STOCKHOLM SWEDEN 2020,
Local Rigidity of Some Lie Group Actions
SVEN SANDFELDT
KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES
Local Rigidity of Some Lie Group Actions
SVEN SANDFELDT
Degree Projects in Mathematics (30 ECTS credits) Degree Programme in Mathematics (120 credits) KTH Royal Institute of Technology year 2020 Supervisor at KTH: Danijela Damjanovic Examiner at KTH: Danijela Damjanovic
TRITA-SCI-GRU 2020:044 MAT-E 2020:012
Royal Institute of Technology School of Engineering Sciences KTH SCI
SE-100 44 Stockholm, Sweden URL: www.kth.se/sci
Sven Sandfeldt
Abstract
In this paper we study local rigidity of actions of simply connected Lie groups. In particular, we apply the Nash-Moser inverse function theorem to give sufficient conditions for the action of a simply connected Lie group to be locally rigid. Let G be a Lie group, H ă G a simply connected subgroup and Γ ă G a cocompact lattice. We apply the result for general actions of simply connected groups to obtain sufficient conditions for the action of H on ΓzG by right translations to be locally rigid. We also discuss some possible applications of this sufficient condition.
Lokal rigiditet för några Liegruppverkan
Sammanfattning
I den här texten så studerar vi lokal rigiditet av gruppverkan av enkelt sammanhängande Liegrupper. Mer specifikt, vi applicerar Nash-Mosers inversa funktionssats för att ge tillräckliga villkor för att en gruppverkan av en enkelt sammanhängande grupp ska vara lokalt rigid. Låt G vara en Lie grupp, H ă G en enkelt sammanhängande delgrupp och Γ ă G ett kokompakt gitter. Vi applicerar resultatet för generella gruppverkan av enkelt sammanhängande grupper för att ge tillräckliga villkor för att verkan av H på ΓzG med translationer ska vara lokalt rigid. Vi diskuterar också några möjliga tillämpningar av det tillräckliga villkoret.
1
1 Introduction 4
1.1 Conjugacy, coordinate changes and local rigidity . . . 7
1.2 Cohomological equation . . . 9
1.3 Cohomology and linearization . . . 11
1.4 Structure of the paper . . . 13
2 Preliminaries 14 2.1 Differential Geometry . . . 15
2.2 Tame Frechét spaces . . . 17
2.3 Cohomology . . . 21
3 An abstract inverse function theorem 24 3.1 An inverse function theorem for tame representations . . . 24
3.2 Automorphisms of h . . . 29
3.3 An inverse functions theorem for the adjoint representation . . . 31
4 Perturbations of Lie group actions on compact manifolds 35 4.1 A tame Fréchet Lie group structure on Diff8pMq . . . 35
4.2 Subgroups of Diff8pMq . . . 37
4.3 An inverse function theorem on Diff8pMq . . . 41
4.4 Parabolic homogeneous actions . . . 43
4.4.1 General homogeneous actions . . . 43
4.4.2 Ergodicity for homogeneous actions . . . 47
2
4.4.3 Parabolic actions . . . 48 4.4.4 Cohomology with coefficients in constant vector fields . . 55
5 On Possible Applications 56
5.1 Partially hyperbolic actions . . . 56 5.2 Nilmanifolds . . . 58 5.3 Action of root subgroups of SLp2n, Rq . . . 58
1 Introduction
The notion of equivalence between dynamical systems is formalized by conju- gacy (see Definition 1.2). A conjugacy have a regularity (that is, we can consider measurable, continuous or Ck´conjugacies) and the conjugacy preserves all of the dynamical properties in the category where it is defined. For example, if we have a continuous conjugacy between two dynamical systems, then every dynamical property that we define in terms of topology coincide for the two systems. Since a smooth conjugacy is in particular continuous and measurable, we are particularly interested in classifying dynamical systems up to smooth conjugacy. A particular type of classification we can consider is the local clas- sification about some system. That is we want to classify all actions that are close (in some appropriate topology) to a given action. Generally we want to show that every action that is close to a given action is conjugated to a (coor- dinate change of) that action. We call actions with this property locally rigid, see Definition 1.4. Intuitively, if an action is locally rigid then the dynamics is invariant under perturbations. For flows (R´actions) this is generally not the case. Indeed, even if we only allow perturbations that preserve the orbits of the flow a conjecture by Katok states that most flows are not invariant under such perturbations (assuming the conjecture the only flows with this property are linear flow on tori with diophantine slope). If we consider action of larger groups, however, then the situation changes and we expect a local classification of a lot of actions. See for example [21]. To see why this happens, note that to each smooth Rk´action we can associate k vector fields. Since the action is abelian these vector fields must commute. When k “ 1 this does not give us new information, but when k ą 1 this gives us restrictions on which k´tuples of vector fields that generate Rk´actions. In particular, if we perturb an action by adding a perturbation to the vector fields then, since any perturbation is still a Rk´action, the perturbations necessarily solve a differential equation guaran- teeing that the new perturbed vector fields still commute. These restrictions on possible perturbations can then be used to show local rigidity, and giving a local classification of actions.
Let π : Γ Ñ G be a homomorphism of a finitely generated group Γ into a Lie group G. We say that π is locally rigid if every homomorphism π1 : Γ Ñ G sufficiently close to π is conjugated to π by some small element of G. That is, π is locally rigid if every π1 close to π satisfies π1pγq “ gπpγqg´1 “ cgπpγq.
Given the homomorphism π : Γ Ñ G we can also let Γ act on the Lie algebra of G, g, by the adjoint representation. This makes g into a Γ´module, so we define it’s cohomology groups HjpΓ, gq. A theorem of André Weil states that if G is finite dimensional and H1pΓ, hq “ 0 then π : Γ Ñ G is locally rigid.
Weils proof uses the implicit functions theorem, which is not in general valid in infinite dimensions, so the theorem does not immediately generalize to infinite dimensional Lie groups.
In this text we are interested in rigidity of smooth actions of Lie groups. Every smooth action α : Γ ˆ M Ñ M on some compact smooth manifold M can be identified with a homomorphism π : Γ Ñ Diff8pMq, so if the homomorphism π : Γ Ñ Diff8pMq is locally rigid then every action close to α is conjugated to α. So to understand perturbations of actions, we want a generalization of Weils theorem to infinite dimensional Lie groups. In [18] Hamilton proves an inverse functions theorem for exact sequences which was then used by Fisher in [13] to prove:
Theorem 1.1. Let Γ be a finitely presented group, pM, gq a compact Rieman- nian manifold and π : Γ Ñ IsompM, gq Ă Diff8pMq a homomorphism. If H1pΓ, Vect8pMqq “ 0 then π : Γ Ñ Diff8pMq is locally rigid.
This theorem shows that every smooth action α : Γ ˆ M Ñ M acting by isometries which has trivial first cohomology, is locally rigid: Every action α1 : Γ ˆ M Ñ M which is C1´close to α is conjugated to α by some diffeomorphism that is C1´close to idM P Diff8pMq. An action where H1pΓ, Vect8pMqq “ 0 is said to be infinitesimally rigid, so Theorem 1.1 can be stated as saying that for isometric actions infinitesimal rigidity implies local rigidity. This theorem generalizes the theorem of Weil to a specific type of infinite dimensional Lie groups and actions. More results and application of actions by discrete groups can be found for example in [14].
