Chapter 6

Vector Fields, Lie Derivatives, Integral Curves, Flows

Our goal in this chapter is to generalize the concept of a vector field to manifolds, and to promote some standard results about ordinary di↵erential equations to manifolds.

6.1 Tangent and Cotangent Bundles

Let M be a Ck-manifold (with k 2). Roughly speaking, avectorfieldonM is the assignment, p X(p), of a tangent vector X(p) T (M), to a point p7! M. 2 p 2 Generally, we would like such assignments to have some smoothness properties when p varies in M,forexample, to be Cl,forsomel related to k.

401 402 CHAPTER 6. VECTOR FIELDS, INTEGRAL CURVES, FLOWS

Now, if the collection, T (M), of all tangent spaces, Tp(M), was a Cl-manifold, then it would be very easy to define what we mean by a Cl-vector field: We would simply require the map, X : M T (M), to be Cl. ! If M is a Ck-manifold of dimension n,thenwecanindeed k 1 make T (M)intoaC -manifold of dimension 2n and we now sketch this construction.

We find it most convenient to use Version 2 of the def- inition of tangent vectors, i.e., as equivalence classes of triples (U,', x), with x Rn.Recallthat(U,', x)and (V, ,y)areequivalenti2↵

1 ( ' )0 (x)=y. '(p) First, we let T (M)bethedisjointunionofthetangent spaces T (M), for all p M.Formally, p 2 T (M)= (p, v) p M,v T (M) . { | 2 2 p } There is a natural projection,

⇡ : T (M) M, with ⇡(p, v)=p. ! 6.1. TANGENT AND COTANGENT BUNDLES 403 We still have to give T (M)atopologyandtodefinea k 1 C -atlas.

For every chart, (U,'), of M (with U open in M)we 1 2n define the function, ': ⇡ (U) R ,by ! 1 '(p, ve)=('(p),✓U,',p(v)),

1 where (p, v) ⇡e (U)and✓U,',p is the isomorphism be- n 2 tween R and Tp(M)describedjustafterDefinition5.10.

1 It is obvious that ' is a bijection between ⇡ (U)and '(U) Rn,anopensubsetofR2n. ⇥ e We give T (M)theweakesttopologythatmakesallthe ' continuous, i.e., we take the collection of subsets of the 1 n form ' (W ), where W is any open subset of '(U) R , ⇥ ase a basis of the topology of T (M). e 404 CHAPTER 6. VECTOR FIELDS, INTEGRAL CURVES, FLOWS One easily checks that T (M)isHausdor↵andsecond- countable in this topology. If (U,')and(V, )areover- lapping charts, then the transition function

1 n n ' : '(U V ) R (U V ) R \ ⇥ ! \ ⇥ is givene bye

1 1 1 ' (z,x)=( ' (z), ( ' )0 (x)), z e with (z,x)e '(U V ) Rn. 2 \ ⇥

1 k 1 It is clear that ' is a C -map. Therefore, T (M) k 1 is indeed a C -manifold of dimension 2n,calledthe tangent bundlee. e 6.1. TANGENT AND COTANGENT BUNDLES 405 Remark: Even if the manifold M is naturally embed- ded in RN (for some N n =dim(M)), it is not at all obvious how to view the tangent bundle, T (M), as em- N bedded in R 0,forsomesuitableN 0.Hence,weseethat the definition of an abtract manifold is unavoidable.

Asimilarconstructioncanbecarriedoutforthecotan- gent bundle.

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

T ⇤(M)= (p, !) p M,! T ⇤(M) . { | 2 2 p }

We also have a natural projection, ⇡ : T ⇤(M) M. ! We can define charts as follows: 406 CHAPTER 6. VECTOR FIELDS, INTEGRAL CURVES, FLOWS For any chart, (U,'), on M,wedefinethefunction 1 2n ': ⇡ (U) R by ! e @ @ '(p, !)= '(p),! ,...,! , @x @x ✓ 1◆p! ✓ n◆p!! e 1 @ where (p, !) ⇡ (U)andthe are the basis of 2 @xi p Tp(M)associatedwiththechart(⇣U,'⌘).

k 1 Again, one can make T ⇤(M)intoaC -manifold of di- mension 2n,calledthecotangent bundle.

