LemmaStatement: AI-generatedProof: AI-generated

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Transverse hitting maps have local differentiable extensions

Statement

Work in ZFC. Let $C$ be a smooth Hausdorff second-countable surface (that is, a two-dimensional smooth manifold) and $M$ a smooth manifold. Let $s : M \to R$ be smooth with $0$ a regular value, and give $Σ = s^{- 1} (0)$ its regular-level embedded smooth structure when nonempty. Let $G : O \to M$ be a supplied $C^{1}$ family on an open set $O \subset C \times R$ , with the product smooth structure. Let $A \subset C$ be Borel.

Suppose for each $x \in A$ there is a unique $t_{x}$ with $(x, t_{x}) \in O$ and $s (G (x, t_{x})) = 0$ , and $\partial_{t} (s \circ G) (x, t_{x}) \neq 0$ . Then $F (x) = G (x, t_{x})$ has local $C^{1}$ extensions from $A$ into $Σ$ , and a countable cover by such extensions. These extensions can be restricted to smaller source coordinate balls covering $A$ so that each coordinate representative in a target chart of $Σ$ is Lipschitz. No regularity of $F (A)$ is assumed.

Facts & Assumptions

[A1]

Work in ZFC, including AC. Only its countable-choice consequence is used: for the countable-subcover theorem and for selecting the extension and chart data attached to the countably many selected neighborhoods. The Axiom of Choice

[F1]

The Euclidean implicit function theorem applies to a scalar $C^{1}$ equation on an open subset of $R^{2} \times R$ when its time derivative at a zero is nonzero. The Euclidean implicit function theorem with derivative formula

[F2]

The Borel subsets of a topological space form the sigma-algebra generated by its open sets. Applied to the manifold topology of $C$ , this defines the Borel hypothesis on $A$ without a Riemannian metric or any openness or boundary regularity of $A$ . The Borel sigma-algebra of a topological space

[F3]

Smooth manifolds have smooth coordinate charts; $C \times R$ has its product smooth structure, and $C^{1}$ regularity between manifolds is tested by coordinate representatives. Smooth manifolds and their smooth charts Products of smooth manifolds have a canonical product smooth structure Cʳ and smooth maps between smooth manifolds

[F4]

A regular value has surjective derivative at every point of its fibre. A nonempty regular fibre of a smooth real-valued map is an embedded codimension-one submanifold. Its slice charts, restricted and projected onto the tangential coordinates, supply its smooth atlas and subspace topology. Regular and critical points and values Codimension and hypersurfaces Embedded submanifolds and slice charts A regular level set is an embedded submanifold Slice-chart restrictions form a smooth atlas

[F5]

Continuous first partial derivatives give total differentiability with the Jacobian derivative. Total derivatives obey the chain rule; for $C^{1}$ coordinate maps the resulting finite matrix products are continuous, so their compositions are $C^{1}$ . If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)

[F6]

Bases trace to subspaces, and an ambient-continuous map with values in a subspace is continuous into that subspace. Second countability means existence of an at most countable basis. Under countable choice every second-countable space is Lindelof. Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace Second countability: an at most countable basis for the topology Assuming countable choice, every second countable space is Lindelöf

[F7]

The vector mean-value inequality bounds the endpoint difference on $[0, 1]$ by a uniform derivative bound. Lipschitz means a finite constant bounds every output distance by that constant times the input distance. Here coordinate distances are Euclidean. The mean value inequality: if f : [a,b] → ℝᵐ is continuous and differentiable on (a,b) with ‖ f'‖₂ ≤ M, then ‖ f(b)-f(a)‖₂ ≤ M(b-a) Lipschitz map, α-Hölder map for rational 0 < α ≤ 1, and contraction

Proof

Given: The family, unique hits and transversality in the Statement. Goal: Produce the stated extensions and countable Lipschitz restrictions.

1.1givenF4

If $A = \emptyset$ , the empty family of extensions proves every assertion. Otherwise any hit belongs to $Σ$ , so $Σ$ is nonempty and [F4] applies to the given smooth map $s : M \to R$ and regular value $0$ . Thus $Σ$ has the asserted embedded smooth structure; writing $m = \dim M$ , its dimension is $k = m - 1 \geq 0$ .

