5.3 special coordinates
5.3.2. Diffeomorphisms on the Cylinder
i=1
B˜i.
Now applying Lemma5.3.19to the tuple(Um, ˜Um, ˜gm, ˜fm)and(Bm+1, ˜Bm+1,gm+1,fm+1,Fm+1) we get a new tuple(Um+1, ˜Um+1, ˜gm+1, ˜fm+1)fulfilling property (i) to (iii). Moreover, we have
U˜m+1=
m [
i=1
B˜i
!
∪B˜m+1=
m+1 [
i=1
B˜i.
The induction stops form=k. The mapg:=g˜kthen is the desired one.
This section can be summarised as follows
Proposition5.3.21. Let f : Sn → Sn be a Cm-diffeomorphism admissibly close to the identity. Then there is a Cm-map g:Sn→so(n+1)such that
f(x) =exp(g(x))·x.
We will now state some consequences of this result. First we draw our attention to the fact that if we choosee small enough, then every map with kf −idk∞,Sn < e is admissibly close to the identity. This can be traced back to the fact that the very convex neighbourhood U(1) of1 ∈ SO(n+1) exists. Forκ small enough, U2κ(1) := {A∈SO(n+1)k kA−1k<2κ}is a subset ofU(1). Hence every diffeomorphism f : Sn → Sn with kf −idk∞,Sn < e ≤ κ will be admissibly close to the identity and so has a representation of the above form. As a consequence we have the following weaker corollary.
Corollary5.3.22. Let K ⊂Diffm(Sn)be a connected component in the space of Cmdiffeomorphisms of Sn with topology induced by the uniform norm. Then either for all or for no f ∈K there exists a Cm-map G:Sn →SO(n+1)such that f(x) =G(x)·x.
Proof: By Proposition5.3.21there is an open ballBκ(id):={f ∈Diffm(Sn)| kf−idk∞,Sn <κ}for which the claim holds. Now let ˜K ⊂Diffm(Sn)be the subset such that for all f ∈K˜ there exists aCm-mapGf : Sn → SO(n+1) such that f(x) = Gf(x)·x. It suffices to show that this set is closed and open with respect to the uniform norm. First we observe that iff,h∈Diffm(Sn)with kf −hk∞,Sn < κ, then
f◦h−1−id
∞,Sn < κ since
f◦h−1(x)−x
= (f−h) h−1(x) ≤ kf −hk∞,Sn. Hence f ◦h−1(x) can be written as Gf◦h−1(x)·x. Consequently if h ∈ K˜ then
f(x) = f ◦h−1
(h(x)) =Gf◦h−1(h(x))·Gh(x)·x and withGf(x) :=Gf◦h−1(h(x))·Gh(x)we have f ∈ K. Hence˜ h∈ K˜ if and only ifBκ(h)⊂K. For the complement we have˜ h ∈K˜Cif and only if there is someκ>0 such thatBκ(h)⊂K˜C. Otherwise for f ∈Bκ(h)∩K˜ 6=∅one also has h∈Bκ(f)and by the previous considerationshitself will have to be an element of ˜K, which is a
contradiction.
5.3.2. Diffeomorphisms on the Cylinder
The same methods used for characterising diffeomorphisms on the sphere will now be used to characterise special diffeomorphisms on the compact cylinder
Z := [−1, 1]×Sn⊂[−1, 1]×Rn+1.
120 Chapter 5: almost einstein structures with vanishing almost scalar curvature
Definition5.3.23. A diffeomorphism f : Z → Z will be said to be a sphere-preserving diffeo-morphism, if there exists an fSn : Z → Sn such that f(t,x) = (t,fSn(t,x)). Hence f|{t}×Sn : {t} ×Sn → {t} ×Snprovides a diffeomorphism on the sphere for a fixed parametert∈[0, 1]. Definition5.3.24. ACm-diffeomorphism f :Z → Zwill said to beadmissibly closeto the identity (with respect toe), if it is sphere-preserving and there is ane<1 and a very convex neighbour-hood U(1) ⊂ SO(n+1) such that for each t ∈ [−1, 1] the map (Sn3 x7→ fSn(t,x)∈Sn) is admissibly close to the identity with respect toeandU(1). The preimage ofU(1)will again be denotedU(0) =exp−1(U(1)).