Another method using Version 3 of the definition of tan- gent vectors is presented in Section ??. 6.1. TANGENT AND COTANGENT BUNDLES 407 For each chart (U,')onM,weobtainachart

1 n 2n '⇤ : ⇡ (U) '(U) R R ! ⇥ ✓ on T ⇤(M)givenbye

'⇤(p, !)=('(p),✓U,⇤ ',⇡(!)(!))

1 for all (p, !) e ⇡ (U), where 2 n ✓⇤ = ◆ ✓> : T ⇤(M) R . U,',p U,',p p !

n Here, ✓U,>',p : Tp⇤(M) (R )⇤ is obtained by dualizing n ! n n the map, ✓U,',p : R Tp(M), and ◆:(R )⇤ R is the ! ! isomorphism induced by the canonical basis (e1,...,en) of Rn and its dual basis.

For simplicity of notation, we also use the notation TM for T (M)(resp.T ⇤M for T ⇤(M)). 408 CHAPTER 6. VECTOR FIELDS, INTEGRAL CURVES, FLOWS Observe that for every chart, (U,'), on M,thereisa bijection

1 n ⌧U : ⇡ (U) U R , ! ⇥ given by

1 ⌧U (p, v)=(p, ✓U,',p(v)).

1 Clearly, pr1 ⌧U = ⇡,on⇡ (U)asillustratedbythe following commutative diagram:

1 ⌧U n ⇡ (U) / U R ⇥ ⇡ pr1 $ U z

Thus, locally, that is, over U,thebundleT (M)lookslike the product manifold U Rn. ⇥ We say that T (M)islocally trivial (over U)andwecall ⌧U a trivializing map. 6.1. TANGENT AND COTANGENT BUNDLES 409 For any p M,thevectorspace 1 2 ⇡ (p)= p Tp(M) ⇠= Tp(M)iscalledthe fibre above{ p}. ⇥

1 Observe that the restriction of ⌧U to ⇡ (p)isanisomor- phism between p T (M) = T (M)and { }⇥ p ⇠ p p Rn = Rn,foranyp M. { }⇥ ⇠ 2 Furthermore, for any two overlapping charts (U,')and (V, ), there is a function gUV : U V GL(n, R)such that \ !

1 (⌧ ⌧ )(p, x)=(p, g (p)(x)) U V UV

n for all p U V and all x R ,withgUV (p)givenby 2 \ 2 1 g (p)=(' )0 . UV (p)

n Obviously, gUV (p)isalinearisomorphismofR for all p U V . 2 \ 410 CHAPTER 6. VECTOR FIELDS, INTEGRAL CURVES, FLOWS

The maps gUV (p)arecalledthetransition functions of the tangent bundle.

For example, if M = Sn,then-sphere in Rn+1,wehave two charts given by the stereographic projection (UN ,N ) from the north pole, and the stereographic projection n (US,S)fromthesouthpole(withUN = S N n { } and US = S S ), and on the overlap, UN US = Sn N,S ,thetransitionmaps{ } \ { } 1 1 = = I S N N S

n defined on 'N (UN US)='S(UN US)=R 0 , are given by \ \ { } 1 (x1,...,xn) n 2 (x1,...,xn); 7! i=1 xi P that is, the inversion of center O =(0,...,0) and power 1. I 6.1. TANGENT AND COTANGENT BUNDLES 411 We leave it as an exercice to prove that for every point u Rn 0 ,wehave 2 { }

2 u, h d u(h)= u h 2h i u , I k k u 2 ✓ k k ◆ the composition of the hyperplane reflection about the n hyperplane u? R with the magnification of center O 2✓ and ratio u . k k This is a similarity transformation.Therefore,thetran- sition function gNS (defined on UN US)ofthetangent bundle TSn is given by \

2 S(p),h gNS(p)(h)= S(p) h 2h i S(p) . k k (p) 2 ✓ k S k ◆

All these ingredients are part of being a vector bundle. 412 CHAPTER 6. VECTOR FIELDS, INTEGRAL CURVES, FLOWS For more on bundles, see Lang [23], Gallot, Hulin and Lafontaine [17], Lafontaine [21] or Bott and Tu [5].