1.2givenF1F3F5construct

Fix $a \in A$ and a source chart $ϕ$ about $a$ . The coordinate function $H (z, t) = s (G (ϕ^{- 1} (z), t))$ is $C^{1}$ on the open coordinate preimage of $O$ , by [F3] and [F5]. It satisfies $H (ϕ (a), t_{a}) = 0$ with nonzero scalar time derivative. Apply [F1] with source dimension two and time dimension one. After restricting to a product neighborhood contained in that open preimage, this gives an open neighborhood $U$ of $a$ and a $C^{1}$ function $τ : U \to R$ with $(x, τ (x)) \in O$ and $s (G (x, τ (x))) = 0$ .

2.1F3F4F5F6step 1.1step 1.2construct

Put $\tilde{F} (x) = G (x, τ (x))$ . In source coordinates the graph map $z \mapsto (z, τ (ϕ^{- 1} (z)))$ is $C^{1}$ : its derivative is $v \mapsto (v, D (τ \circ ϕ^{- 1}) (z) v)$ and is continuous. Hence [F5] makes $\tilde{F}$ a $C^{1}$ map into $M$ , with image in $Σ$ . Take a slice chart $ψ$ about $\tilde{F} (a)$ and shrink $U$ so its image stays in that chart. In this chart $ψ \circ \tilde{F} \circ ϕ^{- 1} = (f, 0)$ , where $f$ consists of the first $k$ coordinate components and is $C^{1}$ . The target chart on $Σ$ is $χ = π_{k} \circ ψ ∣_{Σ}$ , so $χ \circ \tilde{F} \circ ϕ^{- 1} = f$ . Continuity into $Σ$ follows from its subspace topology and ambient continuity. Thus [F3] proves that $\tilde{F} : U \to Σ$ is $C^{1}$ , including when $k = 0$ .

3.1givenF2step 1.2step 2.1

For $x \in A \cap U$ , the value $τ (x)$ is a zero in the given fibre $O_{x}$ . Its assumed uniqueness gives $τ (x) = t_{x}$ , hence $\tilde{F} ∣_{A \cap U} = F$ . This proves the local extension assertion. The Borel convention [F2] imposes no additional geometric condition on $A$ and is not needed for this pointwise equality.

3.2F5F7step 2.1algebraconstruct

Shrink the source patch about $a$ to a coordinate ball $B$ whose closure lies in $ϕ (U)$ , retaining the target chart of step 2.1. If $k = 0$ , its coordinate representative $f$ is constant and is Lipschitz with constant zero. If $k \geq 1$ , continuity at $ϕ (a)$ of the finitely many entries of $D f$ permits a further ball on which all entries have absolute value at most a finite $b \geq 1$ . Componentwise estimation gives $∥ D f (z) v ∥_{2} \leq 2 k b ∥ v ∥_{2}$ there. For $u, v \in B$ , convexity keeps $u + t (v - u)$ in $B$ for $0 \leq t \leq 1$ . By [F5], the derivative of $t \mapsto f (u + t (v - u))$ has norm at most $2 k b ∥ v - u ∥_{2}$ . Apply [F7] on $[0, 1]$ to get $∥ f (v) - f (u) ∥_{2} \leq 2 k b ∥ v - u ∥_{2}$ . Thus each such restricted extension is Lipschitz in the specified coordinates.

4.1A1F6step 1.2step 2.1step 3.1step 3.2∎

Take the family of all smaller neighborhoods admitting the extension and chart data from steps 1.2, 2.1, 3.1 and 3.2; no choice indexed by all points of $A$ is necessary to form this family. Their traces on $A$ are an open cover of $A$ in its subspace topology. If $B$ is a countable basis of $C$ , then ${B \cap A : B \in B}$ is a countable basis of $A$ , by [F6]. Apply the Lindelof theorem in [F6] to $A$ , using [A1], to select countably many traces covering $A$ . Countable choice also selects extension and chart witnesses for these countably many selected members. By steps 3.1 and 3.2 the witnesses are $C^{1}$ extensions of $F$ with Lipschitz coordinate representatives on their smaller source neighborhoods. This proves the countable-cover and Lipschitz-piece assertions without any regularity hypothesis on $F (A)$ .

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