5.3 special coordinates
5.3.3. Extension of Special Cone Diffeomorphisms
This section considers the extension of special diffeomorphism, given on a cone. The extension will later correspond to a coordinate transformation. The section is organised as follows. First we will specify the bijective map on the cone inRnthat we would like to expand to a map onRn. The second step is to blow up the vertex of the cone and to define a sufficiently smooth bijective map on the cylinderZ = [−1, 1]×Sn−1, which emerges from the blow up. Using the results of the last section this can be lifted to a sufficiently smooth map on the cylinder with values in the Lie algebraso(n−1). This map can be extend to a map defined on [−1, 1]×Rn−1 with values in the Lie algebra. The map will then be used to define an extension of the original bijection on the cone. Its smoothness properties and its bijectivity are topic of the last part of this section.
Definition5.3.33. Let ζ : Rn ⊃ U →Rn be a Cm≥2-diffeomorphism onto its image. It will be calledcone-preserving diffeomorphism close to the identityif it has the following properties:
(i) ζ(x) =x+o(kxkn)
(ii) ζ(y)∈C⇔y∈C∩UwhereC=ny∈Rn| kyk1,n−1=0o
is the Minkowski cone inR1,n−1. So for the rest of this section letζbe a cone-preserving diffeomorphism close to the identity.
For the moment consider the rescaled map
f : Rn → Rn
y 7→
( kykn
kζ(y)knζ(y) y6=0
0 else.
(5.21)
124 Chapter 5: almost einstein structures with vanishing almost scalar curvature
It is of classC1in a neighbourhood of the origin6. Moreover it is a cone-preserving diffeomor-phism close to the identity. An important property of f is that it preserves spheres of type
{r} ×Sn−2r
on the coneC = {y∈Rn|kyk1,n−1=0}. Aim of this section is to extend the re-striction f|C to a map on a neighbourhood of the origin such that the extension more generally preserves spheres of type {t} ×Srn−2
witht6=r. The map that we are going to construct will have some additional important properties that will be summarised at the end of this section.
Initially consider the map β=β0,β
:(R\ {0})×Rn−13(t,x)7→t(1,x)∈(R\ {0})×Rn−1.
It maps cylinders inRn to cones and preserves the first component (see figure9for a visualisa-tion ofβ). In particular points(t,x)withkxkn−1=c =const. have images with kβ(t,x)kn−1
where the projection is with respect to the lastn−1 components of β−1◦ξ◦β. More explicitly (t,x) is mapped to
t, ξ(t(1,x))
ξ0(t(1,x))
and one can see that for the transformation of two different maps one has the equalityξβ =ξ˜β ifξ =λξ˜for a nowhere vanishing functionλ. We also have the following lemma for the transformation of diffeomorphisms close to the identity.
Lemma5.3.34. Letξ ∈Cm>1(U⊂Rn,Rn)be a diffeomorphism on a star shaped neighbourhood U of
The extension will also be denoted byξβ. Moreover the extension is a local diffeomorphism for points at the hyperplane{0} ×Rn−1.
Proof: The claim is a direct consequence of Taylor’s theorem with remainder (e.g. [Tu11]) as it allows one to write the mapξ asξ(x) = J(x)·x, where J :U → Mat(n,R)is a matrix valued Numerator and denominator in the fraction areCm−1-maps with the denominator being 1 at the origin. Henceξβ at least locally is a composition of Cm−1-maps which provides the first part of the claim. The second part is an immediate consequence, as the fraction is well defined for all(t,x) ∈ U˜ ∪ {0} ×Rn−1
. The differential of the extended ξβ then has full rank along the hyperplane{0} ×Rn−1which makes it a local diffeomorphism along the hyperplane.
The analogue definition for a reverse transformation then gives
6 The map f is defined only for the purpose of motivation of further constructions. So no smoothness issues will be proven explicitly. On the other hand the proofs are kind of hidden in the following work.
7 The mapJis defined byJ(y) = R1 0
dξsyds, wheredξsyis the Jacobian ofξatsy.
