MAXIMAL TORI AND THE WEYL INTEGRATION FORMULA

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1 AXIAL ORI AND HE WEYL INERAION FORULA AKUI URAYAA 1. Introduction Consider a compact, connected Lie group. A maximal torus is a subgroup of that is also a torus k R k /Z k, and is a maximal such subgroup with respect to inclusion. he Weyl group of, defined as W = N( )/ for a maximal torus, depends definitionally on the choice of a maximal torus, but by showing that any two tori are conjugate, we can see that W is unique up to isomorphism. his also shows that every conjugacy class in meets, and so to compute the integral of a class function, i.e., a function invariant under conjugation (such as the inner product of two characters), it should suffice to integrate only over the torus. he formula that allows this is the important Weyl Integration Formula (heorem 4.1), which is fundamental in representation theory and in other areas like random matrix theory. What is in fact remarkable about the conjugacy theorem and the Weyl Integration formula is the variety of approaches one can take in proving them, although the conjugacy theorem was first found by Élie Cartan and the Weyl Integration formula by Hermann Weyl. Some are based on the Hopf-Rinow theorem from differential geometry [4], others are based on André Weil s proof using the Lefschetz fixed-point formula from algebraic topology [1], and still others are based on a (rather tedious, yet enlightening) move back and forth between and its associated Lie algebra, where there is a similar theorem on the conjugacy of Cartan subalgebras, which has an elegant proof due to Hunt [7], [8]. In previous versions of this paper, I have actually written up (most of) the Hopf-Rinow and Lie algebraic proofs, which I can provide for those interested. Here we take yet another approach, mostly following [3], which relies on the notion of mapping degree (see heorem 3.3). I would like to thank asho Kaletha for his guidance throughout our seminar, and Simon Segert for helping me in revising this paper. 2. Definitions and Preliminaries In this paper, Lie group will mean finite dimensional real Lie group, and will refer to a compact connected Lie group, unless stated otherwise. Definition. A Lie subgroup is a maximal torus if is a torus, i.e., a compact, connected, abelian subgroup, and is a maximal such subgroup with respect to inclusion. { e } is a torus. Since tori are compact and connected, if, then dim < dim. Since dim <, this shows maximal tori exist. We also note that a torus is a maximal abelian subgroup, for if A is larger, then A is a closed connected abelian subgroup, and hence a torus, contradicting maximality. he converse does not hold in general. Our notion of tori is not the intuitive one of k R k /Z k, the torus of dimension k. We would like to show they are equivalent. But first, we give two examples in Lie groups we have already studied closely: Example 2.1 (SU(n)). Let = { diag(exp(iθ 1 ),..., exp(iθ n )) θ j = 0 } SU(n) = { X SL(n, C) X X = I }. is compact, connected, and abelian since it is isomorphic to n 1, and maximal since any non-diagonal matrix would not commute with elements in ; thus, is a maximal torus of SU(n). Example 2.2 (Sp(n)). Let = { diag(exp(iθ 1 ),..., exp(iθ n )) } Sp(n) = { X L(n, H) X X = I } Date: January 15,

2 2 AKUI URAYAA is compact, connected, and abelian since it is isomorphic to n, and maximal since any non-diagonal matrix would not commute with elements in, and moreover, any diagonal matrix with entries in H \ C would not commute either; thus, is a maximal torus of Sp(n). We now will show that any maximal torus is isomorphic to k for some k, but we first need a few lemmas. Lemma 2.3. Let be a connected Lie group and U a neighborhood of e. = n=1 U n where U n consists of all n-fold products of elements of U. hen U generates, i.e., Proof. Let V = U U 1, where U 1 is the set of inverses of elements in U; V is open since the inverse map is continuous. Let H = n=1 V n, which is an open subgroup containing e. For g, gh = { gh h H } contains g and is open for any g, since left multiplication by g 1 is continuous. If we pick a representative g α H for each coset in /H, then = α g αh. Connectedness of implies = eh. We recall our definitions for the various adjoint actions. Recall from 10/12 that for any g, we have the conjugation action: Ad(g) :, h ghg 1. he differential of Ad at e provides a linear map Ad(g) : g g. From the chain rule, Ad(gh) = Ad(g) Ad(h), and so we have a homomorphism of Lie groups Differentiating this homomorphism we obtain Ad : L(g). ad : g End(g). We then have the