Régimen General de Renta

For any set S, element x∈XS _{and affine map g ∈A(X) we define g.x:= (g(x} v))v∈S.

With this notation we define for any S the map

φ: Isom(X)×XS →XS, (g, x)7→g.x.

If |S|<∞ then this is a Lie group action of Isom(X) on XS_{; we shall always refer to}

this group action if we mention Isom(X) acting on XS_.

Lemma 1.2.10. For any normed space X and finite set S, the group of isometries

Isom(X) acts properly onXS.

Proof. Let ((gn.pn, pn))n∈_N be a convergent sequence in the image of

with limit (q, p). By Mazur-Ulam’s theorem (Theorem 1.1.28), for eachn ∈_N there

exists Gn ∈ IsomLin(X) and xn ∈ X such that gn = Txn ◦ Gn, where Txn is the

translation mapy7→y+xn. As ((gn.pn, pn))n∈Nconverges then (gn.pn)n∈N and (p

n₎ n∈N

are bounded inXS, thus (xn)n∈N is bounded as

∥xn∥=∥(xn)v∈S∥S ≤ ∥Gn.pn+ (xn)v∈S∥S+∥Gn.pn∥S =∥gn.pn∥S+∥pn∥S.

It follows by the Bolzano-Weierstrass theorem that there exists a convergent subsequence (xnk)k∈N of (xn)n∈N with limit x ∈ X. Since Isom

Lin_{(X) is compact, there exists a}

convergent subsequence (Gn_kl)l∈N of (Gnk)k∈N with limitG∈Isom

Lin_{(X). This implies}

(gn_kl)l∈_N converges to g :=Tx◦G, thus ((gn, pn))n∈_N has a convergent subsequence.

Choose a compact set C ⊂XS_×XS _{and any sequence ((g}

n, pn))n∈_N in the closed

set θ−1(_C). As _C is compact then ((_g

n.pn, pn))n∈_N has a convergent subsequence

((gnk.p

nk_{, p}nk))

k∈_N. As shown previously, it follows that ((gnk, p

nk))

n∈_Nhas a convergent

subsequence, thus ((gn, pn))n∈N has a convergent subsequence. Since θ

−1_{(C) is closed}

then the convergent subsequence of ((gn, pn))n∈N converges inφ

−1_{(C). It now follows}

that φ−1_{(C) is compact, thus asC was chosen arbitrarily, θ is proper as required.}

Corollary 1.2.11. Let (p, S) be a placement in a normed space X, then Stabp is

a compact subgroup of Isom(X). Further, if pv = 0 for some v ∈ S then Stabp ≤

IsomLin_(X).

Proof. It is immediate that Stabp is a subgroup of Isom(X).

Suppose p is a finite placement, then by Lemma 1.2.10, φ is a proper map. As

Stabp :={g : (g, p)∈φ−1[(p, p)]} then Stabp is compact. If p is not finite then we note

that for any finite subplacement q, Stabp is a closed subgroup of the compact group

Stabq, thus Stabp is compact for any placement p. We finally note that if pv = 0 then

1.2 Frameworks and placements 47 Lemma 1.2.12. Letp be a placement of a finite set in a normed spaceX, then Op is

a closed smooth submanifold of XV(G) and the map

˜

φp : Isom(X)/Stabp → Op, gStabp 7→g.p

is a smooth diffeomorphism.

Proof. By Lemma 1.2.10 and Lemma 1.1.26 it follows that Op is a closed smooth

submanifold of XV(G) diffeomorphic to Isom(X)/Stabp under the diffeomorphism

˜

φp.

For a placement (p, S) in a normed space X define the finite dimensional linear

space Ap :={h.p:h∈A(X)} with the topology inherited fromXS. We note that the

topology on Ap is equivalent to the norm topology, thus Ap is a smooth manifold. We

also note that if α: (−δ, δ)→Ap is a differentiable path then α′(t) = (α′v(t))v∈S.

Lemma 1.2.13. Let (q, T) ⊂ (p, S) be placements in X where the affine span of

{pv :v ∈S}is equal to the affine span of {qv :v ∈T}. If|T|<∞then Op is a smooth

manifold that is diffeomorphic to Oq and the restriction map

ρ:Op → Oq, (xv)v∈S 7→(xv)v∈T

is a smooth diffeomorphism.