5.3 special coordinates 125
forξ0(t,x)6=0, which along the coneCcoincides with kξ(y)kkykn
nξ(y)ifξpreserves the coneC. Now consider the cone-preserving diffeomorphism ζ we have started with and let fZ := ζβ the mapξ. As fZ is the restriction of a localCm−1-diffeomorphism in a neighbourhood of the hyperplane{0} ×Rn−1 by Lemma 5.3.34, it is a local diffeomorphism if eis has been chosen small enough. Moreover it then is bijective by Lemma5.3.2, as it preserves spheres and hence is aCm−1-diffeomorphism onZe. Lemma5.3.34also provides that fZ is of classCmfor(t,x)∈ Ze witht6=0. We now may assumee>1. If not, we may continue the following calculations with a scaled fZ, i.e. a map defined by fZ,δ(y):= 1
δfZ(δy)forδbeing sufficiently small. So far we have shown that fZ is a diffeomorphism on the compact cylinderZ : [−1, 1]×Sn−2 which is admissibly close to the identity and fulfils the requirements of Proposition5.3.32. Hence, there is a Cm−1-map g : Z → so(n−1), which is of classCm away from {0} ×Sn−2 and it holds
Remark 5.3.36. Roughly speaking in cylindrical coordinatesκ is a bump function that depends only on the radius. The last three requirements also imply two more properties that are impor-tant for later considerations
(iv) The support ofκ is subset of the compact set n
x∈Rn−1| kxk ∈h12,32io We now define the extension of fZ as follows
FZ= F1Z,F2Z the identity in a neighbourhood of that line and hence well defined.
Now we are able to define the extension of f|C:C→CbyF=: FZβ−1 this section we will point out the important properties of this map.
126 Chapter 5: almost einstein structures with vanishing almost scalar curvature ver-tex at the origin. In particular F is the identity there and as long as x0 6= 0, extending F to points (0,x0)by the identity gives a C∞-smooth map there. It remains to checkC1smoothness in(t,x) =0.
We will show the claim for the lastn−1 components of F, as it is obvious for the first one.
The norm ofF(t,x)independently oftis
For C1-smoothness we derive the differential ofF away from t = 0 and then determine its limit as(t,x)goes to zero.
3/2(0) exists and is zero. Combining this consequence with the first consideration on the complement ofB3/2 then provides uniform convergence all overRn−1. Now we take care of the case where (V1,V2) = (1, 0). By using Equation (5.22) we get away
t is bounded and hence the first term converges to(1, 0)in any limit where t is sent to zero. Also by Remark 5.3.36(v) the factor dκx vanishes ifx=0. Hence for convergence of the second term in (5.23) it suffices to show vanishing
8 For the direct calculation we first observeκ(y)g is of classC1then gives the claimed equation.
5.3 special coordinates 127 The first factor on the right-hand side clearly has operator norm1. So it suffices to show that (y7→g(t,y)·y) uniformly converges to zero in the limit t → 0. This is provided, as by
This coincides with the differential of F if derived using partial derivatives in 0. In particular dF0(0,V2) =V2sinceF(0,x) = (0,x)anddF0(1, 0) = (1, 0)sinceF(t, 0) = (t, 0)for allt∈[−1, 1] andx∈Rn−1.
Now as we know F to be the identity in a neighbourhood of the punctured hyperplane {0} ×Rn−1\ {0}and to be a diffeomorphism in a neighbourhood of the origin, we can conclude that ifeis chosen small enough, thenF:(−e,e)×Rn−1→(−e,e)×Rn−1is a diffeomorphism.
As announced in the beginning of this section, we will now point out the important properties ofF.
Proposition5.3.37. Let ζ : Rn → Rn be a cone-preserving Cm≥2-diffeomorphism that is close to the identity. Then there exists ane>0and a cone-preserving C1-diffeomorphism F:(−e,e)×Rn−1→Rn close to the identity, which has the following properties. It
(i) is Cm-smooth away from(t,x) =0.
One may equivalently replace (iia) and (iib) by a point(ii) and instead observe that F pre-serves spheres of type{t} ×Srn−2for(t,r)∈(−e,e)×R+.