Chapter 2 Introduction To Manifolds And Lie Groups
Chapter 2Introduction to Manifolds and LieGroups2.1The Derivative of a Function Between Normed Vector SpacesIn most cases, E Rn and F Rm. However, it issometimes necessary to allow E and F to be infinite dimensional.Let E and F be two normed vector spaces, let A Ebe some open subset of E, and let a 2 A be some elementof A. Even though a is a vector, we may also call it apoint.55
56CHAPTER 2. INTRODUCTION TO MANIFOLDS AND LIE GROUPSThe idea behind the derivative of the function f at a isthat it is a linear approximation of f in a small openset around a.The difficulty is to make sense of the quotientf (a h)hf (a)where h is a vector.We circumvent this difficulty in two stages.A first possibility is to consider the directional derivativewith respect to a vector u 6 0 in E.We can consider the vector f (a tu) f (a), where t 2 R(or t 2 C). Now,f (a tu)tmakes sense.f (a)
2.1. THE DERIVATIVE OF A FUNCTION BETWEEN NORMED VECTOR SPACES57The idea is that the map from (r, s) to F given byt 7! f (a tu)defines a curve (segment) in F , and the directional derivative Duf (a) defines the direction of the tangent line at ato this curve.Definition 2.1. Let E and F be two normed spaces, letA be a nonempty open subset of E, and let f : A ! Fbe any function. For any a 2 A, for any u 6 0 in E,the directional derivative of f at a w.r.t. the vector u,denoted by Duf (a), is the limit (if it exists)limt!0, t2Uf (a tu)tf (a),where U {t 2 R a tu 2 A, t 6 0}(or U {t 2 C a tu 2 A, t 6 0}).Since the map t 7! a tu is continuous, and since A {a}is open, the inverse image U of A {a} under the abovemap is open, and the definition of the limit in Definition2.1 makes sense.
58CHAPTER 2. INTRODUCTION TO MANIFOLDS AND LIE GROUPSThe directional derivative is sometimes called the Gâteauxderivative.In the special case where E R and F R, and we letu 1 (i.e., the real number 1, viewed as a vector), it isimmediately verified that D1f (a) f 0(a).When E R (or E C) and F is any normed vectorspace, the derivative D1f (a), also denoted by f 0(a), provides a suitable generalization of the notion of derivative.However, when E has dimension 2, directional derivatives present a serious problem, which is that their definition is not sufficiently uniform.A function can have all directional derivatives at a, andyet not be continuous at a. Two functions may have alldirectional derivatives in some open sets, and yet theircomposition may not.
2.1. THE DERIVATIVE OF A FUNCTION BETWEEN NORMED VECTOR SPACES59Thus, we introduce a more uniform notion.Definition 2.2. Let E and F be two normed spaces, letA be a nonempty open subset of E, and letf : A ! F be any function. For any a 2 A, we say that fis di erentiable at a 2 A if there is a linear continuousmap, L : E ! F , and a function, (h), such thatf (a h) f (a) L(h) (h)khkfor every a h 2 A, wherelimh!0, h2U (h) 0,with U {h 2 E a h 2 A, h 6 0}.The linear map L is denoted by Df (a), or Dfa, or df (a),or dfa, or f 0(a), and it is called the Fréchet derivative,or derivative, or total derivative, or total di erential ,or di erential , of f at a.
60CHAPTER 2. INTRODUCTION TO MANIFOLDS AND LIE GROUPSSince the map h 7! a h from E to E is continuous,and since A is open in E, the inverse image U of A {a}under the above map is open in E, and it makes sense tosay thatlim (h) 0.h!0, h2UNote that for every h 2 U , since h 6 0, (h) is uniquelydetermined since (h) f (a h)f (a)khkL(h),and the value (0) plays absolutely no role in this definition.It does no harm to assume that (0) 0, and we willassume this from now on.
2.1. THE DERIVATIVE OF A FUNCTION BETWEEN NORMED VECTOR SPACES61Note that the continuous linear map L is unique, if itexists.The following proposition shows that our new definition isconsistent with the definition of the directional derivative.Proposition 2.1. Let E and F be two normed spaces,let A be a nonempty open subset of E, and letf : A ! F be any function. For any a 2 A, if Df (a)is defined, then f is continuous at a and f has a directional derivative Duf (a) for every u 6 0 in E.Furthermore,Duf (a) Df (a)(u).The uniqueness of L follows from Proposition 2.1.