If the acting group is no longer assumed to be discrete, but instead assumed to be a simply connected Lie group, we would like to apply the theorem of Hamilton as in the proof of Theorem 1.1 to obtain a result like Theorem 1.1.
It turns out that when the acting group is not discrete, then results of local rigidity can not only be phrased in terms of the cohomology of the action. One reason for this is the occurrence of arbitrarily small coordinate changes, which we introduce in the next section, which can give arbitrarily small perturbations not conjugated to the original action. Instead of considering only one action we can look at a smooth family of actions, αλ, that are parametrized on some smooth manifold λ P Λ, and instead of considering any perturbation, we can consider only perturbations that preserve some foliation F . If αλ : H ˆ M Ñ M is a smooth family of actions, and ρλ : h Ñ Vect8pMq is the induced homomorphism, then let Aλ : TλΛ Ñ Vect8pMq be the derivative of ρλ with respect to λ. The existence of coordinate changes suggests that we should have a different notion of infinitesimal rigidity when the acting group is a simply
connected Lie group. We prove the following result in this direction:
Theorem C. Let H be a simply connected Lie group with Lie algebra h, αλ : H ˆ M Ñ M a smooth tame family of actions of some closed manifold M and let H1ph, Vect8pMqq be the group from Definition 2.7. If Aλ has a tame inverse on it’s image, ImpAλq Ñ H1ph, Vect8FpMqq is surjective for all λ P Λ and if the complex:
C0ph, Vect8FpMqq
d0ρλ
ÝÝÑ C1ph, Vect8FpMqq
d1ρλ
ÝÝÑ C2ph, Vect8FpMqq
is tamely split then for every action, ˜α, that preserves F and is sufficiently close, as in Definition 1.3, to αλ there is some ˜λ P Λ, f P Diff8FpMq and an automorphism φ : H Ñ H such that:
αph, pq “ f ˝ α˜ ˜λpφphq, f´1ppqq.
Suppose that F “ tMu is the trivial foliation and the family of actions only contain one action in Theorem C. Then we allow any perturbation (since F “ tMu) and the conclusion is that any perturbation is conjugated to a coordinate change of the original action (since Λ “ t0u). So in this case Theorem C reduces to a statement about local rigidity. Namely if H1ph, Vect8pMqq “ 0 and if the sequence in Theorem C is tamely split, then the action is locally rigid.
If G is a Lie group, Γ ă G a lattice in G and H ă G a non-compact simply connected subgroup then we can act with H on M “ ΓzG. We call actions of this type homogeneous actions. In the case of homogeneous actions we can say more then Theorem C. We have the following theorem:
Theorem D. If the action of Hλ is; a pP q´action with respect to F , ergodic, cocycle rigid, ImpAλq Ñ H1phλ, gq is surjective and the sequence:
C0phλ, C8pMqq d
0
ÝÑ C1phλ, C8pMqq d
1
ÝÑ C2phλ, C8pMqq
is tamely split for all λ P Λ, then for any action ˜α that preserves F and is sufficiently close to the natural right action of H on M we have some λ P Λ close to λ0, f P Diff8FpMq close to idM and φ : H Ñ H close to idH such that:
αph, pq “ f ˝ α˜ λpφphq, f´1ppqq.
That is ˜α is conjugated to a coordinate change of the natural right action of Hλ
on M.
In the next section we describe the connection we will use between dynamics and cohomology and introduce the idea of how to prove local rigidity.
1.1 Conjugacy, coordinate changes and local rigidity
For a finitely generated group Γ we say that a homomorphism π : Γ Ñ G is locally rigid if every homomorphism π1 : Γ Ñ G close to π is conjugated to π with some small element in G. This definition carries over to homomorphism from simply connected Lie groups. However it turns out that this does not give us the right definition of local rigidity when the acting group is connected.
The reason is that when the acting group is connected there exists a continu- ous family of perturbations called coordinate changes which are not in general conjugated to the original action. Before defining local rigidity we make three preliminary definitions. For the remainder of this section, let M be a smooth closed connected manifold and let H be a simply connected Lie group.
Definition 1.1. Let α, β : H ˆ M Ñ M be two actions. We say that β is a coordinate change of α if there is a automorphism of H, ψ : H Ñ H, such that:
βph, xq “ αpψphq, xq.
Remark 1. If the acting group is finitely generated we can still make sense of a coordinate change as defined above. The reason we don’t need coordinate changes when defining local rigidity for finitely generated groups is that there is in general a neighbourhood about the action where the only coordinate change is the trivial coordinate change.
Definition 1.2. Let α : H ˆ M Ñ M and β : H ˆ N Ñ N be two actions.
We say that β is conjugated to α if there is a diffeomorphism f : M Ñ N such that:
βph, xq “ f`α `h, f´1pxq˘˘ .
Remark 2. If α : H ˆ M Ñ M and β : H ˆ N Ñ N are actions, then they give homomorphisms:
πα: H Ñ Diff8pMq, πβ: H Ñ Diff8pN q.
If β is conjugated to α according to Definition 1.2 then we find some f P Diff8pM, N q such that πβ“ cfπαwhere cfpgq “ f gf´1is conjugation.
Remark 3. A conjugacy between between two actions is the dynamical notion of isomorphism between dynamical systems. That is, conjugacies preserve all the dynamical information in the category where the conjugacy is defined.
For an action α : H ˆ M Ñ M we can define the local generators by:
ρα: h Ñ Vect8pMq, ραpXqppq :“ d dt
ˇ ˇ ˇ ˇt“0
αpexpptXq, pq.
Using local generators we can topologize the space of actions.
Definition 1.3. Let α and β be smooth actions. We define the C`´distance between α and β by:
sup
kXk“1
kραpXq ´ ρβpXqk`
where the C`´norm on Vect8pMq is given in Definition 2.3.
Remark 4. We say that a property holds for every action β C`´close to α if there is a open ball in the C`´distance about α such that the property holds.
We say that a property holds for every β close to α if there is some ` such that the property holds for every β that is C`´close to α.
We need to introduce coordinate changes because it is generally not true that a coordinate change of some action α is conjugated to α. A simple example of this is given by linear flow on a tori.
Example 1.1. Let ϕt, ψt: R2{Z2Ñ R2{Z2 be the flows (R´action) given by:
ϕtpxq “ x ` tω ` Z2, ω “ˆω1
ω2
˙ , ψtpxq “ x ` tpω
qω1` Z2, p, q P Z
where ω1, ω2and 1 are linearly independent over Q. Note that ψtis a coordinate change of ϕt. We claim that there exists no homeomorphism f : R2{Z2Ñ R2{Z2 such that ψt“ f ˝ ϕt˝ f´1, or ψt˝ f “ f ˝ ϕt. Indeed, if f : R2{Z2Ñ R2{Z2is continuous and f ˝ ϕt“ ψt˝ f then:
f px ` tωq “ f pxq ` tp q
ω ω1
“ f pxq ` t
˜ p
q p q
ω2 ω1
¸ .