following theorem from 10/12 concerning the exponential map and its relationship with these adjoint actions: heorem 2.4. here exists a smooth map exp : g with the following properties: (a) d exp (e) = id g. (b) Ad exp = exp L(g) ad. (c) Ad(x) exp = exp Ad(x), x. from which we have the easy consequence Lemma 2.5. Let be a compact Lie group, and g = Lie(). (a) he map exp : g is a local diffeomorphism near 0. (b) When is connected, exp g generates. Proof. (b) follows from (a) and Lemma 2.3, and so it suffices to show (a). Since d exp(e) = id g from heorem 2.4, we are done by the Inverse apping theorem. Note that this does not prove that exp is onto; for this to hold, it is sufficient to have exp be a homomorphism, which we prove in the lemma below. he lemma gives a relationship between the commutativity of a Lie group and its Lie algebra. iven the examples above, it is not too surprising that, as a result, we can establish a one-to-one correspondence between Cartan subalgebras and maximal tori; see [8], heorem 5.4, or one of my previous drafts. Lemma 2.6. Let be a compact Lie group, and g = Lie(). If X, Y g, [X, Y ] = 0, then e X and e Y commute. oreover, g is abelian if and only if 0 is abelian, and in this case e X+Y = e X e Y. Proof. Suppose [X, Y ] = 0. From heorem 2.4 we have Ad(exp X)(exp Y ) = exp(ad(e X )Y ) = exp(e ad X Y ) = exp(y ), since ad(x)y = 0 = e ad X (Y ) = Y. It follows that if ad = 0, then the group generated by exp g, which equals 0 by Lemmas 2.3 and 2.5, is abelian. Conversely, if 0 is abelian, then Ad(x) 0 = id 0, and so its differential Ad(x) = I on g. Differentiating with respect to x at x = 1 we get ad = 0. he last statement is a consequence of the Campbell-Baker-Hausdorff formula: ( e X e Y = exp X + Y [X, Y ] + 1 ) 1 [X, [X, Y ]] [Y, [Y, X]] +, 12 12

3 AXIAL ORI AND HE WEYL INERAION FORULA 3 where the remaining terms involve high-order iterations of bracket operations; see [6], heorem I.5.5. Finally, we have the result desired: heorem 2.7. If is a torus with dim = k, then R k /Z k k, as Lie groups. Proof. Let t = Lie( ). exp is a homomorphism t by Lemma 2.6. Since exp is surjective by Lemma 2.5 and Lemma 2.6, this implies t/ ker(exp). Since t R k as a vector space and since ker(exp) is discrete by Lemma 2.5(a) (it is a local bijection at the origin), ker(exp) Z r for r k. But since R k /Z r k R k r must be compact, k = r, and so R k /Z k k. Now let ϕ Aut( ). his induces the following commutative diagram of homomorphisms: exp 0 Z k R k k 0 L ϕ L ϕ ϕ exp 0 Z k R k k 0 where we identify with k R k /Z k and Lie( ) with R k. he pullback of ϕ is L ϕ Aut(R k ), so is an invertible matrix with integral entries by the fact that it preserves Z. hus, we can define L : Aut( k ) Aut(Z k ) = L(k, Z), ϕ L ϕ Z k. In a similar fashion, we can analyze a function f : S 1 ; this gives us a way to represent as a subgroup generated by some element. his is useful later in making our analysis of the torus easier, and so we make it precise: Definition. An element t k is a generator if the subgroup t generated by t is dense in k. heorem 2.8 (Kronecker). Let (t 1,..., t k ) R k, and let t be its image in k = (R/Z) k. hen t is a generator of k if and only if 1, t 1,..., t k are linearly independent over Q. Proof. A homomorphism f : k S 1 induces the commutative diagram 0 Z k R k k 0 L f L f exp 0 Z R S 1 0 and so L f(t 1,..., t k ) = α 1 t α k t k for α i Z. Now the following are equivalent: i) 1, t 1,..., t k are linearly dependent over Q. ii) q i t i Q for some n-tuple 0 (q 1,..., q k ) Q k. iii) α i t i Z for some n-tuple 0 (α 1,..., α k ) Z k. iv) t mod Z n is in the kernel of a nontrivial homomorphism f : k S 1. v) t mod Z n is not a generator. i ii iii are trivial, and iii iv follows from what was said. It remains to show iv v. Let t = t mod Z k ker f. If f is nontrivial, this kernel is not all of k and hence is a proper closed subgroup of k, so t is not a generator. Conversely, a nongenerator t is contained in a proper closed subgroup H k, and the quotient group k /H is a nontrivial compact connected abelian Lie group. hus k /H is a torus r, r > 0, and t is in the kernel of exp f k k /H r = S 1 S 1 proj S 1. Corollary 2.9. Every torus has a generator; indeed, they are dense in. Proof. By heorem 2.8, it suffices to show that k-tuples (t 1,..., t k ) such that 1, t 1,..., t k are linearly independent over Q are dense in R k. If 1, t 1,..., t i 1 are linearly independent, then linear independence of 1, t 1,..., t i excludes only countably many t i, and the result follows since R is uncountable. We now define the Weyl group mentioned in the introduction.

