andit can be shown thatTp (M) is isomorphic to mM,p/m

2M,p

(see Warner [?], Lemma 1.16).

Actually, if we let m(k)M,p denote theC

k germs that vanish

atpand s(k)

M,pdenote the stationaryCk-germs that vanish

atp, it is easy to show that

O(k)M,p/S

(k)M,p

= m(k)M,p/s

(k)M,p.

(Given any f O(k)

M,p, send it to f f(p) m(k)M,p.)

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Clearly, (m(k)M,p)

2 consists of stationary germs (by the deriva-tion property) and when k = , Proposition 6.2.10shows that every stationary germ that vanishes at p be-longs to m2M,p. Therefore, whenk=, we have

s()M,p= m

2M,pand so,

Tp (M) =O()M,p/S

()M,p

= mM,p/m2M,p.

Remark:The ideal

m(k)

M,pis in fact the unique maximalideal ofO

(k)M,p.

Thus, O(k)

M,p is a local ring (in the sense of commutative

algebra) called the local ring of germs ofCk-functionsatp. These rings play a crucial role in algebraic geometry.

Yet one more way of defining tangent vectors will makeit a little easier to define tangent bundles.

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Definition 6.2.12(Tangent Vectors, Version 3)GivenanyCk-manifold,M, of dimensionn, withk 1, for anypM, consider the triples, (U,,u), where (U, ) is any

chart atpanduis any vector in Rn. Say that two suchtriples (U,,u) and (V , , v) areequivalent iff

( 1)(p)(u) =v.

A tangent vector to M at p is an equivalence class oftriples, [(U,,u)], for the above equivalence relation.

The intuition behind Definition 6.2.12 is quite clear: Thevectoruis considered as a tangent vector to Rn at(p).

If (U, ) is a chart onMatp, we can define a natural iso-morphism,U,,p:R

n Tp(M), between Rn andTp(M),

as follows: For anyu Rn,

U,,p: u[(U,,u)].

One immediately check that the above map is indeed lin-ear and a bijection.

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The equivalence of this definition with the definition interms of curves (Definition 6.2.2) is easy to prove.

Proposition 6.2.13Let M be any Ck-manifold ofdimensionn, withk1. For anypM, letxbe anytangent vector (version 1) given by some equivalenceclass ofC1-curves, : ] , +[ M, throughp (i.e.,p=(0)). The map

x[(U,, ( )

(0))]is an isomorphism betweenTp(M)-version 1 andTp(M)-version 3.

For simplicity of notation, we also use the notationTpM

forTp(M) (resp. T

p M forT

p (M)).

After having explored thorougly the notion of tangentvector, we show how aCk-map,h: MN, betweenCk

manifolds, induces a linear map,dhp: Tp(M)Th(p)(N),for everypM.

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We find it convenient to use Version 2 of the definition ofa tangent vector. So, letuTp(M) be a linear derivation

onO(k)M,p that vanishes onS

(k)M,p.

We would likedhp(u) to be a linear derivation on O(k)N,h(p)

that vanishes onS(k)

N,h(p).

So, for every germ, g O(k)N,h(p), set

dhp(u)(g) =u(g h).

For any locally defined function, g, at h(p) in the germ,g (at h(p)), it is clear that g h is locally defined at pand isCk, so g his indeed a Ck-germ atp.

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Moreover, ifgis a stationary germ ath(p), then for somechart, (V, ) onN atq=h(p), we have(g 1)((q)) = 0 and, for some chart (U, ) atp on

M, we get

(gh1)((p)) = (g1)((q))((h1)((p)))

= 0,

which means that g his stationary atp.

Therefore, dhp(u) Th(p)(M). It is also clear that dhpis a linear map. We summarize all this in the followingdefinition:

Definition 6.2.14Given any two Ck-manifolds,MandN, of dimensionmandn, respectively, for any Ck-map,h: M N, and for every p M, the differential ofh atp or tangent map, dhp: Tp(M) Th(p)(N), is thelinear map defined so that

dhp(u)(g) =u(g h),

for every u Tp(M) and every germ, g O(k)

N,h(p)

. The

linear map dhp is also denoted Tph (and sometimes, hporDph).

