We are now going to provide a full construction scheme for a suitable inverse atlas that meets all the requirements we have for our inverse atlases on ESMs.
9.1.1 The Exponential Map
The first concept for the ESM itself we will require and use is that of the exponential
map of an ESM. We present it along with some basic results on how to employ this
map for the construction of a useful inverse atlas to an ESM (cf. [9, §3], [13, Prop. 88]):
9.1 Definition Let Ϻ ∈ 𝕄𝑘bd(ℝ𝑑), 𝑥 ∈ Ϻ and T𝑥Ϻ ≅ ℝ𝑘 the tangent space in 𝑥. Then for some suitably small open ball B𝑘𝑟
𝑥(0) ∈ ℝ
𝑘 ≅ T
𝑥Ϻ the exponential map exp𝑥,Ϻ∶ B𝑘𝑟𝑥(0) ⊆ ℝ𝑘→ Ϻ is defined by exp𝑥,Ϻ(𝑧) ∶= ⎧ { ⎨ { ⎩ 𝑥 if 𝑧 = 0 γ𝑥,𝑧(||𝑧||2) otherwise ,
where γ𝑥,𝑧is the uniquely defined arc-length parameterised geodesic that contains 𝑥 and has direction 𝑣𝑧 for some suitable identification 𝑣𝑧 in T𝑥Ϻ of 𝑧 in ℝ𝑘.
9.2 Proposition For a smooth Ϻ ∈ 𝕄𝑘bd(ℝ𝑑), the exponential map exp 𝑥 is a
smooth diffeomorphism in a suitably small ball B𝑘𝑟
𝑥(0) for any 𝑥 ∈ Ϻ and 𝑟𝑥 ∈]0,
∞
]called injectivity radius. If Ϻ is topologically complete, then 𝑟𝑥 = 𝑟(𝑥) depends
continuously on 𝑥.
We can also define a distance function dϺ(⋅, ⋅) on Ϻ: If we consider curves γ ∶ [0, 1] → Ϻ, then we obtain a distance function on Ϻ as
dϺ(𝑥, 𝑧) ∶= inf ⎧ { ⎨ { ⎩
∫
[0,1] ∣∣γ′∣∣ 2∶ γ(0) = 𝑥, γ(1) = 𝑧 ⎫ } ⎬ } ⎭ .Because the relative closure of any nonclosed Ϻ is compact in our case as it is a subdomain of a compact ESM, this is well defined and finite even in that case. And if additionally 𝑧 ∈ exp(B𝑟𝑥(0)), then it is well known that the respective curve realising the distance is a geodesic from 𝑥 to 𝑧. Even better, the following holds:
9.3 Corollary Within exp(B𝑟
𝑥(0)), the squared distance function dϺ(𝑥, 𝑧)
2 is as
smooth as Ϻ.
Proof: This is clear because of the smoothness of the exponential map and the one-to-one correspondence between Euclidean distance of exp-1
𝑥(𝑧) to 0 and the intrinsic distance of 𝑧 to 𝑥 that is directly implied by the definition. q
9.1.2 The Construction of an Atlas
First of all, we can now find a finite inverse atlas of a compact ESM that is made up just of balls as parameter spaces and exponential maps as parameterisations: The exponential map exp𝑥,Ϻ of Ϻ is injective in a radius 𝑟(𝑥) around the origin, and that radius is a continuous function of 𝑥. So by compactness of Ϻ there must be a minimal 𝑟∗ > 0 over all of Ϻ. Thus we can take the exponential map and any 0 < 𝑟 < 𝑟∗ to obtain an open cover of Ϻ by considering the images of the open balls B𝑘𝑟(0) under exp𝑥,Ϻ to all 𝑥 ∈ Ϻ. By compactness, there is a finite subcover, and exp𝑥,Ϻ for arbitrary 𝑥 ∈ Ϻ is C-bounded on any closed ball B𝑘𝑟(0).
9.4 Remark: If we want to have fixed functions T, N that map any point 𝑧 ∈ B𝑘𝑟(0) to an orthonormal frame of the tangent or normal space in exp𝑥,Ϻ(𝑧), then we can easily achieve this by application of Gram-Schmidt orthonormalisation. To make these functions well-defined on all of B𝑘𝑟(0), we can be required to reduce 𝑟 further, but by compactness we can still find a sufficiently small 𝑟 > that works on all of Ϻ and gives a finite subcover.
This construction would suffice for our desired inverse atlas if we have only com- pact ESMs. But as we do have subdomains with boundary as an option, we have to invest considerably more work. So let us have a closer look at the parameterisa- tions of our ESMs. To achieve our goal of a finite inverse atlas where any inverse of a chart is suitably C-bounded and any parameter space is a Lipschitz domain also in the case with boundary, we will need some auxiliary results that are also interesting in their own right:
9.5 Theorem — Intrinsic Tubular Neighbourhood Theorem —
Γ → TϺ be a smooth unit vector field that is normal to Γ. Then there is 𝛿 > 0 such
that on the set Γ × ]−𝛿, 𝛿[ the map ψΓ,Ϻis injective, where
ψΓ,Ϻ∶ (𝑥, 𝑡) ↦ exp𝑥,Ϻ(𝑡 ⋅ ν(𝑥)) .
The image of this set under the respective map is the intrinsic tubular neighbour-
hood UϺ
𝛿(Γ).