62CHAPTER 2. INTRODUCTION TO MANIFOLDS AND LIE GROUPSAlso, when E is of finite dimension, it is easily shownthat every linear map is continuous, and this assumptionis then redundant.If Df (a) exists for every a 2 A, we get a mapDf : A ! L(E; F ),called the derivative of f on A, and also denoted bydf . Here, L(E; F ) denotes the vector space of continuouslinear maps from E to F .When E is of finite dimension n, for any basis (u1, . . . , un)of E, we can define the directional derivatives with respectto the vectors in the basis (u1, . . . , un)This way, we obtain the definition of partial derivatives,as follows.
2.1. THE DERIVATIVE OF A FUNCTION BETWEEN NORMED VECTOR SPACES63Definition 2.3. For any two normed spaces E and F , ifE is of finite dimension n, for every basis (u1, . . . , un) forE, for every a 2 E, for every function f : E ! F , thedirectional derivatives Duj f (a) (if they exist) are calledthe partial derivatives of f with respect to the basis(u1, . . . , un). The partial derivative Duj f (a) is also de@fnoted by @j f (a), or(a).@xj@f(a) for a partial derivative, although@xjcustomary and going back to Leibniz, is a “logical obscenity.”The notationIndeed, the variable xj really has nothing to do with theformal definition.This is just another of these situations where tradition isjust too hard to overthrow!
64CHAPTER 2. INTRODUCTION TO MANIFOLDS AND LIE GROUPSProposition 2.2. Given two normed spaces E andF , if f : E ! F is a constant function, thenDf (a) 0, for every a 2 E. If f : E ! F is a!continuous affine map, then Df (a) f , for every!a 2 E, where f is the linear map associated with f .Proposition 2.3. Given a normed space E and anormed vector space F , for any two functionsf, g : E ! F , for every a 2 E, if Df (a) and Dg(a)exist, then D(f g)(a) and D( f )(a) exist, andD(f g)(a) Df (a) Dg(a),D( f )(a) Df (a).Proposition 2.4. Given three normed vector spacesE1, E2, and F , for any continuous bilinear mapf : E1 E2 ! F , for every (a, b) 2 E1 E2, Df (a, b)exists, and for every u 2 E1 and v 2 E2,Df (a, b)(u, v) f (u, b) f (a, v).
2.1. THE DERIVATIVE OF A FUNCTION BETWEEN NORMED VECTOR SPACES65We now state the very useful chain rule.Theorem 2.5. Given three normed spaces E, F , andG, let A be an open set in E, and let B an open setin F . For any functions f : A ! F and g : B ! G,such that f (A) B, for any a 2 A, if Df (a) existsand Dg(f (a)) exists, then D(g f )(a) exists, andD(g f )(a) Dg(f (a)) Df (a).Proposition 2.6. Given two normed spaces E andF , let A be some open subset in E, let B be someopen subset in F , let f : A ! B be a bijection from Ato B, and assume that Df exists on A and that Df 1exists on B. Then, for every a 2 A,Df1(f (a)) (Df (a)) 1.Proposition 2.6 has the remarkable consequence that thetwo vector spaces E and F have the same dimension.
66CHAPTER 2. INTRODUCTION TO MANIFOLDS AND LIE GROUPSIn other words, a local property, the existence of a bijection f between an open set A of E and an open set Bof F , such that f is di erentiable on A and f 1 is differentiable on B, implies a global property, that the twovector spaces E and F have the same dimension.If both E and F are of finite dimension, for any basis(u1, . . . , un) of E and any basis (v1, . . . , vm) of F , everyfunction f : E ! F is determined by m functionsfi : E ! R (or fi : E ! C), wheref (x) f1(x)v1 · · · fm(x)vm,for every x 2 E.Then, we getDf (a)(uj ) Df1(a)(uj )v1 · · · Dfi(a)(uj )vi · · · Dfm(a)(uj )vm,that is,Df (a)(uj ) @j f1(a)v1 · · · @j fi(a)vi · · · @j fm(a)vm.