Lemma5.3.25. Let f :Z → Z be a sphere-preserving Cm-diffeomorphism. Then there is neighbourhood U(Z) ⊂ [−1, 1]×Rn+1of the cylinder and an Cm-extension F = (FR,FRn+1) : U(Z) → [−1, 1]× Rn+1such that
(i) rankdFp=n+2for all p∈ Z (iia) F(p) = f(p)for all p∈ Z
(iib) kFRn+1(t,x)kRn+1 6=kxkRn+1 for all p= (t,x)∈U(Zn)\ Zn (iii) dF(t,x)(0,x) =2(0,fSn(t,x))for all p= (t,x)∈ Z,
where fSn := πRn+1 ◦ f . In the last line(0,fSn(t,x)) is interpreted as a vector in the tangent space TpRn+2'Rn+2.
The neighbourhood in the last lemma is understood as open set with respect to the induced topology of[−1, 1]×Rn+1.
Proof: First we define the neighbourhood asU(Z):=(t,x)∈[−1, 1]×Rn+2kxk>1/2 and we further define the extension by
F(t,x):=
t,(2kxk −1)fSn
t, x
kxk
. Then the map Ft :
x∈Rn+1 kxk>1/2 → Rn+1 defined by (x7→FRn+1(t,x)) is a sphere-preserving extension of ftin the sense of Definition5.3.4, where ft(x):= fSn(t,x). The Jacobian ofFis given by
dF(t,x)=∂tF 1 0
Rn+1(t,x)dFxt
.
The claim then is a corollary of Lemma5.3.3.
Definition5.3.26. Such an extension will be calledcylinder-preserving extensionof f. Definition5.3.27. Let f =id[−1,1],fSn
:Z → Zbe a map admissibly close to the identity. Let U(1)be the corresponding very convex neighbourhood and consider F:U(Z)→R×Rn+1to be acylinder-preserving extensionof f. We then define
G˜ : U(Z)×U(0) → Rn+1
(t,x,g) 7→ FRn+1(t,x)−exp(g)·x and
M˜ :=G˜−1(0), whereU(0) =exp−1(U(1))
Lemma5.3.28. With the assumptions of the last definition M˜ ⊂ U(Z)×so(n+1) is an embedded submanifold with boundary anddim(M˜) = n(n+1)2 +1. Moreover,M˜ ⊂ Z ×U(0).
Proof: The proof is an analogue to that of Lemma5.3.8. We observe that ˜G(t,x,g) = 0 implies kFRn+1(t,x)k = kxk. By property (iib) of cylinder-preserving extensions this is equivalent to
5.3 special coordinates 121
requiring(t,x)∈ Z and hence ˜G⊂ Z ×U(0). It suffices to show thatdG˜p has full rank for all p= (t,x,g)∈ Z ×U(0). The differentialdG˜(t,x,g):R×Rn+1×Tgso(n+1)→Rn+1is given by
dG˜(t,x,g)(vt,vx,vg) = d ds
s=0
G˜(t+svt,x+svx,g+svg)
= (dFRn+1)(t,x)(vt,vx)−(dexp)g(vg)·x−exp(g)·vx. Restricting to(vt,vx) = (0, 0)and using (1.88) gives
dG˜(t,x,g)(0, 0,Tgso(n+1)) =Texp(g)·xSn=TfSn(t,x)Sn
for all(t,x)∈ Z. Now choosing(vt,vx,vg) = (0,x, 0)and using that f is admissibly close to the identity gives for each(t,x)∈ Z
dG˜(t,x,g)(0,x, 0) =dFxt(x)−exp(g)·x= fSn(t,x)6=0
On the left-hand side, fSn(t,x) interpreted as a vector in the tangent space is transversal to the sphere. Hence fSn(t,x) and TfSn(t,x)Sn span the full Rn+1 and we conclude that dG˜(t,x,g) is surjective for all(t,x,g) ∈ Z ×U(0). By the regular value theorem ˜G−1(0) is a n(n+1)2 +1 dimensional submanifold ofU(Z)×U(0)if it is not empty. Non-emptiness is provided by the requirement that f is admissibly close to the identity and so for each(t,x)∈ Zthere is at least
oneg∈U(0)such that ˜G(t,x,g) =0.