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

Given a Ck-map, h: M N,betweentwoCk-manifolds, we can define the function,! dh: T (M) T (N), (also de- noted Th,orh ,orDh)bysetting ! ⇤ dh(u)=dh (u), i↵ u T (M). p 2 p We leave the next proposition as an exercise to the reader (A proof can be found in Berger and Gostiaux [4]).

Proposition 6.1. Given a Ck-map, h: M N, be- tween two Ck-manifolds M and N (with k ! 1), the k 1 map dh: T (M) T (N) is a C map. !

We are now ready to define vector fields. 6.2. VECTOR FIELDS, LIE DERIVATIVE 413 6.2 Vector Fields, Lie Derivative

Definition 6.1. Let M be a Ck+1 manifold, with k 1. For any open subset, U of M,avector field onU is any section X of T (M) over U,thatis,any function X : U T (M)suchthat⇡ X =id (i.e., ! U X(p) Tp(M), for every p U). We also say that X is a lifting2 of U into T (M). 2

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

The set of Ck-vector fields over U is denoted (k)(U, T (M)).

Given a curve, :[a, b] M,avector field X along is any section of T (M!)over,i.e.,aCk-function, X :[a, b] T (M), such that ⇡ X = .Wealsosay that X lifts! into T (M).

Clearly, (k)(U, T (M)) is a real vector space. 414 CHAPTER 6. VECTOR FIELDS, INTEGRAL CURVES, FLOWS 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)).

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

If M = Rn and U is an open subset of M,then T (M)=Rn Rn and a section of T (M)overU is simply afunction,X⇥,suchthat

n X(p)=(p, u), with u R , 2 for all p U.Inotherwords,X is defined by a function, 2 f : U Rn (namely, f(p)=u). ! This corresponds to the “old” definition of a vector field in the more basic case where the manifold, M,isjustRn. 6.2. VECTOR FIELDS, LIE DERIVATIVE 415 For any vector field X (k)(U, T (M)) and for any p U,wehaveX(p)=(2p, v)forsomev T (M), and2 2 p it is convenient to denote the vector v by Xp so that X(p)=(p, Xp).

In fact, in most situations it is convenient to identify X(p) with X T (M), and we will do so from now on. p 2 p This amounts to identifying the isomorphic vector spaces p T (M)andT (M). { }⇥ p p Let us illustrate the advantage of this convention with the next definition.

Given any Ck-function, f k(U), and a vector field, X (k)(U, T (M)), we define2C the vector field, fX,by 2 (fX) = f(p)X ,p U. p p 2 416 CHAPTER 6. VECTOR FIELDS, INTEGRAL CURVES, FLOWS Obviously, fX (k)(U, T (M)), which shows that (k)(U, T (M)) is2 also a k(U)-module. C For any chart, (U,'), on M it is easy to check that the map

@ p ,p U, 7! @x 2 ✓ i◆p is a Ck-vector field on U (with 1 i n). This vector   field is denoted @ or @ . @xi @xi ⇣ ⌘ 6.2. VECTOR FIELDS, LIE DERIVATIVE 417 Definition 6.2. Let M be a Ck+1 manifold and let X be a Ck vector field on M.IfU is any open subset of M and f is any function in k(U), then the Lie derivative C of f with respect to X,denotedX(f)orLXf,isthe function on U given by

X(f)(p)=X (f)=X (f),pU. p p 2 Observe that

X(f)(p)=df p(Xp), where df p is identified with the linear form in Tp⇤(M) defined by

df (v)=v(f),vT M, p 2 p by identifying Tt0R with R (see the discussion following Proposition 5.14).

The Lie derivative, LXf,isalsodenotedX[f]. 418 CHAPTER 6. VECTOR FIELDS, INTEGRAL CURVES, FLOWS As a special case, when (U,')isachartonM,thevector field, @ ,justdefinedaboveinducesthefunction @xi @ p f, f U, 7! @x 2 ✓ i◆p denoted @ (f)or @ f. @xi @xi ⇣ ⌘ k 1 It is easy to check that X(f) (U). 2C As a consequence, every vector field X (k)(U, T (M)) induces a linear map, 2

k k 1 L : (U) (U), X C ! C given by f X(f). 7! 6.2. VECTOR FIELDS, LIE DERIVATIVE 419

It is immediate to check that LX has the Leibniz property, i.e.,

LX(fg)=LX(f)g + fLX(g).