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The chain rule is easily generalized to manifolds.

Proposition 6.2.15Given any two C

k

-mapsf: MNandg: NPbetween smoothCk-manifolds,for anypM, we have

d(g f)p=dgf(p) dfp.

In the special case where N= R, aCk-map between themanifoldsM and Ris just aCk-function onM.

It is interesting to see whatdfpis explicitly. SinceN= R,germs (of functions on R) att0=f(p) are just germs ofCk-functions,g:R R, locally defined att0.

Then, for anyu Tp(M) and every germ gatt0,

dfp(u)(g) =u(g f).

If we pick a chart, (U, ), onM at p, we know that the xi

p

form a basis ofTp(M), with 1in.

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Therefore, it is enough to figure out what dfp(u)(g) is

whenu=

xip

. In this case,

dfp

xi

p

(g) =

(g f 1)

Xi

(p)

.

Using the chain rule, we find that

dfp

xip (g) =

xip fdg

dt t0 .Therefore, we have

dfp(u) =u(f) d

dt

t0

.

This shows that we can identify dfp with the linear form

inTp (M) defined by

dfp(v) =v(f).

This is consistent with our previous definition ofdfp as

the image of f in Tp (M) = O(k)M,p/S

(k)M,p (as Tp(M) is

isomorphic to (O(k)

M,p

/S(k)

M,p

)).

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In preparation for the definition of the flow of a vectorfield (which will be needed to define the exponential mapin Lie group theory), we need to define the tangent vector

to a curve on a manifold.

Given a Ck-curve, : ]a, b[ M, on a Ck-manifold,M,for any t0 ]a, b[ , we would like to define the tangentvector to the curve at t0 as a tangent vector to M atp=(t0).

We do this as follows: Recall that ddt

t0is a basis vector

ofTt0(R) = R.

So, define the tangent vector to the curveatt, denoted(t0) (or

(t), or ddt (t0)) by

(t) =dtd

dt

t0

.

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Sometime, it is necessary to define curves (in a manifold)whose domain is not an open interval.

A map, : [a, b] M, is a Ck-curve in M if it is therestriction of some Ck-curve,: ]a , b+[ M, forsome (small) >0.

Note that for such a curve (ifk1) the tangent vector,(t), is defined for allt[a, b],

A curve, : [a, b] M, is piecewise Ck iff there a se-quence,a0=a, a1, . . . , am=b, so that the restriction ofto each [ai, ai+1] is aC

k-curve, fori= 0, . . . , m 1.

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6.3 Tangent and Cotangent Bundles, Vector Fields

Let Mbe a Ck-manifold (with k2). Roughly speaking,

a vector field on M is the assignment, p (p), of atangent vector,(p)Tp(M), to a pointp M.

Generally, we would like such assignments to have somesmoothness properties whenpvaries inM, for example,to beCl, for somelrelated tok.

Now, if the collection,T(M), of all tangent spaces,Tp(M),was a Cl-manifold, then it would be very easy to definewhat we mean by a Cl-vector field: We would simplyrequire the maps,: MT(M), to beCl.

IfM is aC

k

-manifold of dimensionn, then we can indeeddefine makeT(M) into aCk1-manifold of dimension 2nand we now sketch this construction.

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6.3. TANGENT AND COTANGENT BUNDLES, VECTOR FIELDS 357

We find it most convenient to use Version 3 of the def-inition of tangent vectors, i.e., as equivalence classes oftriple (U,,u).

First, we let T(M) be the disjoint union of the tangentspacesTp(M), for allpM. There is anatural projec-tion,

: T(M)M, where (v) =p if vTp(M).