Again we would likedπZto have full rank.
Lemma5.3.29. The projection mapπZ : ˜M3(t,x,g)7→(t,x)∈ Z is a surjective submersion.
Proof: Surjectivity of πZ is due to the fact that f is admissibly close to the identity. To show surjectivity of its differentialdπZ, we calculate its kernel. On the one hand vt andvgwill have to vanish for(vt,vx,vg) ∈ kerdπZ ⊂ T(t,x,g)M. On the other hand˜ (0, 0,vg) will have to be an element of the kernel ofdG˜(t,x,g), which is true if and only if dexpg(vg) ∈ Texp(g)SO(n+1) = exp(g)·so(n+1)annihilatesx. Hence we find
(vt,vx,vg)∈ker
dπZ(t,x,g)
⇐⇒
( (vt,vx) = (0, 0)
vg ∈ dexp−1g (exp(g)·stab(x)) and dim
kerπZ(t,x,g)
= dim(so(n)). The dimension of the image of dπZ then is dim
im(πZ)(t,x,g)=dim(M˜)−dim(so(n)) =n+1=dim(Z), which makesdπZ a surjective
map.
We use the previous lemma to find the following local result.
Lemma5.3.30. For each(t,x) ∈ Z there is a neighbourhood U(t,x) ⊂ Z and a map g : U(t,x) → so(n+1)such that
(i) (s,y,g(s,y))∈M for all˜ (s,y)∈U(t,x)and (ii) g is Cm-smooth.
Proof: We will only sketch the proof here since it essentially agrees with that of Lemma5.3.10. Let (t,x,g)∈ M˜ be an arbitrary point. SinceZ ×so(n+1)is a manifold with boundary and ˜Mis a submanifold of dimension n(n+1)2 +1=:n+1+d, there is aCm-chartϕ:U⊂ Z ×so(n+1)→ R+0 ×Rn×Rd×Rn for a neighbourhood of (t,x,g) such that ϕ(t,x,g) = 0 and ϕ(s,y,h) ∈ R+0 ×Rn×Rd× {0}for all(s,y,h)∈M˜ ∩U. Only the lastRn-part in the decomposition refers to the special property of ˜Mof being a submanifold. The remaining decomposition is arbitrary at the moment and will be justified later. In case where−1 < t < 1, R+0 may be replaced by R. It is a reference to the boundary of the interval[−1, 1]. By Lemma5.3.29the composition
122 Chapter 5: almost einstein structures with vanishing almost scalar curvature
πZ◦ϕ−1 : Rn×R+0 ×Rd× {0} → Z is a surjective submersion. Hence we may assume ϕ to be such that ker
dπZ◦ϕ−1
ϕ(t,x,g)
= {0} × {0} ×Rd× {0}. This may be achieved by a rotation inRn+1+d. The restriction ˜ϑ := πZ◦ϕ−1
R+
0×Rn×{0}×{0} then is a diffeomorphism for some neighbourhoodU0 ⊂R+0 ×Rn× {0} × {0}of the origin. Altogether the previous sketch motivates the existence of a well defined mapg in the following diagram for a neighbourhood of(t,x)∈ Z.
U0
R+0 ×Rn× {0} × {0}
ϕ−1(U0) Z ×so(n+1)
Z so(n+1)
ϕ−1
ϑ˜
⊂
πZ
πso(n+1)
⊂ g
The map is defined by
g: ϑ˜(U0) → so(n+1)
(s,y) 7→ πso(n+1)◦ϕ−1◦ϑ˜−1(s,y).
Again the idea of the proof was to show that the dashed line indeed represents a bijective map.
Making this a global result gives the following proposition
Proposition 5.3.31. Let f : Z → Z be a Cm-smooth map admissibly close to the identity, M the˜ corresponding manifold defined above. Then there is a Cm-smooth map g:Z →so(n+1)such that
(p,g(p))∈ M˜ (5.19)
for all p= (t,x)∈ Z. Consequently f can be written as
f(t,x) = (t, exp(g(t,x))·x).