Linear maps with this property are called derivations.

Thus, we see that every vector field induces some kind of di↵erential operator, namely, a derivation.

Unfortunately, not every derivation of the above type arises from a vector field, although this turns out to be true in the smooth case i.e., when k = (for a proof, see Gallot, Hulin and Lafontaine [17] or Lafontaine1 [21]).

In the rest of this section, unless stated otherwise, we assume that k 1. The following easy proposition holds (c.f. Warner [33]): 420 CHAPTER 6. VECTOR FIELDS, INTEGRAL CURVES, FLOWS Proposition 6.2. Let X be a vector field on the Ck+1- manifold, M, of dimension n. Then, the following are equivalent: (a) X is Ck.

(b) If (U,') is a chart on M and if f1,...,fn are the functions on U uniquely defined by n @ X U = f , i @x i=1 i X k then each fi is a C -map. (c) Whenever U is open in M and f k(U), then k 1 2C X(f) (U). 2C

Given any two Ck-vector field, X, Y ,onM,foranyfunc- tion, f k(M), we defined above the function X(f) and Y (f2). C

Thus, we can form X(Y (f)) (resp. Y (X(f))), which are k 2 in (M). C 6.2. VECTOR FIELDS, LIE DERIVATIVE 421 Unfortunately, even in the smooth case, there is generally no vector field, Z,suchthat Z(f)=X(Y (f)), for all f k(M). 2C This is because X(Y (f)) (and Y (X(f))) involve second- order derivatives.

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

Intuitively, XY YXmeasures the “failure of X and Y to commute.”

Proposition 6.3. Given any Ck+1-manifold, M, of dimension n, for any two Ck-vector fields, X, Y , on k 1 M, there is a unique C -vector field, [X, Y ], such that

k 1 [X, Y ](f)=X(Y (f)) Y (X(f)), for all f (M). 2C 422 CHAPTER 6. VECTOR FIELDS, INTEGRAL CURVES, FLOWS Definition 6.3. Given any Ck+1-manifold, M,ofdi- mension n,foranytwoCk-vector fields, X, Y ,onM,the k 1 Lie bracket,[X, Y ], of X and Y ,istheC vector field defined so that

k 1 [X, Y ](f)=X(Y (f)) Y (X(f)), for all f (M). 2C

An an example, in R3,ifX and Y are the two vector fields, @ @ @ X = + y and Y = , @x @z @y then @ [X, Y ]= . @z 6.2. VECTOR FIELDS, LIE DERIVATIVE 423 We also have the following simple proposition whose proof is left as an exercise (or, see Do Carmo [12]):

Proposition 6.4. Given any Ck+1-manifold, M, of dimension n, for any Ck-vector fields, X, Y, Z, on M, for all f,g k(M), we have: 2C (a) [[X, Y ],Z]+[[Y,Z],X]+[Z, X],Y]=0 (Jacobi identity). (b) [X, X]=0. (c) [fX,gY]=fg[X, Y ]+fX(g)Y gY (f)X. (d) [ , ] is bilinear. 424 CHAPTER 6. VECTOR FIELDS, INTEGRAL CURVES, FLOWS Consequently, for smooth manifolds (k = ), the space ( ) 1 of vector fields, 1 (T (M)), is a vector space equipped with a bilinear operation, [ , ], that satisfies the Jacobi identity.

( ) This makes 1 (T (M)) a Lie algebra.

Let h: M N be a di↵eomorphism between two man- ifolds. Then,! vector fields can be transported from N to M and conversely. 6.2. VECTOR FIELDS, LIE DERIVATIVE 425 Definition 6.4. Let h: M N be a di↵eomorphism between two Ck+1 manifolds.! For every Ck vector field, Y ,onN,thepull-back of Y along h is the vector field, h⇤Y ,onM,givenby

1 (h⇤Y ) = dh (Y ),pM. p h(p) h(p) 2

For every Ck vector field, X,onM,thepush-forward of X along h is the vector field, h X,onN,givenby ⇤ 1 h X =(h )⇤X, ⇤ that is, for every p M, 2