4 4 AKUI URAYAA Definition. Let be a maximal torus in and N = N( ) := { g g g 1 = }, the normalizer of in. hen, the group W = N/ is called the Weyl group of. Although it appears that the Weyl group depends on choice of, we will show all maximal tori are conjugate, implying that different yield isomorphic Weyl groups. Note the normalizer N is compact since if t is a generator, N is the preimage of under g gtg 1. he normalizer N of operates on by conjugation N, (n, t) ntn 1, but since the operation of on is trivial, we have the induced action of the Weyl group W, (n, t) ntn 1, which is well-defined since if n, n are in the same coset, say n, then n = nt for some t, and so t maps to n tn 1 = nt tt 1 n 1 = ntn 1 since is abelian. his gives us the following result: heorem he Weyl group is finite. Proof. Using the isomorphism above, we view the action of N above on as the continuous map N Aut( ) L L(k, Z), n Ad(n) Lie( ). hen, the identity map id Im(N 0 ) by definition, where N 0 is the connected component of the identity in N, and moreover since Im(N 0 ) is a connected subset of the discrete space L(k, Z) Aut(Lie( )), Im(N 0 ) = { id }, so N 0 acts trivially on. Now let α be a one-parameter subgroup, i.e., a homomorphism R N 0. hen, since α(r) N 0, and conjugation by N 0 acts trivially on, conjugation by α(r) also acts trivially on. hus, α(r) is a connected abelian group containing (α(r) is path connected by definition and is connected to e, which is in ), but since is a maximal connected abelian subgroup of, then α(r) =. hus, α(r). Now, since the exponential maps exp(tx) are one-parameter subgroups which are locally bijective by Lemma 2.5, the groups α(r) cover an open neighborhood of the identity in N 0 and hence generate N 0. herefore, = N 0. his implies W = N/N 0, and so W is compact and discrete since N is compact, i.e., W is finite. his is important since otherwise, the Weyl Integration Formula (heorem 4.1) would not make sense. We also remark that given w W, H Lie( ), λ Lie( ), the dual space of Lie( ), we can define the action w(h) = Ad(w)H, [w(λ)](h) = λ(w 1 (H)) = λ(ad(w 1 )H), and this action is faithful on Lie( ), Lie( ), making our definition of the Weyl group similar to that from 11/30; note that this is not absolutely precise, see [8], heorem 6.36, or one of my previous drafts, for details. 3. Conjugacy of aximal ori he goal of this section is to prove the following theorem: heorem 3.1 (aximal orus heorem). Any two maximal tori in a compact connected Lie group are conjugate, and every element of is contained in a maximal torus. he proof of this theorem relies on the notion of mapping degree, which relies first on orientations on and / (recall the latter has a natural smooth structure by requiring / /, (g, g ) gg is smooth). o begin, we need to know how the canonical volume form dg is transformed by ψ : /, (g, t) gtg 1, which is well-defined since if g, g are in the same coset, say g, then g = gt for some t, and so t maps to g tg 1 = gt tt 1 g 1 = gtg 1 since is abelian. Choose an inner product (, ) on Lie() which is unitary relative to Ad, which exists from 11/9 since the representation (Ad, Lie()) is finite-dimensional. Relative to (, ), we can decompose Lie() into Lie() = Lie(/ ) Lie( ), Lie(/ ) := Lie( ), where we are justified in identifying Lie( ) with Lie(/ ) since the projection π : / induces a map between Lie( ) Lie( ) and e (/ ) (the tangent space to / at e ), which is an isomorphism