So in particular, for n P Z we have:
f px ` nqωq “ f pxq `
˜ 0 n ¨ pωω2
1
¸ .
But tx ` npω : n P Zu is dense in R2{Z2 if ω1, ω2 and 1 are independent over Q. It follows by continuity of f that f takes values in f pxq ` t0u ˆ R{Z so f is in particular not a homeomorphism.
We are now ready to define local rigidity:
Definition 1.4. A smooth action α : H ˆM Ñ M is said to be C`´locally rigid if every smooth action α1 C`´close to α is conjugated to a coordinate change of α by a smooth diffeomorphism. If there is some ` such that α is C`´locally rigid, then we say that α is locally rigid.
As we saw in Example 1.1 we need the freedom of conjugating back to a coor- dinate change, since a coordinate change may not be conjugated to the original action.
1.2 Cohomological equation
Let H be a simply connected Lie group and M a closed smooth connected manifold. Let α : H ˆ M Ñ M be some action, we shall denote this action αph, xq “ hx in this section to simplify notation. Let G be some group. We say that a map β : H ˆ M Ñ G is a cocycle if β satisfies the cocycle equation:
βph1h2, xq “ βph1, h2xqβph2, xq. (1.1) Note that if π : H Ñ G is a homomorphism then βph, xq “ πphq satisfies βph1h2, xq “ πph1h2q “ πph1qπph2q “ βph1, h2xqβph2, xq so ph, xq ÞÑ πphq is a cocycle. We call such cocycles constant cocycles, and in particular if πphq “ e we call the cocycle βph, xq “ πphq “ e the trivial cocycle. If β1 and β2 are cocycles we say that β2is cohomologous to β1if there is a function f : M Ñ G such that:
β2ph, xq “ f phxqβ1ph, xqf pxq´1
where a calculation shows that f phxqβ1ph, xqf pxq´1 is a cocycle. In particular, we say that a cocycle β is a coboundary if β is cohomologous to the trivial cocycle. We are interested in real valued cocycles, that is when G “ R. In this case the cocycle equation becomes:
βph1h2, xq “ βph1, h2xq ` βph2, xq (1.2) and β is a coboundary if there is some f : M Ñ R such that β satisfies the cohomological equation:
βph, xq “ f phxq ´ f pxq. (1.3)
The cocycle equation and the cohomological equation occur frequently in dy- namics. Apart from applications to rigidity, which we will focus on in this text,
a few examples are; cocycles are used to formulate Oseledecs multiplicative er- godic theorem, see for example [22, Theorem S.2.9], in this case the group where the cocycle take values is the general linear group. Bounds on solutions to the cohomological equation was used in [12] to obtain bounds on ergodic averages, cocycles are important in the study of time changes as is seen in [15], more applications are found in [20] and [27].
Since a real cocycle is a map from a smooth manifold into the reals we can consider the regularity of a cocycle, and in particular we can consider smooth cocycles. That is we say that a cocycle β : H ˆ M Ñ R is smooth if it is smooth as a map between smooth manifolds. If h is the Lie algebra of H, then we can differentiate a smooth cocycle at h “ e to obtain the infinitesimal generator of the cocycle:
u : h ˆ M Ñ M, upX, xq :“ d dt
ˇ ˇ ˇ ˇt“0
βpexpptXq, xq
where upX ` Y, xq “ upX, xq ` upY, xq, that is u is a linear map from h into C8pMq. Since the H´action is smooth, we can act with h on C8pMq by letting:
Xpgqpxq “ d dt
ˇ ˇ ˇ ˇt“0
gpexpptXqxq.
If u : h Ñ C8pMq is the infinitesimal generator of some cocycle (or cobound- ary), then one can check that p1.2q (or p1.3q) implies that:
XpupY qq ´ Y pupXqq “ uprX, Y sq, for a cocycle, (1.4)
upXq “ Xpf q, for a coboundary. (1.5)
Equation p1.4q and p1.5q are also called the cocycle and cohomological equation respectively. By integrating the infinitesimal generator one can check that two cocycles coincide if and only if their infinitesimal generators coincide. So in the case of smooth cocycles it often suffices to consider infinitesimal generator of the cocycle.
Note that the equation in p1.4q and p1.5q makes sense as long as we have an additive structure, and if we can differentiate with respect to h. It follows in particular that we can define cocycles and coboundaries with coefficients among vector fields, where differentiation is the Lie derivative. This will be used in the next section, since it will turn out that the cocycle equation and coboundary equation are the linearized versions of the equation defining an action and a conjugacy.
Solving the cohomological equation is in general a difficult problem. The goal when solving the equation is to show which cocycles are coboundaries and ob- taining regularity to solutions of equation p1.5q. If the action of H preserves a Borel probability measure, dµ, then we obtain an obstruction to solving the co- homological equation. Indeed, from p1.3q we see that the integral of a cobound- ary βph, xq with respect to x must be 0 if the action of H preserve dµ. If the only obstruction to solving the cohomological equation is an invariant measure, then the action is said to be cocycle rigid. That is, an action is cocycle rigid if every cocycle that integrate to zero under dµ is a coboundary. A few examples where the cohomological equation is studied can be found in [21] for hyper- bolic actions, [11] for the horocycle flow on quotients of SLp2, Rq by lattices, in [6] the cohomological equation is studied for certain homogeneous action on SLp2, Rq ˆ SLp2, Rq and in [26] the cohomological equation is studied for cer- tain homogeneous actions on SLpn, Rq for n ě 3. All examples above studies the cohomological equation using representation theory and harmonic analysis, namely the cohomological equation is reduced to a cohomological equation in every irreducible representation. In [7] the cohomological equation is studied for hyperbolic and partially hyperbolic actions and the methods used do not rely on harmonic analysis.
1.3 Cohomology and linearization
Let α : H ˆM Ñ M be an action of some simply connected Lie group on a closed connected manifold M. This gives us a homomorphism πα: H Ñ Diff8pMq.
We can differentiate πα at e P H to obtain a Lie algebra homomorphism ρα : h Ñ Vect8pMq given explicitly by:
ραpXqpxq “ d dt
ˇ ˇ ˇ ˇt“0
αpexpptXq, xq,
that is the derivative of παis the local generator of α. If πβ : H Ñ Diff8pMq is another homomorphism, and β is conjugated to a coordinate change of α then one can check explicitly that there is f P Diff8pMq and ψ P Autphq such that:
ρβpXq “ Adpf qρα˝ ψ (1.6)
where Adpf qV “ D|f´1V ˝ f´1where V is some vector field. One can also check that if some action, β, satisfy p1.6q then β is conjugated to a coordinate change of α. If we let Cjph, Vect8pMqq be the space of j´linear alternating maps from hj to Vect8pMq and define the map q : C1ph, Vect8pMqq Ñ C2ph, Vect8pMqq:
qpωqpX, Y q “ rωpXq, ωpY qs ´ ω prX, Y sq .