2.1. THE DERIVATIVE OF A FUNCTION BETWEEN NORMED VECTOR SPACES67The linear map Df (a) is determined by the m n-matrixJ(f )(a) (@j fi(a)), or@fiJ(f )(a) (a) :@xj01@1f1(a) @2f1(a) . . . @nf1(a)B @1f2(a) @2f2(a) . . . @nf2(a) CCJ(f )(a) B.@ .A@1fm(a) @2fm(a) . . . @nfm(a)or0@f1B @x1 (a)BB @f2B(a)BJ(f )(a) B @x1B .B@ @fm(a)@x1@f1(a)@x2@f2(a)@x2.@fm(a)@x21@f1(a) C@xn C@f2 C.(a) C.@xn CC. C.C@fm A.(a)@xn.
68CHAPTER 2. INTRODUCTION TO MANIFOLDS AND LIE GROUPSThis matrix is called the Jacobian matrix of Df at a.When m n, the determinant, det(J(f )(a)), of J(f )(a)is called the Jacobian of Df (a).We know that this determinant only depends on Df (a),and not on specific bases. However, partial derivativesgive a means for computing it.When E Rn and F Rm, for any functionf : Rn ! Rm, it is easy to compute the partial derivatives@fi(a).@xjWe simply treat the function fi : Rn ! R as a function ofits j-th argument, leaving the others fixed, and computethe derivative as the usual derivative.
2.1. THE DERIVATIVE OF A FUNCTION BETWEEN NORMED VECTOR SPACES69Example 2.1. For example, consider the functionf : R2 ! R2, defined byf (r, ) (r cos , r sin ).Then, we haveJ(f )(r, ) cos r sin sin r cos and the Jacobian (determinant) has valuedet(J(f )(r, )) r.
70CHAPTER 2. INTRODUCTION TO MANIFOLDS AND LIE GROUPSIn the case where E R (or E C), for any functionf : R ! F (or f : C ! F ), the Jacobian matrix ofDf (a) is a column vector. In fact, this column vector isjust D1f (a). Then, for every 2 R (or 2 C),Df (a)( ) D1f (a).Definition 2.4. Given a function f : R ! F(or f : C ! F ), where F is a normed space, the vectorDf (a)(1) D1f (a)is called the vector derivative or velocity vector (in thereal case) at a. We usually identify Df (a) with its Jacobian matrix D1f (a), which is the column vector corresponding to D1f (a).By abuse of notation, we also let Df (a) denote the vectorDf (a)(1) D1f (a).
2.1. THE DERIVATIVE OF A FUNCTION BETWEEN NORMED VECTOR SPACES71When E R, the physical interpretation is that f definesa (parametric) curve that is the trajectory of some particlemoving in Rm as a function of time, and the vector D1f (a)is the velocity of the moving particle f (t) at t a.Example 2.2.1. When A (0, 1), and F R3, a functionf : (0, 1) ! R3 defines a (parametric) curve in R3. Iff (f1, f2, f3), its Jacobian matrix at a 2 R is01@f1B @t (a)CBCB @f2 CJ(f )(a) B(a)CB @tC@ @fA3(a)@t
72CHAPTER 2. INTRODUCTION TO MANIFOLDS AND LIE GROUPS2. When E R2, and F R3, a function ' : R2 ! R3defines a parametric surface. Letting ' (f, g, h),its Jacobian matrix at a 2 R2 is01@f@fB @u (a) @v (a)CBC@g CB @gJ(')(a) B (a)(a) CB @u@v C@ @h@h A(a)(a)@u@v3. When E R3, and F R, for a functionf : R3 ! R, the Jacobian matrix at a 2 R3 is @f@f@fJ(f )(a) (a)(a)(a) .@x@y@z
2.1. THE DERIVATIVE OF A FUNCTION BETWEEN NORMED VECTOR SPACES73More generally, when f : Rn ! R, the Jacobian matrixat a 2 Rn is the row vector @f@fJ(f )(a) (a) · · ·(a) .@x1@xnIts transpose is a column vector called the gradient of fat a, denoted by gradf (a) or rf (a).Then, given any v 2 Rn, note that@f@fDf (a)(v) (a) v1 · · · (a) vn gradf (a) · v,@x1@xnthe scalar product of gradf (a) and v.When E, F , and G have finite dimensions, if A is an opensubset of E, B is an open subset of F , for any functionsf : A ! F and g : B ! G, such that f (A) B, for anya 2 A, letting b f (a), and h g f , if Df (a) existsand Dg(b) exists, by Theorem 2.5, the Jacobian matrixJ(h)(a) J(g f )(a) is given by