5 AXIAL ORI AND HE WEYL INERAION FORULA 5 Lie( ) e (/ ) = Lie(/ ). his decomposition is invariant under Ad, since the torus acts trivially on Lie( ) and since acts nontrivially on every nonzero vector of Lie(/ ) by the fact that is a maximal abelian subgroup. he induced action on Lie(/ ) is Ad / : Aut(Lie(/ )). Now recall from 11/9 that there exist left-invariant volume forms d(g ), dt, dg on /,, respectively with volume 1. Let n = dim, k = dim, and π : / the projection map. We obtain a leftinvariant differential form π d(g ) Ω n k () from the volume form d(g ) on /, and the alternating form dt e Alt k (Lie( )) gives rise to the alternating form pr 2 dt e Alt k (Lie()), where pr 2 : Lie() = Lie(/ ) Lie( ) Lie( ) is projection onto the second summand. But pr 2 dt e determines a left-invariant differential form dτ Ω k such that dτ = dt by its group structure, and π d(g ) dτ is a left-invariant volume form on. We may choose our signs such that π d(g ) dτ = c dg with c > 0. We want to show c = 1. We first give a version of Fubini s theorem which is useful in our context: heorem 3.2 ( Fubini ). Let be a compact Lie group, H a closed subgroup, and d(gh) a left-invariant normalized volume form on /H. For any continuous real-valued function f on, ( ) f(g) dg = f(gh) dh d(gh). /H Proof. he right-hand side defines a normalized left-invariant integral on, so the theorem follows from the uniqueness of such an integral from 11/9. Now consider a bundle chart ϕ : π 1 U U of the -principal bundle π : / and let f be a nonzero nonnegative continuous real-valued function on with support in π 1 (U). hen, ( ) ( ) 0 ψ dg = f(gt) dt d(g ) = f(gt) dt d(g ), / using Fubini s theorem (heorem 3.2). We now change coordinates based on the following diagram associated to the bundle chart ϕ: π 1 (U) U H U ϕ pr 2 π / U π Note that, for fixed g π 1 (U), ϕ(g) = (u, s) with u = π(g) and some s, and ϕ(gt) = (u, ts), since ϕ is a right -map. We can arrange that the decomposition Lie() = Lie(/ ) Lie( ) coincides with that induced by the bundle chart ϕ, so that pr 2 dt = (ϕ 1 ) dτ. hen, ( ) ( ) f(gt) dt d(g ) = fϕ 1 (u, t) pr 2 dt pr 1 d(g ) = fϕ 1 (u, t) pr 1 d(g ) pr 2 dt U U U = fϕ 1 (ϕ 1 ) (π d(g ) dτ) = c fϕ 1 (ϕ 1 ) dg = c f dg, U so c = 1. We now have the volume form on and the volume form on /. dg = π d(g ) dτ, dτ = dt, α = pr 1 d(g ) pr 2 dt Definition. he determinant det(ψ) : / R of ψ is defined by the equation ψ dg = det(ψ) α. Definition. p / is a regular point of ψ if dψ(p) : p ( / ) ψ(p) is bijective. q is a regular value of ψ if dψ(p i ) : pi ( / ) q is bijective for each p i ψ 1 (q), i.e., if each p i ψ 1 (q) is a regular point. U

6 6 AKUI URAYAA heorem 3.3 (heorem on apping Degrees). here exists an integer deg(ψ) assigned to the homotopy class of ψ, called the mapping degree of ψ, such that, for every form α Ω n (), ψ α = deg(ψ) α. / If q is a regular value, then the mapping degree is given by deg(ψ) = sgn(det(ψ)(p)), In particular, if deg(f) 0, f is surjective. p ψ 1 (q) Proof. For clarity, we omit the proof for now. See section A. he proof of heorem 3.1, then, is easily proved by the following: Lemma 3.4 (Weyl Covering heorem). Let be a maximal torus in. hen the map ψ : /, (g, t) gtg 1, has mapping degree deg(ψ) = W, where W is the order of the Weyl group associated to. In particular, ψ is surjective. Proof of heorem 3.1 assuming Lemma 3.4. Let, be maximal tori and t a generator of by Corollary 2.9. By Lemma 3.4, there is a g with t g g 1. Hence g g 1, and by maximality = g g 1. Since ψ is surjective every element of is contained in some conjugate of. he statement for Lemma 3.4 is slightly different from Weyl Covering heorem found in, say, [5], heorem 3.7.2, which chooses to consider ψ through the factorization / ψ / W where / W denotes the orbit