Then ker q is the space of Lie algebra homomorphism from h into Vect8pMq.
We also define p : Diff8pMq ˆ Autphq Ñ C1ph, VectpMqq:
ppf, ψqpXq “ Adpf qρα˝ ψpXq.
Since ppf, ψq is a Lie algebra homomorphism we have q ˝ p “ 0, which gives us a complex:
Diff8pMq ˆ AutphqÝÑ Cp 1ph, Vect8pMqqÝÑ Cq 2ph, Vect8pMqq. (1.7) We can now state local rigidity in terms of the sequence p1.7q, namely if the sequence in p1.7q is exact in some open neighbourhood about ppidM, idhq then every action close to α is conjugated to a coordinate change of α, so α is locally rigid.
Since local rigidity is a local classification of actions, it makes sense to differ- entiate the complex in p1.7q. If we differentiate the non-linear complex in p1.7q about pidM, idhq we obtain a linear complex:
Vect8pMq ˆ DerphqÝPÑ C1ph, Vect8pMqqÝQÑ C2ph, Vect8pMqq (1.8) with P “ D|pidM,idhqp and Q “ D|ppidM,idhqq. One can check explicitly that P and Q are given by:
P pH, φqpXq “LραpXqpHq ` ρα˝ φpXq, (1.9) QpωqpX, Y q “LραpXqpωpY qq ´LραpY qpωpY qq ´ ωprX, Y sq (1.10) whereL is the Lie derivative. If we set Qpωq “ 0 then we note that we obtain a equation like equation p1.4q, that is we obtain a cocycle equation with values in vector fields. If we set P pH; φq “ ω then we almost obtain a cohomological equation except that we have an extra term ρα˝ φpXq. Since q ˝ p “ 0 we have Q ˝ P “ 0 so we can define H1ph, Vect8pMqq “ kerpQq{ImpP q, which is the group in Theorem C. We can now give the steps of proving local rigidity:
(1) Show that among functions, every cocycle is a coboundary, except for an additative constant. That is if u : h Ñ C8pMq is a cocycle then upXq “ Xpf q`CXwhere CXis some constant, we call this cocycle rigidity.
Furthermore we want to obtain tame estimates, that is if upXq “ Xpf q then kf krď Crkukr`r
0 for some constant r0 and constant Cr.
(2) Use cocycle rigidity and tame estimates to show that H1ph, Vect8pMqq “ 0 and that the sequence p1.8q splits tamely. That is we find mappings P and pp Q that split p1.8q and satisfy
P ωp
r ď Crkωkr`r
0 and Qηp
r ď Crkηkr`r
0.
(3) Use p2q to show that the sequence p1.7q is locally exact. That is, there is some neighbourhood U in C1ph, Vect8pMqq such that kerpqq X U “ Imppq X U .
In this text we only deal with p2q and p3q above. There are two different ap- proaches one can take in part p3q above. The first is to apply the KAM method.
The idea of the KAM method is to use that p1.8q is exact to iteratively construct approximate solutions to the equation Adpf qρα˝ ψ “ ρβ which converge to a solution. This approach have been used in [4, 5, 6] to prove rigidity results for both discrete and continuous actions of higher rank. It has also been used in [8] to prove a general perturbation result. The second approach is to use the Nash-Moser inverse functions theorem for exact sequences, see [18], to obtain the result. This approach has been used in [13, 1] to show local rigidity when the acting group is discrete, it has also been used in [3] for Rk´actions.
1.4 Structure of the paper
Section 2 goes through a few prelimenaries which will be used throughout the text simultaneously fixing notation. We go through some definitions from dif- ferential geometry. We define tame Fréchet spaces which are the appropriate functional analytic framework to deal with smooth functions and vector fields.
It is also in the framework of tame Fréchet spaces the Nash-Moser inverse func- tions theorem is stated. We also introduce the Chevalley-Eilenberg cohomology.
In section 3 we apply the Nash-Moser inverse functions theorem to prove our first inverse function theorem:
Theorem A. Consider the complex:
G ˆ ΛÝÑ Vp ÝÑ Wq (3.6)
where p is defined as ppg, λq “ rpgquλ. If:
(1) Qpe, uλq has a smooth and tame inverse of it’s image for all λ P Λ.
(2) Rpe, uλq has a smooth and tame inverse on it’s image for all λ P Λ.
(3) ker Qpe, uλq “ ImpRpe, uλqq ‘ ImpAλq for all λ P Λ.
Then the sequence p3.6q is locally exact. More precisely, there is some open U Ă V and a smoothp : U Ñ G ˆ Λ such that pp ppvq “ v for every v such thatp qpvq “ 0.
Theorem B. If the sequence:
C0ph, Vq
d0ρλ
ÝÝÑ C1ph, Vq
d1ρλ
ÝÝÑ C2ph, Vq
is tamely split for all λ P Λ and ImpAλq Ñ H1ph, Vq is surjective for all λ P Λ, then the sequence:
Λ ˆ pG ˆ AutphqqÝÝÑ Cpρλ 1ph, VqÝÑ Cq 2ph, Vq
is exact in some neighbourhood about ρλ0. More precisely, there is some open ρλ0 P U Ă C1ph, Vq and some smooth ppρλ : U Ñ Λ ˆ G ˆ Autphq satisfying pρλ0pppρλpgqq “ g if qpgq “ 0.
where the spaces involved are defined in section 3.
We apply our result from section 3 in section 4 to concrete cases of actions of Lie groups on compact Riemannian manifolds. We begin by describing the Fréchet topology on the group of diffeomorphism of a compact Riemannian manifold.
We then show that this group satisfies the assumptions to apply the results from section 3 proving Theorem C. We end this section by considering the special case of homogeneous spaces and proving Theorem D.
In section 5 we discuss possible application of the results from the previous sections.
Acknowledgements
My deepest gratitude to my advisor Danijela Damjanovic, for agreeing to su- pervise this thesis, for all the help she has given me during the writing of this thesis and for introducing me to this to the subject of local rigidity.
2 Preliminaries
In this section we introduce some background and simultaneously fix notation.
In section 2.1 we go through some notions from differential geometry and smooth manifolds, our main reference is [16]. In section 2.2 we introduce the framework in which we will work for the remainder of the text. That is, tame Fréchet spaces. The main reference for tame Fréchet spaces we’re using is [18]. We also introduce the cohomology in which we shall be interested. As we’ve seen in
p1.9q and p1.10q this cohomology is important because cocycles, coboundaries and the cohomology group describe the linearization of conjugation.
We shall assume that the reader have some knowledge of general smooth mani- fold theory and dynamics. Sufficient background in dynamics and smooth man- ifolds can be found in, for example, [22] and [23] respectively.
2.1 Differential Geometry
Let M be a smooth closed manifold with an affine connection ∇. For a vector bundle E Ñ M we denote the space of global sections by ΓpEq, or in the special case of the tangent bundle we denote ΓpT Mq “ Vect8pMq. Recall that an affine connection is a bilinear map ∇ : Vect8pMq ˆ Vect8pMq Ñ Vect8pMq, denoted pX, Y q ÞÑ ∇XY , satisfying:
(1) ∇f XY “ f ∇XY for all f P C8pMq.