(h X)h(p) = dhp(Xp), ⇤ or equivalently,

(h X)q = dhh 1(q)(Xh 1(q)),qN. ⇤ 2 426 CHAPTER 6. VECTOR FIELDS, INTEGRAL CURVES, FLOWS It is not hard to check that

1 Lh Xf = LX(f h) h , ⇤ for any function f Ck(N). 2 One more notion will be needed to when we deal with Lie algebras.

Definition 6.5. Let h: M N be a Ck+1-map of manifolds. If X is a Ck vector! field on M and Y is a Ck vector field on N,wesaythatX and Y are h-related i↵

dh X = Y h.

Proposition 6.5. Let h: M N be a Ck+1-map of manifolds, let X and Y be Ck!vector fields on M and k let X1,Y1 be C vector fields on N.IfX is h-related to X1 and Y is h-related to Y1, then [X, Y ] is h-related to [X1,Y1]. 6.3. INTEGRAL CURVES, FLOW, ONE-PARAMETER GROUPS 427 6.3 Integral Curves, Flow of a Vector Field, One-Parameter Groups of Di↵eomorphisms

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

k 1 k Definition 6.6. Let X be a C vector field on a C - manifold, M,(k 2) and let p0 be a point on M.An integral curve (or trajectory) for X with initial con- k 1 dition p is a C -curve, : I M,sothat 0 !

˙ (t)=X , for all t I and (0) = p , (t) 2 0 where I =]a, b[ R is an open interval containing 0. ✓

What definition 6.6 says is that an integral curve, ,with initial condition p0 is a curve on the manifold M passing through p0 and such that, for every point p = (t)on this curve, the tangent vector to this curve at p,i.e.,˙ (t), coincides with the value, Xp,ofthevectorfieldX at p. 428 CHAPTER 6. VECTOR FIELDS, INTEGRAL CURVES, FLOWS Given a vector field, X,asabove,andapointp M, 0 2 is there an integral curve through p0?Issuchacurve unique? If so, how large is the open interval I?

We provide some answers to the above questions below.

k 1 k Definition 6.7. Let X be a C vector field on a C - manifold, M,(k 2) and let p be a point on M.A 0 local flow for X at p0 is a map,

': J U M, ⇥ ! where J R is an open interval containing 0 and U is an ✓ open subset of M containing p0,sothatforeveryp U, the curve t '(t, p)isanintegralcurveofX with initial2 condition p7!.

Thus, a local flow for X is a family of integral curves for all points in some small open set around p0 such that these curves all have the same domain, J,independently of the initial condition, p U. 2 6.3. INTEGRAL CURVES, FLOW, ONE-PARAMETER GROUPS 429 The following theorem is the main existence theorem of local flows.

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

Theorem 6.6. (Existence of a local flow) Let X be a k 1 k C vector field on a C -manifold, M,(k 2) and let p be a point on M. There is an open interval J 0 ✓ R containing 0 and an open subset U M containing p , so that there is a unique local flow✓': J U M 0 ⇥ ! for X at p0.

What this means is that if '1 : J U M and '2 : J U M are both local flows with⇥ domain! J U, then⇥ ! k 1 ⇥ '1 = '2. Furthermore, ' is C . 430 CHAPTER 6. VECTOR FIELDS, INTEGRAL CURVES, FLOWS Theorem 6.6 holds under more general hypotheses, namely, when the vector field satisfies some Lipschitz condition, see Lang [23] or Berger and Gostiaux [4].

Now, we know that for any initial condition, p0,thereis some integral curve through p0.

However, there could be two (or more) integral curves : I M and : I M with initial condition p . 1 1 ! 2 2 ! 0

This leads to the natural question: How do 1 and 2 di↵er on I I ?Thenextpropositionshowstheydon’t! 1 \ 2

k 1 Proposition 6.7. Let X be a C vector field on a k C -manifold, M,(k 2) and let p0 be a point on M. If : I M and : I M are any two integral 1 1 ! 2 2 ! curves both with initial condition p0, then 1 = 2 on I I . 1 \ 2 6.3. INTEGRAL CURVES, FLOW, ONE-PARAMETER GROUPS 431 Proposition 6.7 implies the important fact that there is a unique maximal integral curve with initial condition p.