74CHAPTER 2. INTRODUCTION TO MANIFOLDS AND LIE GROUPSJ(h)(a) J(g)(b)J(f )(a) 0@g1@g1(b)(b) . . .B @y1@y2BB @g2@g2BB @y (b) @y (b) . . .2B 1B .B .B@ @gm@gm(b)(b) . . .@y1@y210@f1@g1(a)(b) C B@yn C B @x1B @f2@g2 CB(a)(b) CC@yn C BB @x1B . CB. CCB .@gm A @ @fn(a)(b)@x1@yn@f1(a) . . .@x2@f2(a) . . .@x2.@fn(a) . . .@x21@f1(a)@xp CC@f2 C(a)C@xp CC. C. CC@fn A(a)@xpThus, we have the familiar formulak nX @gi @fk@hi(a) (b)(a).@xj@yk @xjk 1Given two normed spaces E and F of finite dimension,given an open subset A of E, if a function f : A ! F isdi erentiable at a 2 A, then its Jacobian matrix is welldefined.
2.1. THE DERIVATIVE OF A FUNCTION BETWEEN NORMED VECTOR SPACES75One should be warned that the converse is false. Thereare functions such that all the partial derivatives existat some a 2 A, but yet, the function is not di erentiableat a, and not even continuous at a.However, there are sufficient conditions on the partialderivatives for Df (a) to exist, namely, continuity of thepartial derivatives.If f is di erentiable on A, then f defines a functionDf : A ! L(E; F ).It turns out that the continuity of the partial derivativeson A is a necessary and sufficient condition for Df toexist and to be continuous on A.To prove this, we need an important result known as themean value theorem.
76CHAPTER 2. INTRODUCTION TO MANIFOLDS AND LIE GROUPSIf E is a vector space (over R or C), given any two pointsa, b 2 E, the closed segment [a, b] is the set of all pointsa (b a), where 0 1, 2 R, and the opensegment ]a, b[ is the set of all points a (b a), where0 1, 2 R.The following result is known as the mean value theorem.Proposition 2.7. Let E and F be two normed vectorspaces, let A be an open subset of E, and let f : A !F be a continuous function on A. Given any a 2 Aand any h 6 0 in E, if the closed segment [a, a h] iscontained in A, if f : A ! F is di erentiable at everypoint of the open segment ]a, a h[, and ifsupx2]a,a h[for some MkDf (x)k M0, thenkf (a h)f (a)k M khk.As a corollary, if L : E ! F is a continuous linearmap, thenkf (a h)f (a)L(h)k M khk,where M supx2]a,a h[ kDf (x)Lk.
2.1. THE DERIVATIVE OF A FUNCTION BETWEEN NORMED VECTOR SPACES77Proposition 2.8. Let f : A ! F be any function bewteen two normed vector spaces E and F , where A is anopen subset of E. If A is connected and if Df (a) 0for all a 2 A, then f is a constant function on A.The mean value theorem also implies is the following important result.Theorem 2.9. Given two normed affine spaces E andF , where E is of finite dimension n and where(u1, . . . , un) is a basis of E, given any open subset Aof E, given any function f : A ! F , the derivativeDf : A ! L(E; F ) is defined and continuous on A i @fevery partial derivative @j f (or) is defined and@xjcontinuous on A, for all j, 1 j n. As a corollary,if F is of finite dimension m, and (v1, . . . , vm) is abasis of F , the derivative Df : A ! L(E; F ) is definedandon A i every partial derivative @j fi continuous @fioris defined and continuous on A, for all i, j,@xj1 i m, 1 j n.