(2) ∇Xpf Y q “ Xpf qY ` f ∇XY for all f P C8pMq.
If γ : r0, T s Ñ M is a smooth curve and X a vector field, then we say that X is parallel along γ if ∇γptq9 X “ 0. If xj is a choice of coordinates, and if we define the Christoffel symbols:
∇BiBj “ ΓkijBk
where Bi“ B{Bxi then we have:
∇YX “∇YiBiXjBj “ Yi ˆ
Xj∇BiBj`BXj Bxi Bj
˙
“
“ ˆ
YiXjΓkij` YiBXk Bxi
˙ Bk.
In particular with Y “ 9γptq and ∇γptq9 X “ 0 we have:
0 “∇γptq9 X “ r 9γptqsiXjΓkij` r 9γptqsiBXk
Bxi “ r 9γptqsiXjΓkij` d
dtXkpγptqq where we’ve used the chain rule. From the Picard-Lindelöf theorem it follows that if X is a vector field parallel along γ then X is uniquely determined by it’s value at t “ 0. So we obtain a map P0,t : Tγp0qM Ñ TγptqM called the parallel transport along γ. Note that the parallel transport is linear since the ODE defining a parallel vector field in coordinates is linear. By smooth depen- dence on initial conditions we also know that the parallel transport is smooth.
Furthermore if γr : r0, T s Ñ M is a smooth family of curves, then the parallel
transport depends smoothly on the parameter as well. If γ : r0, T s Ñ M is a curve such that ∇γptq9 γptq then we say that γ is a geodesic. A geodesic is deter-9 mined by it’s initial position and velocity, so we obtain a map exp : T M Ñ M mapping an element vp P TpM to the geodesic with initial position p and initial velocity vp at time 1. We denote expp “ exp |TpM : TpM Ñ M, this map is a local diffeomorphism so we denote it’s inverse logp: M Ñ TpM when it exists.
If M is a Riemannian manifold with metric x¨, ¨y, then let ∇ be the Levi-Civita connection. If N Ă M is a submanifold then by restricting x¨, ¨y to T N we obtain a Riemannian structure on N , so we have a Levi-Civita connection on N .
Definition 2.1. If every geodesic in N is a geodesic in M, then we say that N is a totally geodesic submanifold.
Definition 2.2. Let F be a family of p´dimensional submanifolds of M such that:
M “ ğ
LPF
L.
Furthermore, suppose that for every p P M we can find coordinates xi : U Ă M Ñ Rn such that L X U “ txj “ 0 : j “ p ` 1, ..., nu where p P L P F , then we say that F is a foliation of M. The elements of F are called leaves.
If F is a foliation of M and p P M is some point, then we denote the unique leaf containing p by F ppq. By differentiating a foliation we obtain a subbundle of the tangent bundle (or a distribution) which we denote T F . Given a subbundle of the tangent bundle, we can integrate and obtain a foliation if and only if the subbundle is closed under the Lie bracket. This follows from Frobenius theorem, see [23, Theorem 14.5].
The Levi-Civita connection defines a map ∇ : Vect8pMq Ñ ΓpT M b T˚Mq by letting the first vector field in ∇ : Vect8pMq ˆ Vect8pMq Ñ Vect8pMq be considered a variable. More generally, for any pk, `q tensor field we can define
∇ : ΓpT`kpMqq Ñ ΓpT`k`1pMqq where:
T`kpMqp“ pTpMqb`b`Tp˚M˘bk
see for example [23]. We define a isomorphism 5 : TpM Ñ Tp˚M by X ÞÑ X5“ xX, ¨y, we also denote the inverse by 7 : Tp˚M Ñ TpM. For ω, ν P Tp˚M we define:
xω, νy :“ xω7, ν7y.
This in combination with xu b v, u1b v1y “ xu, u1yxv, v1y we can define x¨, ¨y : T`kpMq ˆ T`kpMq Ñ R for all pk, `q. We can now topologize ΓpT`kpMqq.
Definition 2.3. Let X P ΓpT`kpMqq, then:
kXkr“
r
ÿ
j“0
sup
pPM
ˇˇ∇jXppqˇ ˇ
where |Xppq| “ a
xXppq, Xppqy. If dµ is a volume form on M and X, Y P ΓpT`kpMqq we also define:
xX, Y yWr,2 :“
ż
M
xXppq, Y ppqypdµppq, kXkWr,2 “a
xX, XyWr,2.
Note that this definition makes sense for an arbitrary vector bundle E Ñ M with a metric and a connection. In particular Definition 2.3 gives us norms on C8pMq (since C8pMq “ ΓpM ˆ Rq “ ΓpT00pMqq) and Vect8pMq. Further- more these norms clearly satisfy k¨krě k¨ks is r ě s.
2.2 Tame Frechét spaces
Recall that a Fréchet space, V, is a topological vector space where the topology is induced by a countable family of seminorms such that V is Hausdorff and V is complete with respect to the seminorms. That is, if a sequence vn P V is Cauchy in every seminorm then vn converge to some v P V. We begin by an example which will be important when we define tame Fréchet spaces.
Example 2.1. Let B be a Banach space with norm k¨k. We define:
ΣpBq :“
#
pfjqjPN0 : for n P N0 ÿ
jPN0
enjkfjk ă 8 +
and make ΣpBq into a Fréchet space by defining the countable collection of norms:
kpfjqjPN0kn:“ ÿ
jPN0
enjkfjk .
We call ΣpBq the space of exponentially decreasing sequences in B.
Before introducing tame Fréchet spaces we need to define graded Fréchet spaces.
This is essentially a Fréchet space where the seminorms are ordered such that
k¨kt ď k¨ks if t ď s. More formally, let V be a Fréchet space and pk¨krqrPN0 a collection of seminorms on V. If k¨kr ď k¨ks for all r ď s and pk¨krqrPN0 generates the topology of V, then we say that pk¨krqrPN0 is a grading on V.
Definition 2.4. If V is a Fréchet space and pk¨krqrPN0 is a grading on V, then pV, pk¨krqrPN0q is a graded Fréchet space. If the grading is understood we simply call V a graded Fréchet space.
If we have a Fréchet space V with two gradings k¨kr and k¨k1r then we say that the gradings are tamely equivalent if k¨kr ď C k¨k1r`d and k¨k1r ď C k¨kr`d for all r and some d ě 0.
Example 2.2. Let M be a compact smooth manifold. We define the norm k¨kron C8pMq to be either the Cr´norm or the Sobolev norm from Definition 2.3. The space C8pMq is a graded Fréchet space with respect to both of these families of norms. These Fréchet spaces are tamely equivalent by the Sobolev embedding theorem, so in the sense of Fréchet spaces it does not matter if one considers Cr´norms or Sobolev norms.
Example 2.3. Let M be a compact smooth manifold. Let Vect8pMq be the space of smooth vector fields over M. Let k¨kr be either the Cr´norm or the Sobolev norm on Vect8pMq from Definition 2.3. The space Vect8pMq is a graded Fréchet space with both of these families of norms.