Indeed, if j : Ij M j K is the family of all integral curves with{ initial! condition} 2 p (for some big index set,

K), if we let I(p)= j K Ij,wecandefineacurve, 2 p : I(p) M,sothat ! S (t)= (t), if t I . p j 2 j Since and agree on I I for all j, l K,thecurve j l j \ l 2 p is indeed well defined and it is clearly an integral curve with initial condition p with the largest possible domain (the open interval, I(p)).

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

Note that Proposition 6.7 implies that any two distinct in- tegral curves are disjoint, i.e., do not intersect each other. 432 CHAPTER 6. VECTOR FIELDS, INTEGRAL CURVES, FLOWS

Consider the vector field in R2 given by @ @ X = y + x @x @y and shown in Figure 6.1.

Figure 6.1: A vector field in R2

If we write (t)=(x(t),y(t)), the di↵erential equation, ˙ (t)=X((t)), is expressed by

x0(t)= y(t) y0(t)=x(t), or, in matrix form,

x0 0 1 x = . y 10 y ✓ 0◆ ✓ ◆✓ ◆ 6.3. INTEGRAL CURVES, FLOW, ONE-PARAMETER GROUPS 433 0 1 If we write X = x and A = ,thentheabove y 10 ✓ ◆ equation is written as

X0 = AX.

Now, as A A2 An etA = I + t + t2 + + tn + , 1! 2! ··· n! ··· we get

2 3 n d tA A A 2 A n 1 tA (e )=A+ t+ t + + t + = Ae , dt 1! 2! ··· (n 1)! ···

tA so we see that e p is a solution of the ODE X0 = AX with initial condition X = p,andbyuniqueness, X = etAp is the solution of our ODE starting at X = p. 434 CHAPTER 6. VECTOR FIELDS, INTEGRAL CURVES, FLOWS

x0 Thus, our integral curve, p,throughp = is the circle y0 given by

x cos t sin t x0 = . y sin t cos t y ✓ ◆ ✓ ◆✓ 0◆

Observe that I(p)=R,foreveryp R2. 2 Here is an example of a vector field on M = R that has integral curves not defined on the whole of R.

Let X be the vector field on R given by @ X(x)=(1+x2) . @x

It is easy to see that the maximal integral curve with initial condition p0 =0isthecurve :( ⇡/2,⇡/2) R given by !

(t)=tant. 6.3. INTEGRAL CURVES, FLOW, ONE-PARAMETER GROUPS 435 The following interesting question now arises: Given any p0 M,ifp0 : I(p0) M is the maximal integral curve2 with initial condition! p and, for any t I(p ), if 0 1 2 0 p1 = p0(t1) M,thenthereisamaximalintegralcurve, : I(p ) 2M,withinitialconditionp ; p1 1 ! 1

What is the relationship between p0 and p1,ifany?

The answer is given by

k 1 Proposition 6.8. Let X be a C vector field on k a C -manifold, M,(k 2) and let p0 be a point on M.If : I(p ) M is the maximal integral p0 0 ! curve with initial condition p0, for any t1 I(p0), if p = (t ) M and : I(p ) M is the2 maximal 1 p0 1 2 p1 1 ! integral curve with initial condition p1, then

I(p1)=I(p0) t1 and p (t)= (t )(t)=p (t+t1), 1 p0 1 0 for all t I(p ) t 2 0 1 436 CHAPTER 6. VECTOR FIELDS, INTEGRAL CURVES, FLOWS

Proposition 6.8 says that the traces p0(I(p0)) and

p1(I(p1)) in M of the maximal integral curves p0 and p1 are identical; they only di↵er by a simple reparametriza- tion (u = t + t1).

It is useful to restate Proposition 6.8 by changing point of view.