Proof: We note that by the required orientability of Γ it is easily verified that such
normal field exists always. Almost copying the ideas of [44], we then note that the Jacobian of ψΓ,Ϻ is clearly nonsingular in any (𝑥, 0), and that map is smooth: Ϻ is a smooth ESM, exp is smooth and ν as well. Consequently, the inverse function theorem and the compactness of Γ give the existence of a suitable 𝛿 > 0. q
Figure 9.1: Depicted on the left is a tubular neighbourhood of a curve in ℝ2 with Bɴ
ԑ(⋅) to two
points. Depicted on the right is an intrinsic tubular neighbourhood of a curve in 𝕊2, again with orbits of expϺ,𝑥(𝑡 ⋅ ν(𝑥)) for fixed 𝑥 and varying 𝑡.
The following theorem now gives us the existence of an inverse atlas with Lipschitz parameter spaces and C-bounded parameterisations that will lead to our ”inverse atlas of choice” for this thesis:
9.6 Theorem If Ϻ ∈ 𝕄𝑘bd(ℝ𝑑), then there is a finite inverse atlas 𝔸
Ϻ= (ψ𝑖, ω𝑖)𝑖∈𝐼
such that any ω𝑖 ∈ 𝕃𝕚𝕡𝑘. Moreover ψ𝑖is C-bounded on ω𝑖with a constant indepen-
dent of 𝑖 ∈ 𝐼.
Proof: The proof for the compact case (so without boundary) is quite short: We
have already defined a suitable inverse atlas by using the exponential map and balls of certain fixed radius 𝑟 > 0, which are obviously Lipschitz.
Turning now to the case with boundary, we have required in this case that Ϻ ⊆ ̂Ϻ with ̂Ϻ ∈ 𝕄𝑘cp(ℝ𝑑) a compact ESM without boundary. Our primary goal is now to parameterise the area near the boundary of Ϻ suitably, because afterwards we can simply fill up with ”exponential” parameterisations in the interior. Therefore, we use the intrinsic tubular neighbourhood of the boundary, which we will param- eterise by using an exponential parameterisation of the boundary and extending this. So let Γ = ∂Ϻ ∈ 𝕄𝑘−1cp (ℝ𝑑) be a hypersurface of some ̂Ϻ ∈ 𝕄𝑘
cp(ℝ𝑑). By conception, we know that Γ is a compact ESM without boundary, whereby there
Figure 9.2: Depicted is the step from one parameter space for the intrinsic tubular neighbourhood
to the respective part of the intrinsic tubular neighbourhood. The boundary is depicted in teal, the dashed green curve implies Ϻ, the dotted green curve completes Ϻ to become ̂Ϻ. Intrinsic tubular
neighbourhood in blue.
is an exponential inverse atlas 𝔸Γ = {( ⏜ψ𝑖, ⏜ω𝑖)}𝑖∈𝐼 with finite index set 𝐼 such that any ⏜ψ𝑖 is C-bounded and any ⏜ω𝑖 is a ball of radius 𝑟Γ > 0. We will now make a parameterisation of the intrinsic tubular neighbourhood out of this. We choose therefore ̂ ψ𝑖 ∶ ⏜ω𝑖× ]−𝛿 2, 𝛿 2[ → ̂Ϻ ̂ψ𝑖(𝑥, 𝑡) = expψ⏜𝑖(𝑥),̂Ϻ(𝑡 ⋅ ν( ⏜ψ𝑖(𝑥))) .
This yields a C-bounded parameterisation of ÛϺ𝛿/2(Γ) if applied on any ⏜ω𝑖 and ⏜ψ𝑖. Moreover, we can demand that the radius 𝑟Γ is the 𝛿 > 0 of the intrinsic tubular neighbourhood theorem without harm, as we just have to shrink one or the other sufficiently. The resulting sets ⏜ω𝑖× ]−𝛿/2, 𝛿/2[ are obviously hypercylinders and thus clearly Lipschitz domains.
We will achieve a parameterisation of ÛϺ𝛿/2(Γ) ∩ Ϻ now by simply restricting the parameter space to ⏜ω𝑖× ]0, 𝛿/2[ or ⏜ω𝑖× ]−𝛿/2, 0[, depending on which side of Γ the desired set Ϻ was. This choice shall be named ̂ω𝑖. Then we have to enhance the resulting inverse atlas {(̂ψ𝑖, ̂ω𝑖)}𝑖∈𝐼 parameterizing UϺ𝛿/2̂ (Γ) ∩ Ϻ by sufficiently many pairs (exp𝜉,Ϻ, B𝑘𝑟(0)) for suitable 𝜉 ∈ Ϻ with dϺ(𝜉, Γ) > 𝛿/2 to obtain an inverse atlas for the whole of Ϻ, which is no problem by the relative compactness
of Ϻ in the compact ESM ̂Ϻ. q
9.7 Remark: (1) We can thereby choose a restrictable Lipschitz inverse atlas for
an ESM ̂Ϻ, which we use in defining Sobolev spaces: We just have to change the last step in the proof of the theorem, where we do not take the upper or lower half of ⏜ω𝑖× ]−𝛿/2, 𝛿/2[, but the whole set, and fill up again by exponentials as parame- terisations and balls as parameter spaces, but this time covering all of ̂Ϻ.
(2) The construction of suitable smooth maps T𝑖 ∶ ω𝑖 → TϺ and N𝑖 ∶ ω𝑖 → NϺ that assign each element of ω𝑖and thereby also each element of ψ𝑖(ω𝑖) to an orthonor-
mal frame of the respective tangent and normal space is again easily achieved: For
T𝑖we just have to apply orthonormalisation to the Jacobian of the parameterisation ψ𝑖, and thereby we can also construct N𝑖 after possibly shrinking the parameter spaces suitably.
(Ʒ) By this construction it is particularly obvious that in the case without boundary we can always choose the radius of the parameter space balls slightly larger or smaller if needed and still obtain an inverse atlas.