Example 2.4. The space of exponentially decreasing sequences for some Ba- nach space is also a graded Fréchet space.
If V and W are graded Frechét spaces and L : V Ñ W is a linear map satisfying kLxkrď Crkxkr`d for all r ě b and some fixed d, then we say that L is a tame map satisfying a tame estimate of order d and base b. Note that compositions of tame maps are tame, indeed if L : V Ñ W and M : W Ñ Z are tame, then:
kM Lxkrď CrkLxkr`d
1 ď Cr`d1 1Crkxkr`d
1`d2
where d1 is the degree of M and d2 the degree of L, and r is assumed to be larger then the base of L and M . If L : V Ñ W is tame and has a tame inverse L´1: W Ñ V then L is a tame isomorphism and V is tamely isomorphic to W.
Note that our definition of equivalent gradings is equivalent to the fact that the identity is a tame isomorphism.
Let V and W be graded Fréchet spaces. We say that V is a direct summand of W if we have tame maps L : V Ñ W and M : W Ñ V such that:
V ÝÑ WL ÝÑ V,M M L “ idV.
We’re now ready to define tame Fréchet spaces.
Definition 2.5. Let V be a graded Fréchet space. If V is a direct summand of ΣpBq for some Banach space B, then we say that V is a tame Fréchet space.
All of our examples of graded Fréchet spaces are also tame Fréchet spaces. That the space of exponentially decreasing sequences is tame follows immediately from the definition since it’s a direct sum of itself. The rest of the examples are proven to be tame by embedding M into Rd and using Fourier analysis, see [18].
Let V and W be tame Fréchet spaces and P : U Ă V Ñ W a map defined on an open subset of V. We define the derivative of P at f P U in direction h P V as
DP |fphq :“ lim
tÑ0
P pf ` thq ´ P pf q t
if this limit exists. If this limit exists for all h then this is a linear map in h. If P is differentiable at all f P U and h P V and the map given by:
pf, hq ÞÑ DP |fphq
is continuous then we say that P is C1 on U . Let L : pU Ă Vq ˆ V Ñ W be linear in the second coordinate. If we can differentiate L with respect to the first coordinate for all f P U , v P V then DL|pf,0q: pU Ă Vq ˆ V Ñ W is bilinear in the last two coordinates. Especially if this can be done for DP |f then we say that the resulting map pU Ă VqˆV ˆV is the second derivative of P and P P C2 if this map is continuous. We can keep constructing higher order derivatives in this manner. If all such derivatives exists are continuous then we say that P is smooth or C8. We have the following theorem [18, Theorem 3.6.4].
Theorem 2.1. If P and Q are Cn then the composition P ˝ Q is also Cn, and the chain rule holds.
Let P : pU Ă Vq Ñ W be a (not necessarily linear) map between graded Fréchet spaces. We say that P is tame if kP pf qkr ď Crp1 ` kf kr`dq for some fixed d and r ě b for some b ě 0. If P is linear this definition coincide with the one given for linear maps, see [18, Theorem 2.1.5]. We also have that composition of tame maps are tame. We say that a map P : pU Ă Vq Ñ W is a smooth tame map if P is smooth and DnP is tame for every n.
Having defined tame Fréchet spaces and smooth and tame maps it is natural to define smooth and tame Fréchet manifolds. We say that M is a smooth and tame Fréchet manifold if we can chart M on a tame Fréchet space V where the transition maps are smooth and tame.
We define tame Fréchet Lie groups as tame Fréchet manifolds who also carry a group structure where multiplication and inversion are smooth tame maps. We
can also define group actions in the category of smooth and tame manifolds.
Let G be a tame Fréchet Lie group, M a tame Fréchet manifold and V a tame Fréchet space. We define tame group actions as mappings r : G ˆ M Ñ M where r is a group action and r is smooth and tame. We can also define a tame representation r : G ˆ V Ñ V where r is a tame group action and rpg, vq is linear in v. For some g P G we shall use the notation Lg : G Ñ G, h ÞÑ gh and Rg : G Ñ G, h ÞÑ hg´1 for left translation and right translation respectively.
Note that these mappings are smooth and tame.
As is hinted in the introduction and will be further investigated in the next section we shall be particularly interested in sequences of tame Fréchet manifolds and spaces. Let A, B and C be tame Fréchet spaces. Let:
AÝFÑ BÝGÑ C
be an exact sequence, where F and G are tame maps. We say that the sequence is exact and tamely split if there are tame maps F˚ : B Ñ A and G˚: C Ñ B satisfying F F˚` G˚G “ IdB. If the sequence is not exact but ImF Ă ker G and we have tame inverses F˚ and G˚ on the image of F and G respectively, then we say that the sequence is tamely split.
Our interest in tame Fréchet manifolds is due to the following theorem of Hamil- ton [18, Theorem 3.1.2]:
Theorem 2.2 (The Nash-Moser inverse function theorem for exact sequences).
Let V, W and H be tame Fréchet manifolds. Let P, Q and R be tame maps V ÝPÑ W ÝQÑ H, V ÝPÑ WÝRÑ H
satisfying Q ˝ P “ R ˝ P . Let T V, T W and T H be the tangent bundle of V, W and H respectively. Furthermore let P˚T W be the pull-back with respect to P and P˚Q˚T W “ P˚R˚T W the pull-back with respect to Q ˝ P “ R ˝ P . Then we have a sequence of maps
T VÝÝÑ PDP ˚T WÝDR´DQÝÝÝÝÝÑ P˚Q˚T H (2.1) satisfying pDQ ´ DRq ˝ DP “ 0. Suppose that the sequence p0.1q is exact and admits a splitting by smooth tame linear maps
T VÐÝÝ PV P ˚T WÐÝÝ PV Q ˚Q˚T H (2.2)
satisfying DP ˝ V P ` V Q ˝ pDQ ´ DRq “ idP˚T W. Then for every f0 P V the image of a neighbourhood about f0 fills up a neighbourhood of g0 “ P pf0q in the subset of W where Qpgq “ Rpgq. Moreover we can find a smooth tame map S defined on a neighbourhood about g0 to a neighbourhood of f0 satisfying P ˝ Spgq “ g for every g such that Qpgq “ Rpgq.
We shall be especially interested when H and W are both tame Fréchet spaces and R is identically 0. In this case Qpgq “ Rpgq reduces to Qpgq “ 0 or g P ker Q.
