Topics In Analysis

28.5.1 An Oriented Manifold

A bounded open subset, Ω, of ℝⁿ,n ≥ 2 has Lipschitz boundary and lies locally on one side of its boundary if it satisfies the following conditions.

For each p ∈ ∂Ω ≡Ω ∖ Ω, there exists an open set, Q, containing p, an open interval

, a bounded open set B ⊆ ℝⁿ⁻¹, and an orthogonal transformation R such that detR = 1,

B × (a,b) = RQ,

and letting W = Q ∩ Ω,

RW = {u ∈ ℝn : a < u1 < g (u2,⋅⋅⋅,un ),(u2,⋅⋅⋅,un ) ∈ B}

where g is Lipschitz continuous on ℝⁿ⁻¹, g

< b for ∈ B, and g vanishing outside some compact set in ℝⁿ⁻¹.

R (∂Ω ∩Q ) = {u ∈ ℝn : u1 = g(u2,⋅⋅⋅,un) ,(u2,⋅⋅⋅,un) ∈ B}.

Note that finitely many of these sets Q cover ∂Ω because ∂Ω is compact. The following picture describes the situation.

Define P₁ : ℝⁿ → ℝⁿ⁻¹ by

P1u ≡ (u2,⋅⋅⋅,un)

and Σ : ℝⁿ → ℝⁿ given by

Σu ≡ u− g (P u)e 1 1

≡ u− g (u2,⋅⋅⋅,un)e1

≡ (u1 − g(u2,⋅⋅⋅,un),u2,⋅⋅⋅,un)

Thus Σ is invertible and

Σ−1u = u+ g (P1u )e1

≡ (u1 + g(u2,⋅⋅⋅,un),u2,⋅⋅⋅,un)

For x ∈ ∂Ω ∩ Q, it follows the first component of Rx is g

. Now define R :W → ℝ_≤ⁿ as

u ≡ Rx ≡ Rx − g(P1 (Rx))e1 ≡ ΣRx

and so it follows

R −1 = R ∗Σ−1.

These mappings R involve first a rotation followed by a variable sheer in the direction of the u₁ axis.

Since ∂Ω is compact, there are finitely many of these open sets, Q₁,

,Q_p which cover ∂Ω. Let the orthogonal transformations and other quantities described above also be indexed by k for k = 1,,p. Also let Q₀ be an open set with Q₀ ⊆ Ω and Ω is covered by Q₀,Q₁,,Q_p. Let u ≡ R₀x ≡ x − ke₁ where k is large enough that R₀Q₀ ⊆ ℝ_<ⁿ. Thus in this case, the orthogonal transformation R₀ equals I and Σ₀x ≡ x − ke₁. I claim Ω is an oriented manifold with boundary and the charts are .

Letting A be an open set contained in R_i

such that ∂A has measure 0 and ∂A separates ℝⁿ into two components, consider

d(u,A,Rj ∘R −i1),u ∕∈ Rj ∘R −i1(∂A)

By convolving g with a mollifier, there exists a sequence of infinitely differentiable functions g_ε which converge uniformly to g on all of ℝⁿ⁻¹ as ε → 0. Therefore, letting Σ_ε be the corresponding functions defined above with g replaced with g_ε, it follows the Σ_ε will converge uniformly to Σ and Σ_ε⁻¹ will converge uniformly to Σ⁻¹. Thus from the above descriptions of R_j⁻¹, it follows R_jε⁻¹ converges uniformly to R_j⁻¹ for each j. Therefore, if ε is small enough, u

and so from properties of the degree, the mappings R_j and R_i⁻¹ can be replaced with smooth ones in computing the degree. To save on notation, I will drop the ε. The mapping involved is

ΣjRjR ∗Σ−1 i i

and it is a one to one mapping. What is the determinant of its derivative? By the chain rule,

D (ΣjRjR ∗iΣ −i1) = D Σj(RjR ∗iΣ−i1)DRj (R∗iΣ−i1)DR ∗i (Σ−i1)D Σ−i1

However,

( ) det(DΣj ) = 1 = det D Σ−j1

and det

= det = 1 by assumption. Therefore, if u ∈, the above degree is 1 and if u is not in this set, the above degree is 0 or undefined if u is on . By Definition 28.1.5 Ω is indeed an oriented manifold.