space of the action of W on / W given by w : / /, (g, t) (gw 1, wtw 1 ), for w W, which is well-defined as in Section 2. he statement is then that the map ψ has mapping degree 1, making the restriction of ψ to regular points/values a diffeomorphism. o prove Lemma 3.4, we proceed in steps. he first step gives a formula for det(ψ) in terms of the determinant on the Lie algebras. Proposition 3.5. he determinant of ψ is given by where I / is the identity on Lie(/ ). det(ψ)(g, t) = det(ad / (t 1 ) I / ), Proof. he forms dg, d(g ) are left-invariant under the action of, and the form dt is left-invariant under the action of. his allows us to consider the following composition: (g, t) ψ gt 1 g 1 (g, t) ψ (x, y) (gx, ty) (gx)(ty)(gx) 1 Ad(g)(Ad(t 1 )(x) y x 1 ) ψ gt 1 g 1 where ψ : : (g, t) gtg 1. In particular, (e, e) e. hus, the determinant we want to compute is the determinant of the differential of this map at (e, e), restricted to the subspace Lie(/ ) Lie( ) Lie() Lie( ) = Lie( ). Now L Ad(g) = Ad(g) has determinant 1, since Ad(g) is orthogonal and is connected. oreover, the differential of a product is the sum of the differentials, and so the determinant of q is the determinant of the linear endomorphism (X, Y ) Ad / (t 1 )X + Y X

7 AXIAL ORI AND HE WEYL INERAION FORULA 7 of Lie(/ ) Lie( ). In matrix form, this is [ Ad/ (t 1 ) I / I ], whose determinant is det(ad / (t 1 ) I / ) as claimed. o determine the mapping degree, we must count the number of elements in the preimage of a generator t, which are clearly regular values of ψ, and keep track of orientation. Lemma 3.6. Let t be a generator as in Corollary 2.9. hen, (a) ψ 1 (t) consists of W points, (b) det(ψ) > 0 at each of these points Proof. (a). Let N = N( ). hen, ψ(g, s) = t gsg 1 = t s = g 1 tg g 1 tg g 1 g g N. hus, ψ 1 (t) = { (g, g 1 tg) g N }. iven g 1, g 2 N such that g 2 g 1, there exists s such that g 2 = sg 1, so g2 1 tg 2 = g1 1 s 1 tsg 1 = g1 1 ts 1 sg 1 = g1 1 tg 1. Otherwise, g 1 g 2, and so (g 1, g1 1 tg 1) (g 2, g2 1 tg 2), and hence ψ 1 (t) has [N : ] = W elements. (b). he determinant we want is det(ad / (t 1 ) I / ). Since this determinant is real, if we show that Ad / (t 1 ) I / has no real eigenvalues, then eigenvalues will come in complex conjugate pairs, and hence, the determinant will be positive. Suppose Ad / (t 1 ) I / has a real eigenvalue. hen Ad / (t 1 ) does also, and would be ±1 since Ad / (t 1 ) is orthogonal. In this case, Ad / (t 2 ) has 1 as eigenvalue. Since t 2 is also a generator of, it suffices to show Ad / (t) cannot have 1 as an eigenvalue. If Ad / (t)x = X for some X Lie(/ ), then X is fixed by the action of t and hence by all of. But then the one-parameter subgroup H = { exp(sx) s R } is left pointwise fixed under Ad( ). hus H is abelian and connected, implying H and X Lie( ) Lie(/ ) = 0. We record a Corollary for later: Corollary 3.7. If t is a generating element of, then Ad / (t) has no real eigenvalues. Hence dim / is even. Finally, we can prove Lemma 3.4: Proof of Lemma 3.4. Let t be a generator of as in Corollary 2.9. he mapping degree of ψ is W since there are W points in ψ 1 (t), and det(ψ) > 0 implies ψ preserves orientation for all elements in ψ 1 (t). 4. Weyl Integration Formula We finally get the main theorem of this paper. What is amazing is that this formula is a trivial consequence of the Weyl Covering heorem (Lemma 3.4) above, despite being unexpected at first glance. heorem 4.1 (Weyl Integration Formula). Let be a maximal torus of, and f a continuous function on. hen, [ ] W f(g) dg = det(i / Ad / (t 1 )) f(gtg 1 ) dg dt. Proof. Let f t : / R be given by g f(gtg 1 ), so in the last integral f(gtg 1 ) = f t π(g), where π : / is the canonical projection. hen, [ ] det(i / Ad / (t 1 )) f(gtg 1 ) dg dt [ ] [ ] = det(i / Ad / (t 1 )) f t π(g) dg dt = det(i / Ad / (t 1 )) f t (g ) dg dt, /