78CHAPTER 2. INTRODUCTION TO MANIFOLDS AND LIE GROUPSDefinition 2.5. Given two normed affine spaces E andF , and an open subset A of E, we say that a functionf : A ! F is a C 0-function on A if f is continuous onA. We say that f : A ! F is a C 1-function on A if Dfexists and is continuous on A.Let E and F be two normed affine spaces, let U Ebe an open subset of E and let f : E ! F be a functionsuch that Df (a) exists for all a 2 U .If Df (a) is injective for all a 2 U , we say that f isan immersion (on U ) and if Df (a) is surjective for alla 2 U , we say that f is a submersion (on U ).When E and F are finite dimensional with dim(E) nand dim(F ) m, if m n, then f is an immersion i theJacobian matrix J(f )(a), has full rank (n) for all a 2 Eand if nm, then f is a submersion i the Jacobianmatrix J(f )(a), has full rank (m) for all a 2 E.
2.1. THE DERIVATIVE OF A FUNCTION BETWEEN NORMED VECTOR SPACES79A very important theorem is the inverse function theorem. In order for this theorem to hold for infinite dimensional spaces, it is necessary to assume that our normedspaces are complete.Given a normed vector space, E, we say that a sequence,(un)n, with un 2 E, is a Cauchy sequence i for every 0, there is some N 0 so that for all m, n N ,kunumk .A normed vector space, E, is complete i every Cauchysequence converges.A complete normed vector space is also called a Banachspace, after Stefan Banach (1892-1945).Fortunately, R, C, and every finite dimensional (real orcomplex) normed vector space is complete.
80CHAPTER 2. INTRODUCTION TO MANIFOLDS AND LIE GROUPSA real (resp. complex) vector space, E, is a real (resp.complex) Hilbert spacepif it is complete as a normed spacewith the norm kuk hu, ui induced by its Euclidean(resp. Hermitian) inner product (of course, positive, definite).Definition 2.6. Given two topological spaces E andF and an open subset A of E, we say that a functionf : A ! F is a local homeomorphism from A to F iffor every a 2 A, there is an open set U A containinga and an open set V containing f (a) such that f is ahomeomorphism from U to V f (U ).If B is an open subset of F , we say that f : A ! Fis a (global) homeomorphism from A onto B if f is ahomeomorphism from A to B f (A).
2.1. THE DERIVATIVE OF A FUNCTION BETWEEN NORMED VECTOR SPACES81If E and F are normed spaces, we say that f : A ! F isa local di eomorphism from A to F if for every a 2 A,there is an open set U A containing a and an open setV containing f (a) such that f is a bijection from U toV , f is a C 1-function on U , and f 1 is a C 1-function onV f (U ).We say that f : A ! F is a (global) di eomorphismfrom A to B if f is a homeomorphism from A to B f (A), f is a C 1-function on A, and f 1 is a C 1-functionon B.Note that a local di eomorphism is a local homeomorphism.Also, as a consequence of Proposition 2.6, if f is a di eomorphism on A, then Df (a) is a linear isomorphism forevery a 2 A.
82CHAPTER 2. INTRODUCTION TO MANIFOLDS AND LIE GROUPSTheorem 2.10. (Inverse Function Theorem) Let Eand F be complete normed spaces, let A be an opensubset of E, and let f : A ! F be a C 1-function onA. The following properties hold:(1) For every a 2 A, if Df (a) is a linear isomorphism(which means that both Df (a) and (Df (a)) 1 arelinear and continuous),1 then there exist some opensubset U A containing a, and some open subsetV of F containing f (a), such that f is a di eomorphism from U to V f (U ). Furthermore,Df1(f (a)) (Df (a)) 1.For every neighborhood N of a, the image f (N ) ofN is a neighborhood of f (a), and for every openball U A of center a, the image f (U ) of U contains some open ball of center f (a).1Actually, since E and F are Banach spaces, by the Open Mapping Theorem, it is sufficient to assumethat Df (a) is continuous and bijective; see Lang [22].
2.1. THE DERIVATIVE OF A FUNCTION BETWEEN NORMED VECTOR SPACES83(2) If Df (a) is invertible for every a 2 A, thenB f (A) is an open subset of F , and f is a localdi eomorphism from A to B. Furthermore, if fis injective, then f is a di eomorphism from A toB.Part (1) of Theorem 2.10 is often referred to as the “(local) inverse function theorem.” It plays an importantrole in the study of manifolds and (ordinary) di erentialequations.If E and F are both of finite dimension, the case whereDf (a) is just injective or just surjective is also importantfor defining manifolds, using implicit definitions.