So far, we have been focusing on integral curves, i.e., given any p M,welett vary in I(p )andgetanintegral 0 2 0 curve, p0,withdomainI(p0). 6.3. INTEGRAL CURVES, FLOW, ONE-PARAMETER GROUPS 437

Instead of holding p0 M fixed, we can hold t R fixed and consider the set 2 2

(X)= p M t I(p) , Dt { 2 | 2 } i.e., the set of points such that it is possible to “travel for t units of time from p”alongthemaximalintegralcurve, p,withinitialconditionp (It is possible that (X)= ). Dt ;

By definition, if t(X) = ,thepointp(t)iswellde- fined, and so, weD obtain a6 map,; X : (X) M,withdomain (X), given by t Dt ! Dt X t (p)=p(t). 438 CHAPTER 6. VECTOR FIELDS, INTEGRAL CURVES, FLOWS

k 1 k Definition 6.8. Let X be a C vector field on a C - manifold, M,(k 2). For any t R,let 2 (X)= p M t I(p) Dt { 2 | 2 } and (X)= (t, p) R M t I(p) D { 2 ⇥ | 2 } and let X : (X) M be the map given by D ! X (t, p)=p(t).

The map X is called the (global) flow of X and (X) is called its domain of definition. D

For any t R such that t(X) = ,themap,p (X) 2X(t, p)= (t),D is denoted6 ; by X (i.e., 2 Dt 7! p t X X t (p)= (t, p)=p(t)). 6.3. INTEGRAL CURVES, FLOW, ONE-PARAMETER GROUPS 439 Observe that (X)= (I(p) p ). D ⇥{ } p M [2

X Also, using the t notation, the property of Proposition 6.8 reads

X X =X , ( ) s t s+t ⇤ whenever both sides of the equation make sense.

Indeed, the above says

X X X X 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 X X X t are invertible. In fact, the inverse of t is t. 440 CHAPTER 6. VECTOR FIELDS, INTEGRAL CURVES, FLOWS

k 1 k Theorem 6.9. Let X be a C vector field on a C - manifold, M,(k 2). The following properties hold: (a) For every t R, if t(X) = , then t(X) is open (this is trivially2 trueD if (6 X;)= ). D Dt ; (b) The domain, (X), of the flow, X, is open and Dk 1 X the flow is a C map, : (X) M. D ! X k 1 (c) Each t : t(X) t(X) is a C -di↵eomor- D !DX phism with inverse t. (d) For all s, t R, the domain of definition of X X 2 s t is contained but generally not equal to X X s+t(X). However, dom(s t )= s+t(X) if s andD t have the same sign. Moreover, D on dom(X X), we have s t X X =X . s t s+t

We may omit the superscript, X,andwriteinsteadof X if no confusion arises. 6.3. INTEGRAL CURVES, FLOW, ONE-PARAMETER GROUPS 441 The reason for using the terminology flow in referring to the map X can be clarified as follows:

For any t such that (X) = ,everyintegralcurve, , Dt 6 ; p with initial condition p t(X), is defined on some open interval containing [0,t],2 andD we can picture these curves as “flow lines” along which the points p flow (travel) for atimeintervalt.

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

Intuitively, we can imagine the flow of a fluid through M,andthevectorfieldX is the field of velocities of the flowing particles. 442 CHAPTER 6. VECTOR FIELDS, INTEGRAL CURVES, FLOWS Given a vector field, X,asabove,itmayhappenthat t(X)=M,forallt R. D 2 In this case, namely, when (X)=R M,wesaythat the vector field X is completeD . ⇥

X Then, the t are di↵eomorphisms of M and they form agroup.

X The family t t R acalleda1-parameter group of X. { } 2 In this case, X induces a group homomorphism, (R, +) Di↵(M), from the additive group R to the !k 1 group of C -di↵eomorphisms of M.

By abuse of language, even when it is not the case that X t(X)=M for all t,thefamily t t R is called a local 1D-parameter group of X,eventhoughitis{ } 2 not agroup, because the composition X X may not be defined. s t 6.3. INTEGRAL CURVES, FLOW, ONE-PARAMETER GROUPS 443

If we go back to the vector field in R2 given by @ @ X = y + x , @x @y

x0 since the integral curve, p(t), through p = is given x0 by

x cos t sin t x0 = , y sin t cos t y ✓ ◆ ✓ ◆✓ 0◆ the global flow associated with X is given by

cos t sin t X(t, p)= p, sin t cos t ✓ ◆

X and each di↵eomorphism, t ,istherotation, cos t sin t X = . t sin t cos t ✓ ◆ 444 CHAPTER 6. VECTOR FIELDS, INTEGRAL CURVES, FLOWS

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

More generally, if B is an n n invertible matrix that has areallogarithmA (that is,⇥ if eA = B), then the matrix A defines a vector field, X,inRn,with

n @ X = (a x ) , ij j @x i,j=1 i X whose integral curves are of the form, tA p(t)=e p, and we have p(1) = Bp.