Proof. We note that the linear map ˜ρ : Ap → Aq where ˜ρ(x) :=x|T is a continuous

linear isomorphism. This implies the map ˜ρ−1|Oq is a smooth embedding into the

smooth manifold Ap with image Op. By Lemma 1.2.12, Oq is a smooth manifold, thus

Oq is diffeomorphic to Op and ρ := ˜ρ|

Oq

Op is a smooth diffeomorphism.

Let (p, S) be a placement in a normed spaceX. We will define a continuous path in XS _{through p to be a familyα:= (α}

some fixed δ > 0) where αv(0) =pv for all v ∈ S. If α(t) := (αv(t))v∈S ∈ Op for all

t∈(−δ, δ) thenα is a trivial finite motion.

Let u∈ XS. If there exists a trivial finite motion α of p that is differentiable at t= 0 and uv = α′v(0) for allv ∈S then we say that uis atrivial (infinitesimal) motion

of p. For any placement p we shall denoteT(p) to be the set all trivial infinitesimal

motions ofp.

Theorem 1.2.14. Let pbe a placement in a normed space X, thenOp is a smooth

manifold with tangent spaceT(p) at pand

˜

φp : Isom(X)/Stabp → Op, gStabp 7→g.p

is a smooth diffeomorphism.

Proof. Choose a finite subplacement (q, T) of (p, S) so that the pand q affinely span

the same space, then Stabp = Stabq. By Lemma 1.2.12, Oq is a smooth manifold

diffeomorphic to Isom(X)/Stabq under the smooth diffeomorphism ˜φq. By Lemma

1.2.13, Op is a smooth manifold diffeomorphic to Isom(X)/Stabp and the restriction

map ρ : Op → Oq is a smooth diffeomorphism. As ˜φp = ρ−1 ◦φ˜q then it is also a

smooth diffeomorphism. It follows from its definition that T(p) is the tangent space of

Op at p.

Remark 1.2.15. As the set of all trivial infinitesimal motions of a placement is a

tangent space to a manifold, it follows that it must therefore be linear.

Corollary 1.2.16. Let p be a placement in a normed space X. Then the mapφp is a

smooth submersion and dφp(ι) :TιIsom(X)→ T(p) is surjective with dφp(ι)g =g.p

for all g ∈TιIsom(X). Further, kerdφp(ι) = TιStabp.

Proof. By Theorem 1.2.14, ˜φp is a smooth diffeomorphism. We note that φp = ˜φp◦π

1.2 Frameworks and placements 49 π is a smooth submersion, thusφp is a smooth submersion also anddφp(ι) is surjective.

As ˜φp is a smooth diffeomorphism then kerdφp(ι) = kerπ as required.

Corollary 1.2.17. Let (p, S) and (q, T) be placements in a normed space X where

the affine span of {pv : v ∈ S} is equal to the affine span of {qv : v ∈ T}, then the

following hold:

(i) The orbits Op and Oq are diffeomorphic.

(ii) dimT(p) = dimT(q).

(iii) If (q, T)⊆(p, S) then the restriction map

ρ:Op → Oq, (xv)v∈S 7→(xv)v∈T

is a smooth diffeomorphism.

Proof. (iii): Choose a finite subplacement (r, U) ⊆(q, T) ⊆ (p, S) so that the affine

span of {rv :v ∈U}is equal to the affine span of {qv :v ∈T}. Define the restriction

maps ρT : Oq → Or and ρS : Op → Or, then by Lemma 1.2.13, ρT, ρS are smooth

diffeomorphisms. As ρ=ρ−_T1◦ρS then it is also a smooth diffeomorphism.

(i): If (q, T)⊆(p, S) this follows from (iii). Suppose (q, T) is not a subplacement of

(p, S). Define (t, S⊔T) to be the placement whereS⊔T is the disjoint union ofS and T, t|S = pand t|T = q. We note all three placements span the same affine subspace of

X. As (p, S)⊂(t, S⊔T) and (q, T)⊆(t, S⊔T) then by (iii) we have Op ∼=Ot∼=Oq

as required.

(ii): By Theorem 1.2.14,Op has tangent space T(p) at pand Oq has tangent space