84CHAPTER 2. INTRODUCTION TO MANIFOLDS AND LIE GROUPSIf Df : A ! L(E; F ) exists for all a 2 A, then we canconsider taking the derivative DDf (a) of Df at a.If it exists, DDf (a) is a continuous linear map inL(E; L(E; F )), and we denote DDf (a) as D2f (a).It is known that the vector space L(E; L(E; F )) is isomorphic to the vector space of continuous bilinear mapsL2(E 2; F ), so we can view D2f (a) as a bilinear map inL2(E 2; F ).It is also known by Schwarz’s lemma that D2f (a) is symmetric (partial derivatives commute).Therefore, for every a 2 A, where it exists, D2f (a) belongs to the space Sym2(E 2; F ) of continuous symmetricbilinear maps from E 2 to F .
2.1. THE DERIVATIVE OF A FUNCTION BETWEEN NORMED VECTOR SPACES85If E has finite dimension n and F R, with respect toany basis (e1, . . . , en) of E, D2f (a)(u, v) is the value ofthe quadratic form u Hessf (a)v, where 2 @ fHessf (a) (a)@xi@xjis the Hessian matrix of f at a.By induction, if Dmf : A ! Symm(E m; F ) exists form1, where Symm(E m; F ) denotes the vector spaceof continuous symmetric multilinear maps from E m toF , and if DDmf (a) exists for all a 2 A, we obtainthe (m 1)th derivative Dm 1f of f , and Dm 1f 2Symm 1(E m 1; F ), where Symm 1(E m 1; F ) is the vector space of continuous symmetric multilinear maps fromE m 1 to F .
86CHAPTER 2. INTRODUCTION TO MANIFOLDS AND LIE GROUPSFor any m 1, we say that the map f : A ! F is a C mfunction (or simply that f is C m) if Df, D2f, . . . , Dmfexist and are continuous on A.We say that f is C 1 or smooth if Dmf exists and iscontinuous on A for all m 1. If E has finite dimensionn, it can be shown that f is smooth i all of its partialderivatives@ mf(a)@xi1 · · · @ximare defined and continuous for all a 2 A, all mall i1, . . . , im 2 {1, . . . , n}.1, andThe function f : A ! F is a C m di eomorphism between A and B f (A) if f is a bijection from A to Band if f and f 1 are C m.Similarly, f is a smooth di eomorphism between A andB f (A) if f is a bijection from A to B and if f andf 1 are smooth.
2.2. SERIES AND POWER SERIES OF MATRICES2.287Series and Power Series of MatricesSince a number of important functions on matrices aredefined by power series, in particular the exponential, wereview quickly some basic notions about series in a complete normed vector space.Given a normedP1 vector space (E, k k), a series is an infinite sum k 0 ak of elements ak 2 E.We denote by Sn the partial sum of the first n 1 elements,nXSn ak .k 0P1Definition 2.7. We say that the series k 0 ak converges to the limit a 2 E if the sequence (Sn) convergesto a. In this case, we sayP1that the series is convergent.We say that the seriesP1 k 0 ak converges absolutely ifthe series of norms k 0 kak k is convergent.
88CHAPTER 2. INTRODUCTION TO MANIFOLDS AND LIE GROUPSThere are series that are convergent but not absolutelyconvergent; for example, the series1X( 1)k 1.kk 1However, if E is complete (which means that every Cauchysequence converges), the converse is an enormously usefulresult.Proposition 2.11. Let (E,P1k k) be a complete normedvector space. If a series k 0 ak is absolutely convergent, then it is convergent.Remark: It can be shown that if (E, k k) is a normedvector space such that every absolutely convergent seriesis also convergent, then E must be complete.If E C, then there are several conditions that implythe absolute convergence of a series.
2.2. SERIES AND POWER SERIES OF MATRICES89The ratio test is the following test. Suppose there is someN 0 such that an 6 0 for all n N , and eitheran 1r limn7!1 anexists, or the sequence of ratios diverges to infinity, inwhich Pcase we write r 1. Then, if 0 r 1, theseries nk 0 ak converges absolutely, else if 1 r 1,the series diverges.If (rn) is a sequence of real numbers, recall thatlim sup rn lim sup{rk }.n7!1n7!1 k nIf rn 0 for all n, then it is easy to see that r is characterized as follows:For every 0, there is some N 2 N such that rn r for all n N , and rn r for infinitely many n.