We still have to give T(M) a topology and to define aCk1-atlas.

For every chart, (U, ), ofM (with U open in M) wedefine the function

: 1(U) R2n by

(v) = ( (v), 1U,,(v)(v)),wherev 1(U) andU,,pis the isomorphism betweenR

n andTp(M) described just after Definition 6.2.12.

It is obvious that

is a bijection between 1(U) and

(U) R

n

, an open subset ofR

2n

.

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We give T(M) the weakest topology that makes all the

continuous, i.e., we take the collection of subsets of theform1(W), whereWis any open subset of(U)Rn,as a basis of the topology ofT(M).One easily checks that T(M) is Hausdorff and second-countable in this topology. If (U, ) and (V, ) are over-lapping charts, then the transition function

1: (U V) Rn (U V) Rnis given by 1(p, u) = ( 1(p), ( 1)(u)).It is clear that

1 is aCk1-map. Therefore,T(M)

is indeed a Ck1-manifold of dimension 2n, called the

tangent bundle.

Remark: Even if the manifold M is naturally embed-ded in RN (for someN n= dim(M)), it is not at allobvious how to view the tangent bundle, T(M), as em-bedded in RN

, for sone suitable N. Hence, we see that

the definition of an abtract manifold is unavoidable.

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A similar construction can be carried out for the cotan-gent bundle.

In this case, we let T(M) be the disjoint union of thecotangent spacesTp (M).

We also have a natural projection, : T(M)M, andwe can define charts as follows: For any chart, (U, ), onM, we define the function:

1(U) R2n by() = (),

x1

()

, . . . ,

xn

()

,

where1(U) and the

xip

are the basis ofTp(M)

associated with the chart (U, ).

Again, one can makeT(M) into aCk1-manifold of di-mension 2n, called thecotangent bundle.

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Observe that for every chart, (U, ), on M, there is abijection

U: 1(U)U Rn,

given byU(v) = ((v),

1U,,(v)(v)).

Clearly,pr1 U=, on1(U).

Thus, locally, that is, overU, the bundleT(M) looks like

the productUR

n

.

We say thatT(M) islocally trivial(overU) and we callU atrivializing map.

For any p M, the vector space 1(p) = Tp(M) iscalled thefibre abovep.

Observe that the restriction of U to 1(p) is an iso-

morphism betweenTp(M) and{p} Rn = Rn, for anypM.

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All these ingredients are part of being a vector bundle(but a little more is required of the maps U). For moreon bundles, see Lang [?], Gallot, Hulin and Lafontaine

[?], Lafontaine [?] or Bott and Tu [?].

WhenM= Rn, observe thatT(M) =M Rn = Rn Rn, i.e., the bundle T(M) is(globally) trivial.

Given aCk-map,h: MN, between twoCk-manifolds,we can define the function, dh: T(M) T(N), (alsodenotedT h, orh, orDh) by setting

dh(u) =dhp(u), iff uTp(M).

We leave the next proposition as an exercise to the reader(A proof can be found in Berger and Gostiaux [?]).

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Proposition 6.3.1Given aCk-map, h: MN, be-tween two Ck-manifoldsM andN (withk 1), themap dh: T(M)T(N) is aCk1-map.

We are now ready to define vector fields.

Definition 6.3.2LetM be aCk+1 manifold, withk 1. For any open subset,U ofM, a vector field onUis anysection, , ofT(M) overU, i.e., any function,

: UT(M), such that = idU(i.e.,(p)Tp(M),for every p U). We also say that is a lifting of Uinto T(M).

We say thatis aCh-vector field onU iffis a sectionoverUand aCh-map, where 0hk.

The set ofCk-vector fields over Uis denoted (k)(U, T(M)).Given a curve, : [a, b] M, a vector field, , along is any section of T(M) over , i.e., a Ck-function,: [a, b] T(M), such that = . We also saythat lifts into T(M).