2.3 Cohomology
Let h be a (finite dimensional) Lie algebra and X ÞÑ LρpXq, X P h, a repre- sentation of h in some h´module V. We shall assume that V is a tame Fréchet space and that the action of h is a smooth tame action. We define the space:
Ckph, Vq :“ Hom`
^kh, V˘
(2.3) that is Ckph, Vq consists of k´linear and alternating maps from hk to V. Espe- cially we have C0ph, Vq “ V and C1ph, Vq “ tlinear maps from h to Vu. We can make Ckph, Vq into a graded Fréchet space by defining:
kωkr:“ sup
kXkh“1
kωpXqk (2.4)
where k¨kh is some norm on h (h is finite dimensional so, it does not matter which norm we choose). This definition actually makes Ckph, Vq into a graded Fréchet space, see Lemma 3.6. We define an operator:
Ckph, Vq
dkρ
ÝÑ Ck`1ph, Vq (2.5)
by:
dkρpωqpX0, ..., Xkq :“ ÿ
0ďiďk
p´1qiLρpXiqωpX0, ..., xXi, ..., Xkq` (2.6)
` ÿ
0ďiďjďk
p´1qi`jωprXi, Xjs, ..., xXi, ..., xXj, ..., Xkq
where pY indicates that we’re not taking Y as an argument. Note that dkρ is a tame linear operator. We are mainly interested in the 01th and 11st coboundary operator given explicitly by:
d0ρpV qpXq “LρpXqpV q, (2.7)
d1ρpωqpX, Y q “LρpXqpωpY qq ´LρpY qpωpXqq ´ ωprX, Y sq. (2.8) We define:
Zkph, Vq :“ ker dkρ, Bkph, Vq :“ Impdk´1ρ q. (2.9)
One can check that dk`1ρ ˝ dkρ “ 0 so dkρ are coboundary operators and:
... d
k´2
ÝÝÝÑ Cρ k´1ph, Vq
dk´1ρ
ÝÝÝÑ Ckph, Vq
dkρ
ÝÑ Ck`1ph, Vq
dk`1ρ
ÝÝÝÑ ... (2.10) is a cochain complex, the Chevalley-Eilenberg complex. It follows that Zkph, Vq are cocycles and Bkph, Vq coboundaries in this complex. We denote the coho- mology with respect to this complex Hkph, Vq “ Zkph, Vq{Bkph, Vq.
The examples to follow are the main examples of h and V1s we are interested in.
Note that the sequences p1.8q is a special cases of the first parts of the complexes obtained in example 2.7 and 2.6 respectively.
Example 2.5. Let V be a (possibly infinite dimensional) Lie algebra. Let ρ : h Ñ V be an injective Lie algebra homomorphism. We define a representation on V as:
LρpXqpV q :“ rρpXq, V s. (2.11)
Example 2.6. Let H be a (finite dimensional) Lie group with Lie algebra h.
Let α : H ˆ M Ñ M be a smooth action on a compact manifold M. Let ρ : h Ñ Vect8pMq be the associated Lie algebra homomorphism given by:
ρpXqppq :“ d dt
ˇ ˇ ˇ ˇt“0
αpexpptXq, pq. (2.12)
We define a representation on the space of smooth functions as LρpXqpf qppq :“ pρpXqf qppq “ d
dt ˇ ˇ ˇ ˇt“0
f pαpexpptXq, pqq (2.13) that is we act by the Lie derivative of ρpXq. We shall denote the associated coboundary operator, defined in p2.6q, by dkρ.
Note that if the action in example 2.6 preserve a measure, then the constant functions are cocycles but not coboundaries so we have Rdim h ă H1ph, C8pMqq.
Motivated by this we make the following definition:
Definition 2.6. Let α : H ˆM Ñ M be a smooth action and let H1ph, C8pMqq be defined as in example 2.6. If H1ph, C8pMqq “ Rdim h then we say that the action is cocycle rigid.
Example 2.7. Let H, h, ρ and M be as in the previous example. We define a representation on Vect8pMq by:
LρpXqpV q :“ rρpXq, V s
that is we act by the Lie derivative. Note that this is a special case of Example 2.5. If H is a closed subgroup of G, Γ Ă G is a cocompact lattice (that is Γ is discrete and ΓzG is compact), M “ ΓzG and H acts by right translations then ρ is simply the embedding of h to the corresponding left-invariant vector field and we writeLX.
If ρ1 : h Ñ Vect8pMq is given by an action which is a coordinate change of ρ then we have:
ρ1pXq “ d dt
ˇ ˇ ˇ ˇt“0
α1pexpptXq, pq “ d dt
ˇ ˇ ˇ ˇt“0
αpφpexpptXqq, pq “ ρpD|eφpXqq.
So if ρ1 is given by a coordinate change of ρ, then ρ1 takes values in Impρq.
Example 2.8. Let g be a finite dimensional Lie algebra with subalgebra h ă g.
We define the representation of h as the restricted adjoint representation, that is ρ : h Ñ g is the inclusion and:
LρpXqpY q :“LXpY q “ rX, Y s. (2.14)
Note that this is a special case of Example 2.5.
In Example 2.5 (and therefore also in Example 2.7 and 2.8) we are given a linear map ρ : h Ñ V so we may define the space:
Zkph, Vq
Bkph, Vq ` Ckph, Impρqq X Zkph, Vq. (2.15) Comparing with Example 2.7 the elements of Ckph, Vq should be interpreted as coordinate changes of ρ. Indeed, if ω P Ckph, Impρqq then:
ωpX1, ..., Xkq “ ρpεpX1, ..., Xkqq
where ε : ^kh Ñ ^kh. Since ρ is injective and ω and ρ are linear we have that ε : ^kh Ñ ^kh is also linear. So elements of Ckph, Impρqq only differ from ρ by a linear map ε : ^kh Ñ ^kh. We make the following definition:
Definition 2.7. If ρ : h Ñ V is a homomorphism as in Example 2.5 then we define the group:
Hkph, Vq :“ Zkph, Vq
Bkph, Vq ` Ckph, Impρqq X Zkph, Vq.
We have the following:
Lemma 2.1. Let V be a h´module, and V “ V1‘ V2 (where the split is as an h´module). Then Hkph, Vq “ Hkph, V1q ‘ Hkph, V2q. Furthermore, if V is a Lie algebra, the h´module structure is given by a Lie algebra homomorphism and Impρq Ă V1 then Hkph, Vq “ Hkph, V1q ‘ Hkph, V2q.
Proof. Let p1 : V Ñ V1 and p2 : V Ñ V2 be projections. We have ω ÞÑ p1˝ ω ` p2˝ ω an isomorphism between Ckph, Vq and Ckph, V1q ‘ Ckph, V2q.
Since V split as an h´module dkρ will respect the split. It follows that the diagram:
Ckph, Vq Ck`1ph, Vq
Ckph, V1q ‘ Ckph, V2q Ck`1ph, V1q ‘ Ck`1ph, V2q
dkρ
dkρ‘dkρ
commute. From this we see that Hkph, Vq “ Hkph, V1q ‘ Hkph, V2q. The second claim also follows from the commuting diagram, and the observation
Zkph, V2q
Bkph, V2q ` Ckph, Impρqq “ Zkph, V2q
Bkph, V2q “ Hkph, V2q
which holds since Impρq Ă V1 so V2X Impρq “ 0.
3 An abstract inverse function theorem
In this section we prove an inverse function theorem for a certain type of ex- act sequences. The idea is to use the inverse functions theorem of Nash and Moser (Theorem 2.2). But to use Theorem 2.2 we need to be able to invert the linearized sequence at every point in some open set. The main objective of this section is to prove that for certain sequences we only need to invert the linearized sequence at one point.