8 8 AKUI URAYAA by definition of dg. We then use Fubini s theorem (heorem 3.2), Lemma 3.4, and Proposition 3.5 to get [ ] = f ψ(g, t) det(ψ)(g, t) dg dt = f ψ(g, t) det(ψ)(g, t) pr 2 dt pr 1 dg. / / Since dim / is even by Corollary 3.7, pr 1 dg is an even form, and so we have = f ψ(g, t) det(ψ)(g, t) pr 1 dg pr 2 dt = f ψ(g, t) det(ψ)(g, t)α = / / f ψ(g, t) ψ dg = / / ψ (f dg) = deg(ψ) A. heorem on apping Degrees f(g) dg = W f(g) dg. We prove here heorem 3.3, which we omitted the proof for before, in a more general setting. We recall that if ϕ : N is a diffeomorphism, then for α Ω n c () (the space of n-forms with compact support) (A.1) α = ε ϕ α, N where ε is locally constant with value 1 or 1 according to whether ϕ locally preserves or reverses orientation (see [3], I, (5.8)). We note that our proof of the heorem on apping Degrees relies on the non-trivial Sard s theorem from differential topology (see [2], (6.1)). heorem A.1 (heorem on apping Degrees). Let, N be compact, connected, oriented, n-dimensional manifolds. here is an integer deg(f) assigned to each homotopy class of (differentiable) maps f : N, called the mapping degree of f, such that, for every form α Ω n (N), f α = deg(f) α. If q N is a regular value with f 1 { q } consisting of k + l points p 1,..., p k+l such that f preserves orientation at p 1,..., p k but reverses orientation at p k+1,..., p k+l, then deg(f) = k l. In particular, if deg(f) 0, f is surjective. Proof. We first show the left-hand side depends on on the homotopy class of f. If we have a homotopy f : [0, 1] N, where f = f 0 on { 0 } and f = f 1 on { 1 }, then since dα Ω n+1 N = 0, Stokes theorem (see [3], I, (5.17)) and (A.1) gives 0 = f dα = df α = f α = f1 α + f0 α = f1 α f0 α. [0,1] [0,1] ( [0,1]) By Sard s theorem (see [2], (6.1)), for a smooth map f : N, almost every point of N is a regular value. So let q be a regular value and f 1 { q } = { p 1,..., p k+l } as in the statement of the theorem. hen f is a local diffeomorphism around each p i, and the complement of an open neighborhood of f 1 { q } is mapped by f to a compact set not containing q. hus we can choose small neighborhoods B around q and B 1,..., B k+l around p 1,..., p k+l such that f B i : B i B and f 1 (B) = i B i. he orientation of f is constant on each B i and hence equal to its value at p i. If Supp(α) B, the theorem would hold for this particular q by (A.1). Now if we could find a diffeomorphism ϕ : N N homotopic to the identity on N with Supp(α) ϕ(b), the theorem would be proved, since Supp(ϕ α) B, and the integrals are homotopy invariant. We will show below that the sets ϕ(b) cover N, where ϕ runs through diffeomorphisms N N homotopic to the identity. hen we can choose a partition of unity { ψ j j N } subordinate to this covering, and the theorem follows since it is valid for each summand of the splitting α = j ψ j α. o show that the ϕ(b) cover N we show that, given x N, there is a ϕ as above with ϕ(q) = x. If x, q are both contained in a compact ball of a chart domain, it is fairly easy to construct such a diffeomorphism ϕ, e.g., by integrating an appropriate vector field which vanishes outside the ball. From this case we obtain the general case by joining q and x with a chain q = x 0, x 1,..., x r = x such that x j, x j+1 are always contained in a compact ball in some chart domain. N

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