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The above definition gives a precise meaning to the ideathat aCk-vector field onMis an assignment, p(p),of a tangent vector,(p)Tp(M), to a point,pM, so

that(p) varies in aCk-fashion in terms ofp.

Clearly, (k)(U, T(M)) is a real vector space. For short,the space (k)(M, T(M)) is also denoted by (k)(T(M))(or X(k)(M) or even (T(M)) or X(M)).

IfM= Rn andUis an open subset ofM, thenT(M) = Rn Rn and a section ofT(M) overUis simplya function,, such that

(p) = (p, u), with u Rn,

for allpU. In other words, is defined by a function,

f: U Rn

(namely,f(p) =u).

This corresponds to the old definition of a vector fieldin the more basic case where the manifold,M, is just Rn.

Given any Ck-function, f Ck(U), and a vector field,

(k)

(U, T(M)), we define the vector field,f, by(f )(p) =f(p)(p), pU.

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Obviously,f(k)(U, T(M)), which shows that(k)(U, T(M)) is also a Ck(U)-module. We also denote(p) byp.

For any chart, (U, ), on M it is easy to check that themap

p

xi

p

, pU,

is a Ck-vector field on U(with 1 i n). This vector

field is denoted xior xi .IfU is any open subset ofM and f is any function inCk(U), then(f) is the function onUgiven by

(f)(p) =p(f) =p(f).

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As a special case, when (U, ) is a chart onM, the vectorfield, xi , just defined above induces the function

p xi

p

f, pU,

denoted xi (f) or

xi

f.

It is easy to check that (f) Ck1(U). As a conse-

quence, every vector field (k)

(U, T(M)) induces alinear map,

L: Ck(U) Ck1(U),

given byf (f).

It is immediate to check that Lhas the Leibnitz property,

i.e.,L(fg) =L(f)g+f L(g).

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Linear maps with this property are called derivations.

Thus, we see that every vector field induces some kind of

differential operator, namely, a linear derivation.

Unfortunately, not every linear derivation of the abovetype arises from a vector field, although this turns out tobe true in the smooth case i.e., whenk= (for a proof,see Gallot, Hulin and Lafontaine [?] or Lafontaine [?]).

In the rest of this section, unless stated otherwise, weassume thatk1. The following easy proposition holds(c.f. Warner [?]):

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Proposition 6.3.3Letbe a vector field on theCk+1-manifold,M, of dimensionn. Then, the following areequivalent:

(a) isCk.

(b) If(U, ) is a chart onM and iff1, . . . , f n are thefunctions onUuniquely defined by

U=n

i=1fi

xi,

then eachfi is aCk-map.

(c) Whenever U is open in M and f Ck(U), then(f) Ck1(U).

Given any twoCk-vector field,, , onM, for any func-tion,f Ck(M), we defined above the function(f) and(f).

Thus, we can form((f)) (resp. ((f))), which are inCk2(M).

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Unfortunately, even in the smooth case, there is generallyno vector field,, such that

(f) =((f)), for allf Ck

(M).This is because ((f)) (and ((f))) involve second-order derivatives.

However, if we consider ((f)) ((f)), then second-order derivatives cancel out and there is a unique vector

field inducing the above differential operator.

Intuitively, measures the failure of and tocommute.

Proposition 6.3.4Given anyCk+1-manifold, M, of

dimensionn, for any twoCk-vector fields,, , onM,there is a uniqueCk1-vector field, [, ], such that

[, ](f) =((f)) ((f)), for all f Ck1(M).

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Definition 6.3.5Given any Ck+1-manifold, M, of di-mensionn, for any twoCk-vector fields,, , onM, theLie bracket, [, ], of and , is the Ck1 vector fielddefined so that

[, ](f) =((f)) ((f)), for all f Ck1(M).

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

Proposition 6.3.6Given anyCk+1-manifold, M, ofdimensionn, for anyCk-vector fields, , , , onM,for allf, g Ck(M), we have:

(a)[[, ], ] + [[, ], ] + [[, ], ] = 0 (Jacobi iden-

tity).(b)[, ] = 0.