90CHAPTER 2. INTRODUCTION TO MANIFOLDS AND LIE GROUPSThen, the root test is this. Letr lim sup an 1/nn7!1if the limit exists (is finite),Pn else write r 1. Then, if0 r 1, the series k 0 ak converges absolutely, elseif 1 r 1, the series diverges.The root test also applies if (E, k k) is a complete normedvector space by replacing an by kank.LetP1k 0 akbe a series of elements ak 2 E and letr lim sup kank1/nn7!1if the limit exists (is finite),Pn else write r 1. Then, if0 r 1, the series k 0 ak converges absolutely, elseif 1 r 1, the series diverges.
2.2. SERIES AND POWER SERIES OF MATRICES91A power series with coefficients ak 2 C in the indeterminate z is a formal expression f (z) of the formf (z) 1Xak z k ,k 0For any fixed value z 2 C, the series f (z) may or may notconverge. It always converges for z 0, sincef (0) a0.A fundamental fact about power series is that they havea radius of convergence.Proposition 2.12. Given any power seriesf (z) 1Xak z k ,k 0there is a nonnegative real R, possibly infinite, calledthe radius of convergence of the power series, suchthat if z R, then f (z) converges absolutely, else if z R, then f (z) diverges. Moreover (Hadamard),we have1R .lim supn7!1 an 1/n
92CHAPTER 2. INTRODUCTION TO MANIFOLDS AND LIE GROUPSNote that Proposition 2.12 does not say anything aboutthe behavior of the power series for boundary values, thatis, values of z such that z R.P1Proposition 2.13. Let f (z) k 0 ak z k be a powerseries with coefficients ak 2 C. Suppose there is someN 0 such that an 6 0 for all n N , and eitherR limn7!1anan 1exists, or the sequence on the righthand side divergesto infinity, inPwhich case we write R 1. Then, thekpower series 1k 0 ak z has radius of convergence R.Power series behave very well with respect to derivatives.
2.2. SERIES AND POWER SERIES OF MATRICES93Proposition2.14. Suppose the power series f (z) P1kk 0 ak z (with real coefficients) has radius of convergence R P0. Then, f 0(z) exists if z R, thek 1power series 1has radius of convergencek 1 kak zR, and1Xf 0(z) kak z k 1.k 1P1Let us now assume that f (z) k 0 ak z k is a powerseries with coefficients ak 2 C, and that its radius ofconvergence is R.Given any matrix A 2 Mn(C) we can form the powerseries obtained by substituting A for z,f (A) 1Xak Ak .k 0Let k k be any matrix norm on Mn(C).
94CHAPTER 2. INTRODUCTION TO MANIFOLDS AND LIE GROUPSP1Proposition 2.15. Let f (z) k 1 ak z k be a powerseries with complex coefficients, write R for its radiusof convergence, and assume that R 0. ForP1every ksuch that 0 R, the series f (A) k 1 ak Ais absolutely convergent for all A 2 Mn(C) such thatkAk . Furthermore, f is continuous on the openball B(R) {A 2 Mn(C) kAk R}.Note that unlike the case where A 2 C, if kAk R, wecannot claim that the series f (A) diverges.This has to do with the fact that even for the operatornorm we may have kAnk kAkn. We leave it as anexercise to find an example of a series and a matrix Awith kAk R, and yet f (A) converges.
2.2. SERIES AND POWER SERIES OF MATRICES95As an application of Proposition 2.15, the exponentialpower series1XAkAe exp(A) k!k 0is absolutely convergent for all A 2 Mn(C), and continuous everywhere.Proposition 2.15 also implies that the serieslog(I A) 1X( 1)k 1is absolutely convergent if kAk 1.kk 1 Ak
96CHAPTER 2. INTRODUCTION TO MANIFOLDS AND LIE GROUPSNow, it is known (see Cartan [8]) that the formal powerseries1XAkE(A) k!k 1andL(A) 1Xk 1( 1)kk 1 Akare mutual inverses; that is,E(L(A)) A,L(E(A)) A,Observe that E(A) eA I exp(A)log(I A). It follows thatfor all A.I and L(A) log(exp(A)) A for all A with kAk log(2)exp(log(I A)) I A for all A with kAk 1.