X The one-parameter group, t t R,generatedbyX is tA { } 2 given by e t R. { } 2 6.3. INTEGRAL CURVES, FLOW, ONE-PARAMETER GROUPS 445 When M is compact, it turns out that every vector field is complete, a nice and useful fact.

k 1 Proposition 6.10. Let X be a C vector field on a Ck-manifold, M,(k 2). If M is compact, then X is complete, i.e., (X)=R M. Moreover, the map X D ⇥ t t is a homomorphism from the additive group 7! k 1 R to the group, Di↵(M), of (C ) di↵eomorphisms of M.

Remark: The proof of Proposition 6.10 also applies when X is a vector field with compact support (this means that the closure of the set p M X(p) =0 is compact). { 2 | 6 } If h: M N is a di↵eomorphism and X is a vector field on M,itcanbeshownthatthelocal1-parametergroup! associated with the vector field, h X,is ⇤ X 1 h t h t R. { } 2 446 CHAPTER 6. VECTOR FIELDS, INTEGRAL CURVES, FLOWS Apointp M where a vector field vanishes, i.e., X(p)=0,iscalleda2 critical point of X.

Critical points play a major role in the study of vec- tor fields, in di↵erential topology (e.g., the celebrated Poincar´e–Hopf index theorem) and especially in Morse theory, but we won’t go into this here.

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

On S2,itsaysthatyoucan’tcombyourhairwithout having a singularity somewhere. Try it, it’s true!). 6.3. INTEGRAL CURVES, FLOW, ONE-PARAMETER GROUPS 447 Let us just observe that if an integral curve, ,passes through a critical point, p,then is reduced to the point p,i.e.,(t)=p,forallt.

Then, we see that if a maximal integral curve is defined on the whole of R,eitheritisinjective(ithasnoself- intersection), or it is simply periodic (i.e., there is some T>0sothat(t + T )=(t), for all t R and is injective on [0,T[),oritisreducedtoasinglepoint.2

We conclude this section with the definition of the Lie derivative of a vector field with respect to another vector field. 448 CHAPTER 6. VECTOR FIELDS, INTEGRAL CURVES, FLOWS Say we have two vector fields X and Y on M.Forany p M,wecanflowalongtheintegralcurveofX with 2 initial condition p to t(p)(fort small enough) and then evaluate Y there, getting Y (t(p)).

Now, this vector belongs to the tangent space Tt(p)(M), but Y (p) T (M). 2 p

So to “compare” Y (t(p)) and Y (p), we bring back Y (t(p)) to Tp(M)byapplyingthetangentmap,d t,att(p), to Y (t(p)). (Note that to alleviate the notation, we use the slight abuse of notation d t instead of d( t) (p).) t 6.3. INTEGRAL CURVES, FLOW, ONE-PARAMETER GROUPS 449

Then, we can form the di↵erence d t(Y (t(p))) Y (p), divide by t and consider the limit ast goes to 0.

Definition 6.9. Let M be a Ck+1 manifold. Given any two Ck vector fields, X and Y on M,foreveryp M, the Lie derivative of Y with respect to X at p,denoted2 (LX Y )p,isgivenby

d t(Y (t(p))) Y (p) (LX Y )p =lim t 0 t !d = (d t(Y (t(p)))) . dt t=0 It can be shown that (LX Y )p is our old friend, the Lie bracket, i.e.,

(LX Y )p =[X, Y ]p.

(For a proof, see Warner [33] or O’Neill [30]). 450 CHAPTER 6. VECTOR FIELDS, INTEGRAL CURVES, FLOWS In terms of Definition 6.4, observe that

( t) Y (p) Y (p) (LX Y )p =lim ⇤ t 0 t ! ⇤Y (p) Y (p) =lim t t 0 t ! d = ⇤Y (p) , dt t t=0 1 since ( t) =t.