(c)[f,g] =f g[, ] +f (g) g(f).

(d)[, ] is bilinear.

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Consequently, for smooth manifolds (k =), the spaceof vector fields, ()(T(M)), is a vector space equippedwith a bilinear operation, [, ], that satisfies the Jacobi

identity.

This makes ()(T(M)) aLie algebra.

One more notion will be needed when we deal with Liealgebras.

Definition 6.3.7Let : M N be a Ck+1-map ofmanifolds. If is a Ck vector field on M and is a Ck

vector field onN, we say that and are-related iff

d = .

Proposition 6.3.8Let: MN be aCk+1-map ofmanifolds, let and1 beC

k vector fields onM andlet, 1 beC

k vector fields onN. If is-related to1 and is-related to 1, then [, ] is-related to[1, 1].

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6.4. SUBMANIFOLDS, IMMERSIONS, EMBEDDINGS 371

6.4 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 differentdefinitions appear in books and that it is not obvious atfirst glance that these definitions are equivalent.

What is important is that a submanifold, N of a givenmanifold,M, not only have the topology inducedMbutalso that the charts ofNbe somewhow induced by thoseofM.

(Recall that ifXis a topological space and Yis a subset of

X, then thesubspace topology onY ortopology inducedbyX onYhas for its open sets all subsets of the formY U, whereU is an arbitary subset ofX.).

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Given m, n, with 0 m n, we can view Rm as asubspace ofRn using the inclusion

Rm

=R

m

{(0, . . . , 0) nm

}R

m

R

nm

=R

n

,

given by

(x1, . . . , xm)(x1, . . . , xm, 0, . . . , 0

nm

).

Definition 6.4.1Given a Ck-manifold, M, of dimen-sionn, a subset,N, ofMis anm-dimensional subman-ifold ofM(where 0 mn) iff for every point,pN,there is a chart, (U, ), ofM, withpU, so that

(U N) =(U) (Rm {0nm}).

(We write 0nm= (0, . . . , 0) nm

.)

The subset,UN, of Definition 6.4.1 is sometimes calledasliceof (U, ) and we say that (U, ) isadapted to N(See ONeill [?] or Warner [?]).

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Other authors, including Warner [?], use the term sub-manifold in a broader sense than us and they use the

wordembedded submanifoldfor what is defined in Defi-

nition 6.4.1.

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

Proposition 6.4.2Given a Ck

-manifold, M, of di-mensionn, for any submanifold, N, ofM of dimen-sionm n, the family of pairs (U N, U N),where(U, )ranges over the charts over any atlas forM, is an atlas forN, whereN is given the subspacetopology. Therefore, Ninherits the structure of aCk-

manifold.

In fact, every chart onNarises from a chart onMin thefollowing precise sense:

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Proposition 6.4.3Given a Ck-manifold, M, of di-mensionnand a submanifold, N, ofMof dimensionmn, for anypNand any chart, (W, ), ofN atp, there is some chart, (U, ), ofM atp so that

(U N) =(U) (Rm {0nm})

andU N= U N,

wherepU NW.

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

Definition 6.4.4Let: NMbe aCk-map of man-

ifolds.(a) The map is an immersion ofN into M iffdp is

injective for allpN.

(b) The set(N) is an immersed submanifold ofM iffis an injective immersion.

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(c) The map is an embedding of N into M iff isan injective immersion such that the induced map,N (N), is a homeomorphism, where (N)

is given the subspace topology (equivalently, is anopen map fromNinto(N) with the subspace topol-ogy). We say that(N) (with the subspace topology)is anembedded submanifold ofM.

(d) The map is a submersion ofN into M iffdp issurjective for allpN.

Again, we warn our readers that certain authors (suchas Warner [?]) call(N), in (b), a submanifold ofM!