2.2. SERIES AND POWER SERIES OF MATRICES97Finally, let us consider thegeneralization of the notionP1of a power series f (t) k 1 ak tk of a real variable t,where the coefficients ak belong to a complete normedvector space (F, k k).Proposition 2.16. Let (F, k k) be a complete normedvector space. Given any power seriesf (t) 1Xak tk ,k 0with t 2 R and ak 2 F , there is a nonnegative real R,possibly infinite, called the radius of convergenceof the power series, such that if t R, then f (t) converges absolutely, else if t R, then f (t) diverges.Moreover, we have1R .1/nlim supn7!1 kank
98CHAPTER 2. INTRODUCTION TO MANIFOLDS AND LIE GROUPSProposition 2.17. Let (F, k k) be a completePnormedkvector space. Suppose the power series f (t) 1k 0 ak t(with coefficients ak 2 F ) has radius of convergence0R.Then,f(t) exists if t R, the power seriesP1k 1kathas radius of convergence R, andkk 1f 0(t) 1Xk 1kak tk 1.
2.3. LINEAR VECTOR FIELDS AND THE EXPONENTIAL2.399Linear Vector Fields and the ExponentialWe can apply Propositions 2.16 and 2.17 to the mapf : t 7! etA, where A is any matrix A 2 Mn(C).This power series has a infinite radius of convergence, andwe have11 k 1 k 1k 1 kXXtAt Af 0(t) k A AetA.k!(k 1)!k 1k 1Note thatAetA etAA.Given some open subset A of Rn, a vector field X on Ais a function X : A ! Rn, which assigns to every pointp 2 A a vector X(p) 2 Rn.Usually, we assume that X is at least C 1 function on A.
100CHAPTER 2. INTRODUCTION TO MANIFOLDS AND LIE GROUPSFor example, if f : A ! R is a C 1 function, then itsgradient defines a vector field X; namely, p 7! grad f (p).If f is C 2, then its second partials commute; that is,@ 2f@ 2f(p) (p),@xi@xj@xj @xi1 i, j n,so this vector field X (X1, . . . , Xn) has a very specialproperty:@Xi @Xj ,@xj@xi1 i, j n.This is a necessary condition for a vector field to be thegradient of some function, but not a sufficient conditionin general.
2.3. LINEAR VECTOR FIELDS AND THE EXPONENTIAL101The existence of such a function depends on the topological shape of the domain A.Understanding what are sufficient conditions to answerthe above question led to the development of di erentialforms and cohomology.Definition 2.8. Given a vector field X : A ! Rn, forany point p0 2 A, a C 1 curve : ( , ) ! Rn (with 0) is an integral curve for X with initial conditionp0 if (0) p0, and0(t) X( (t)) for all t 2 ( , ).Thus, an integral curve has the property that for everytime t 2 ( , ), the tangent vector 0(t) to the curveat the point (t) coincides with the vector X( (t)) givenby the vector field at the point (t).
102CHAPTER 2. INTRODUCTION TO MANIFOLDS AND LIE GROUPSDefinition 2.9. Given a C 1 vector field X : A ! Rn,for any point p0 2 A, a local flow for X at p0 is afunction' : J U ! Rn ,where J R is an open interval containing 0 and U is anopen subset of A containing p0, so that for every p 2 U ,the curve t 7! '(t, p) is an integral curve of X with initialcondition p.The theory of ODE tells us that if X is C 1, then for everyp0 2 A, there is a pair (J, U ) as above such that there isa unique C 1 local flow ' : J U !
64 CHAPTER 2. INTRODUCTION TO MANIFOLDS AND LIE GROUPS Proposition 2.2. Given two normed spaces E and F, if f: E ! F is a constant function, then Df(a) 0, for every a 2 E.Iff: E ! F is a continuous ane map, then Df(a) ! f , for every a 2 E, where! f is the linear map associated with f. Proposition 2.3. Given a normed space E and a
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