Proof: The reasoning of the proof coincides with that done in the last section. So we will only outline it here. LetU(0):=exp(U(1))be the preimage of the very convex neighbourhoodU(1). Then for each (t,x) ∈ Z there is at least one ht,x ∈ U(0) such that f(t,x) = (t,fSn(t,x)) =
t, exp ht,x
·x
. We define a fibration of ˜Mby
{(t,x)} ×F˜(t,x):=π−1Z ((t,x)).
More explicitly we can write ˜F(t,x)=h∈U(0)kexp(h)·x =exp(ht,x)·x . By applying Lemma 5.3.11we find the ˜F(t,x)to be connected and it allows us to write it as
F˜(t,x)=exp−1 exp ht,x
·stab(x)∩U(1).
The next step is to construct a global Cm-smooth section of that fibration. Local sections are provided by Lemma5.3.30. By further shrinking the neighbourhood, we find for eachp∈ Z an open ballBp ⊂ Z and a Cm-map gp : Bp →U(0) such that(q,gp(q)) ∈ M˜ for all q ∈ Bp. By open ball, we mean the restriction of open balls inRn+2to the cylinderZ. Using compactness ofZ we can choose a finite subcoverZ =SiBi and local sections gi : Bi →U(0). We observe that the double partition of unity provided by Lemma5.3.17and Corollary5.3.18and the gluing Lemma5.3.19not works only for maps on the sphere but also for maps on the compact cylinder [−1, 1]×Sn, since the only property needed in the proof is compactness. The remaining proof then can be carried out, using the induction method introduced in the proof of Proposition5.3.20
5.3 special coordinates 123
A minor modification is required if f : Z → Z is a Cm>1-map only away from t = 0, but a Cm−1-map else. In that case Proposition 5.3.31 would provide only a Cm−1-map g : Z → so(n+1). To get a Cm-map away fromt = 0 we have to require that allgi have this property.
Lemma5.3.30can be modified such that it supports that smoothness. Smoothness of the local sectionsgi is inherited from the charts ϕ : U → R×Rn×Rd×Rn. Fort 6= 0 we can choose a neighbourhoodU(t,x) of(t,x)such that f :U(t,x) → Z isCm-smooth and hence ˜Mlocally admits aCm-chart for a neighbourhood of(t,x,g)∈ M. It remains to consider points˜ (0,x,g)∈ M˜ ⊂ Z ×U(0) and to construct a chart for a neighbourhood of them that is Cm−1 for t = 0 andCmelse. The differential of ˜G:U(Z)×U(0)→Rn+1has full rank in(0,x,g). The implicit function theorem then provides a mapψ:U0 →U00such that
(U0⊕U00)∩M˜ = x⊕ψ(x)|x∈U0 U0 ⊂ ker
dG˜(0,x,g)
'Rn(n2+1)+1 U00⊂ ker
dG˜(0,x,g)⊥
'Rn+1.
In particular ˜G(y⊕ψ(y)) = 0 for all y ∈ U0⊕U00 and the differential ofψis given by dψy =
−d00G˜−1y⊕ψ(y)◦d0G˜y⊕ψ(y). Consequently the map dψ : y 7→ dψy inherits differentiability of ˜G, which is of classCm−1as long asy⊕ψ(y)6= (0,·). Finally, we can define the chartφ−1:U0 'M˜ byφ−1(x):=x⊕ψ(x).
Summarising the last paragraphs gives a modification of Proposition5.3.31.
Proposition5.3.32. Let f : Z → Z be a Cm−1-smooth map admissibly close to the identity, which is Cm-smooth for point(t,x) ∈ Z with t 6=0. LetM be the corresponding manifold defined above. Then˜ there is a Cm−1-smooth map g:Z →so(n+1), which is Cm-smooth away from t=0such that
(p,g(p))∈ M˜ (5.20)
for all p= (t,x)∈ Z. Consequently f can be written as
f(t,x) = (t, exp(g(t,x))·x).