We prefer the terminologyimmersed submanifold.

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The notion of immersed submanifold arises naturally inthe framewok of Lie groups.

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

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

It is thus necessary to have a more inclusive notion ofsubmanifold for Lie groups and the concept of immersedsubmanifold is just whats needed.

Immersions ofR into R3 are parametric curves and im-

mersions of R2

into R3

are parametric surfaces. Thesehave been extensively studied, for example, see DoCarmo[?], Berger and Gostiaux [?] or Gallier [?].

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Immersions (i.e., subsets of the form (N), where N isan immersion) are generally neither injective immersions(i.e., subsets of the form (N), where N is an injective

immersion) nor embeddings (or submanifolds).

For example, immersions can have self-intersections, asthe plane curve (nodal cubic): x=t2 1; y=t(t2 1).

Injective immersions are generally not embeddings (or

submanifolds) because(N) may not be homeomorphictoN.

An example is given by the Lemniscate of Bernoulli, aninjective immersion ofRinto R2:

x =

t(1 +t2)

1 +t4 ,

y = t(1 t2)

1 +t4 .

There is, however, a close relationship between submani-folds and embeddings.

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Proposition 6.4.5IfNis a submanifold ofM, thenthe inclusion map, j: NM, is an embedding. Con-versely, if : N M is an embedding, then (N)

with the subspace topology is a submanifold ofM and is a diffeomorphism betweenN and(N).

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

Some authors refer to spaces of the form (N), whereis an injective immersion, asimmersed submanifolds.

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

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

Our next goal is to review and promote to manifolds somestandard results about ordinary differential equations.

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6.5. INTEGRAL CURVES, FLOW, ONE-PARAMETER GROUPS 379

6.5 Integral Curves, Flow of a Vector Field,

One-Parameter Groups of Diffeomorphisms

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

Definition 6.5.1Letbe aCk1 vector field on aCk-manifold, M, (k 2) and let p0 be a point on M. Anintegral curve (or trajectory) forwith initial condi-

tionp0 is aCp1-curve,: IM, so that

(t) =((t)), for allt I and (0) =p0,

whereI= ]a, b[ Ris an open interval containing 0.

What definition 6.5.1 says is that an integral curve, ,

with initial condition p0 is a curve on the manifold Mpassing throughp0and such that, for every pointp=(t)on this curve, the tangent vector to this curve at p, i.e.,(t), coincides with the value, (p), of the vector field atp.

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Given a vector field, , as above, and a point p0 M,is there an integral curve through p0? Is such a curveunique? If so, how large is the open interval I?

We provide some answers to the above questions below.

Definition 6.5.2Letbe aCk1 vector field on aCk-manifold, M, (k 2) and let p0 be a point on M. Alocal flow for atp0 is a map,

: J UM,

whereJ Ris an open interval containing 0 andUis anopen subset ofM containingp0, so that for everypU,the curvet(t, p) is an integral curve ofwith initialconditionp.

Thus, a local flow foris a family of integral curves for allpoints in some small open set around p0 such that thesecurves all have the same domain,J, independently of theinitial condition,pU.

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The following theorem is the main existence theorem oflocal flows.

This is a promoted version of a similar theorem in theclassical theory of ODEs in the case where Mis an opensubset ofRn.

Theorem 6.5.3(Existence of a local flow) Letbe aCk1 vector field on aCk-manifold, M, (k 2) andlet p0 be a point on M. There is an open interval,J R, containing 0 and an open subset, U M,containing p0, so that there is a unique local flow,: J UM, for atp0. Furthermore, isC

k1.

Theorem 6.5.3 holds under more general hypotheses, namely,when the vector field satisfies someLipschitz condition,see Lang [?] or Berger and Gostiaux [?].

Now, we know that for any initial condition, p0, there issome integral curve throughp0.

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However, there could be two (or more) integral curves1: I1M and2: I2Mwith initial conditionp0.