3.1 An inverse function theorem for tame representations
Let G be a (possibly infinite dimensional) tame Fréchet Lie group and let V and W be tame Fréchet spaces. Furthermore, suppose that we have tame represen- tations r and s of G on V and W respectively. That is we have maps:
r : G ˆ V Ñ V, s : G ˆ W Ñ W
which we denote rpgqv and spgqv, and these maps are linear in v and they satisfy rpghqv “ rpgqrphqv, rpeq “ idV and similarly for s. Finally let q : V Ñ W be a G´equivariant smooth and tame map.
We define maps R : T G ˆ V Ñ V, denoted Rpg, uqv where g P G and v P TgG, and Q : pG ˆ Vq ˆ V Ñ W, denoted Qpg, uqv, by:
Rpg, uqv :“ D|pg,uqrpv, 0q (3.1)
Qpg, uqv :“ D|rpgquqpvq. (3.2)
We begin by giving formulas for R and Q.
Lemma 3.1. We have:
Rpg, uqv “ rpgqRpe, uqD|gL´1g v (3.3)
and:
Qpg, uqv “ spgqQpe, uqrpg´1qv (3.4)
Proof. We can show p3.3q and p3.4q by direct computations. Let gtbe a smooth curve such that g0“ g and pd{dtq|t“0gt“ v. Then we have:
Rpg, uqv “d dt
ˇ ˇ ˇ ˇt“0
rpgtqu “ d dt
ˇ ˇ ˇ ˇt“0
rpgqrpg´1gtqu “
“rpgqd dt
ˇ ˇ ˇ ˇt“0
r`L´1g pgtq˘ u “ rpgqRpg´1g0, uqD|g0L´1g d dt
ˇ ˇ ˇ ˇt“0
gt“
“rpgqRpe, uqD|gL´1g v.
Let vtbe a smooth curve in V such that v0“ rpgqu and pd{dtq|t“0vt“ v then:
Qpg, uqv “d dt
ˇ ˇ ˇ ˇt“0
qpvtq “ d dt
ˇ ˇ ˇ ˇt“0
spgqqprpg´1qvtq “
“spgqd dt
ˇ ˇ ˇ ˇt“0
qprpg´1qvtq “ spgqQpe, vqrpg´1qd dt
ˇ ˇ ˇ ˇt“0
vt“
“spgqQpe, vqrpg´1qv where we’ve used that:
d dt
ˇ ˇ ˇ ˇt“0
rpg´1qvt“ rpg´1qv
since rpg´1q is linear.
A consequence of Lemma 3.1 is the following two results.
Lemma 3.2. Let u P V be fixed. If there is a smooth tame map:
Rpe, uq : ImpRpe, uqq Ñ Tp eG
satisfying that Rpe, uq pRpe, uq is identity on ImpRpe, uqq then there are smooth tame maps:
Rpg, uq : ImpRpg, uqq Ñ Tp gG
satisfying that Rpg, uq pRpg, uq is identity on ImpRpg, uqq for all g P G. Further- more, the map pRpg, uq is smooth and tame with respect to g.
Proof. We claim that we can define pRpg, uq with the formula:
Rpg, uq :“ D|p eLgRpe, uqrpgp ´1q. (3.5) First, the formula for pRpg, uq is a composition of maps which are smooth and tame with respect to g, so pRpg, uq as defined above is smooth and tame with respect to g, and the last part of the lemma follows. Second, by Lemma 3.1 we have that ImpRpg, uqq “ rpgqImpRpe, uqq so pRpg, uq is defined on ImpRpg, uqq.
Finally we obtain by a calculation using Lemma 3.1:
Rpg, uq pRpg, uq “ rpgqRpe, uqD|gL´1g D|eLgRpe, uqrpgp ´1q “ idImpRpg,uqq.
Lemma 3.3. Let u P V be fixed. If there is a smooth tame map:
Qpe, uq : ImpQpe, uqq Ñ Vp
satisfying that Qpe, uq pQpe, uq is identity on ImpQpe, uqq then there are smooth tame maps:
Qpg, uq : ImpQpg, uqq Ñ Vp
satisfying that Qpg, uq pQpg, uq is identity on ImpQpg, uqq for all g P G. Further- more, the map pQpg, uq is smooth and tame with respect to g.
Proof. We define pQpg, uq “ rpgq pQpe, uqspg´1q which immediately implies that Qpg, uq is smooth and tame with respect to g. From Lemma 3.1 we see thatp ImpQpg, uqq “ spgqImpQpe, uqq so pQpg, uq is defined on ImpQpg, uqq. Using Lemma 3.1 we obtain by a calculation:
Qpg, uq pQpg, uq “ spgqQpe, uqrpg´1qrpgq pQpe, uqspg´1q “ idImpQpg,uqq.
Let Λ be a smooth tame Fréchet manifold (which we shall interpret as a pa- rameter space), and λ ÞÑ uλ a smooth tame injective map Λ Ñ ker q Ă V. We denote the derivative of this map by Aλ : TλΛ Ñ ker D|uλq Ă V, we assume that Aλis smooth and tame, and has a smooth and tame inverse on it’s image.
Note that this holds, in particular, if Λ is finite dimensional.
Lemma 3.4. Let R and Q be defined as above. If we have:
ker Qpe, uλq “ ImpRpe, uλqq ‘ ImpAλq for all λ P Λ then:
ker Qpg, uλq “ ImpRpg, uλqq ‘ ImprpgqAλq
for all g P G. Furthermore, if Rpe, uλq has a smooth and tame inverse on it’s image then we have a smooth and tame map:
T : ker Qpg, uλq Ñ TgG ‘ TλΛ
satisfying pRpg, uλq ` rpgqAλqT “ idker Qpg,uλq for all g P G and λ P Λ.
Proof. Note that qprpgquλq “ spgqqpuλq “ 0 since uλP ker q. So we can differ- entiate qprpgquλq “ 0 with respect to g and λ to obtain:
Qpg, uλq pRpg, uλq ` rpgqAλq “ 0
so ImpRpg, uλqq ‘ ImprpgqAλq Ă ker Qpg, uλq. To show the converse, if v P ker Qpg, uλq then rpg´1qv P ker Qpe, uλq by Lemma 3.1. So we have some v0P TeG and v1P TλΛ such that:
rpg´1qv “ Rpe, uλqv0` Aλv1 or:
v “rpgqRpe, uλqv0` rpgqAλv1“
“rpgqRpe, uλqD|gL´1g D|eLgv0` rpgqAλv1“
“Rpg, uλqD|eLgv0` rpgqAλv1
where we’ve used Lemma 3.1. So v P ImpRpg, uλqq ‘ ImprpgqAλq. To show the second part of the lemma, note that ker Qpg, uλq “ ImpRpg, uλqq ‘ ImprpgqAλq so we can apply pRpg, uλq from Lemma 3.2 on the first coordinate and A´1λ rpg´1q
on the second coordinate.
We can now prove our main theorem of this subsection.