This leads to the natural question: How do 1 and 2differ onI1 I2? The next proposition shows they dont!

Proposition 6.5.4Let be aCk1 vector field on aCk-manifold, M, (k2) and letp0 be a point onM.If 1: I1 M and 2: I2 M are any two integralcurves both with initial conditionp0, then1=2 onI1 I2.

Proposition 6.5.4 implies the important fact that there isa unique maximal integral curve with initial condition

p.

Indeed, if{k: Ik M}kK is the family of all integralcurves with initial condition p (for some big index set,K), if we let I(p) =

kKIk, we can define a curve,

p: I(p)M, so that

p(t) =k(t), if tIk.

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Sincekandlagree onIk Ilfor allk, lK, the curvepis indeed well defined and it is clearly an integral curvewith initial conditionpwith the largest possible domain

(the open interval,I(p)).

The curvep is called the maximal integral curve withinitial conditionpand it is also denoted(t, p).

Note that Proposition 6.5.4 implies that any two distinct

integral curves are disjoint, i.e., do not intersect eachother.

The following interesting question now arises: Given anyp0M, ifp0: I(p0)Mis the maximal integral curvewith initial condition p0, for anyt1 I(p0), and ifp1 =

p0(t1) M, then there is a maximal integral curve,p1: I(p1) M, with initial condition p1.

What is the relationship betweenp0andp1, if any? Theanswer is given by

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Proposition 6.5.5Let be a Ck1 vector field ona Ck-manifold, M, (k 2) and let p0 be a pointon M. If p0: I(p0) M is the maximal integralcurve with initial conditionp0, for any t1 I(p0), ifp1 = p0(t1) M andp1: I(p1) M is the maximalintegral curve with initial conditionp1, then

I(p1) =I(p0)t1 and p1(t) =p0(t1)(t) =p0(t+t1),

for allt I(p0) t1

It is useful to restate Proposition 6.5.5 by changing pointof view.

So far, we have been focusing on integral curves, i.e., givenanyp0 M, we let t vary in I(p0) and get an integralcurve,p0, with domainI(p0).

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Instead of holdingp0Mfixed, we can holdt Rfixedand consider the set

Dt() ={pM |tI(p)},i.e., the set of points such that it is possible to travel fortunits of time fromp along the maximal integral curve,p, with initial conditionp(It is possible that Dt() =).

By definition, ifDt()=, the pointp(t) is well defined,

and so, we obtain a map,t : Dt()M, with domainDt(), given by

t (p) =p(t).

The above suggests the following definition:

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Definition 6.5.6Letbe aCk1 vector field on aCk-manifold,M, (k2). For any t R, let

Dt() ={pM |tI(p)}and

D() ={(t, p) R M |tI(p)}

and let : D()Mbe the map given by

(t, p) =p(t).

The map is called the(global) flow ofandD() iscalled itsdomain of definition. For anyt Rsuch thatDt() = , the map, p Dt() (t, p) = p(t), is

denoted by t (i.e., t (p) =

(t, p) =p(t)).

Observe thatD() =

pM

(I(p) {p}).

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Also, using the t notation, the property of Proposition6.5.5 reads

s

t =

s+t, ()

whenever both sides of the equation make sense.

Indeed, the above says

s(t (p)) =

s(p(t)) =p(t)(s) =p(s + t) =

s+t(p).

Using the above property, we can easily show that the tare invertible. In fact, the inverse of t is

t.

We summarize in the following proposition some addi-tional properties of the domains D(), Dt() and the

maps t :

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Theorem 6.5.7Letbe aCk1 vector field on aCk-manifold, M, (k2). The following properties hold:

(a) For every t R

, ifDt() = , thenDt() is open(this is trivially true ifDt() =).

(b) The domain,D(), of the flow,, is open and theflow is aCk1 map, : D() M.

(c) Eacht : Dt() Dt()is aCk1-diffeomorphism

with inverse

t.

(d) For alls, t R, the domain of definition ofs

t is contained but generally not equal to Ds+t().

However, dom(s t ) = Ds+t() ifs and t have

the same sign. Moreover, ondom(s t ), we have

s

t

=

s+t

.

The reason for using the terminology flow in referring tothe map can be clarified as follows:

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For any t such that Dt() = , every integral curve, p,with initial conditionp Dt(), is defined on some openinterval containing [0, t], and we can picture these curves

as flow lines along which the points pflow (travel) fora time intervalt.

Then, (t, p) is the point reached by flowing for theamount of time t on the integral curve p (through p)starting fromp.

Intuitively, we can imagine the flow of a fluid throughM, and the vector field is the field of velocities of theflowing particles.

Given a vector field, , as above, it may happen that

Dt() =M, for allt R.

In this case, namely, when D() = R M, we say thatthe vector field iscomplete.

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Then, the t are diffeomorphisms ofMand they form agroup.

The family{t }tRa called a 1-parameter group of.

In this case, induces a group homomorphism,(R, +) Diff(M), from the additive group R to thegroup ofCk1-diffeomorphisms ofM.

By abuse of language, even when it is notthe case thatDt() =M for allt, the family{

t }tR is called a local

1-parameter group of, even though it is nota group,because the composition s

t may not be defined.

WhenM is compact, it turns out that every vector field

is complete, a nice and useful fact.

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Proposition 6.5.8Let be aCk1 vector field on aCk-manifold, M, (k 2). If M is compact, then is complete, i.e., D() = R M. Moreover, the maptt is a homomorphism from the additive group Rto the group, Diff(M), of (Ck1) diffeomorphisms ofM.

Remark: The proof of Proposition 6.5.8 also applies

whenis a vector field with compact support (this meansthat the closure of the set {p M | (p) = 0} is com-pact).

A pointpMwhere a vector field vanishes, i.e.,(p) = 0, is called acritical point of.

Critical points play a major role in the study of vec-tor fields, in differential topology (e.g., the celebratedPoincareHopf index theorem) and especially in Morsetheory, but we wont go into this here (curious readersshould consult Milnor [?], Guillemin and Pollack [?] or

DoCarmo [?], which contains an informal but very clearpresentation of the PoincareHopf index theorem).

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Another famous theorem about vector fields says thatevery smooth vector field on a sphere of even dimension(S2n) must vanish in at least one point (the so-called

hairy-ball theorem.

On S2, it says that you cant comb your hair withouthaving a singularity somewhere. Try it, its true!).

Let us just observe that if an integral curve, , passes

through a critical point,p, thenis reduced to the pointp, i.e.,(t) =p, for allt.

Then, we see that if a maximal integral curve is definedon the whole ofR, either it is injective (it has no self-intersection), or it is simply periodic (i.e., there is some

T > 0 so that (t+T) = (t), for all t R and isinjective on [0, T[ ), or it is reduced to a single point.

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We conclude this section with the definition of the Liederivative of a vector field with respect to another vectorfield.

Say we have two vector fields and on M. For anyp M, we can flow along the integral curve of withinitial conditionpto t (p) (fortsmall enough) and then

evaluate there, getting(t (p)).

Now, this vector belongs to the tangent spaceTt (p)(M),but(p)Tp(M).

So to compare (t (p)) and (p), we bring back (t (p))

toTp(M) by applying the tangent map, dt, at

t (p),

to (t (p)) (Note that to alleviate the notation, we use

the slight abuse of notationdtinstead ofd(t)t (p).)

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Then, we can form the differencedt((t (p))) (p),

divide bytand consider the limit as tgoes to 0.

This is the Lie derivative of with respect to at p,denoted (L)p, and given by

(L)p = limt0

dt((t (p))) (p)

t =

d

dt(dt((

t (p)))t=0 .

It can be shown that (L)p is our old friend, the Liebracket, i.e.,

(L)p= [, ]p.

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