LemmaStatement: AI-generatedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Equal fixed-surface flow heights imply boundary tangency

Statement

Let $M$ be a smooth four-manifold without boundary, $U \subseteq M$ open, and $K$ a smooth vector field on $U$ . Let $C \subseteq U$ be a smooth embedded three-dimensional collar with boundary the smooth surface $S$ , and assume $K ∣_{S} = 0$ . Write $ϕ : D \to U$ for the maximal flow of $K$ on $U$ , with open domain $D \subseteq R \times U$ . Let $f, g : C \to R$ be smooth up to $S$ , with $(h (x), x) \in D$ for every $x \in C$ and $h \in {f, g}$ . Define the variable-time maps directly by $F_{h} (x) = ϕ (h (x), x)$ , and suppose both are smooth embeddings up to $S$ .

For $p \in S$ , let $H_{p}^{+} C \subset T_{p} C$ be the closed inward tangent half-space: in a boundary chart with collar interior $x^{3} > 0$ , it is the set of vectors with third coordinate nonnegative, including $T_{p} S$ . Here the exterior tangent side of $F_{h} (C)$ means $d (F_{h})_{p} (H_{p}^{+} C)$ , the tangent half-space pointing into the collar image, rather than a choice of ambient normal side.

Both maps fix $S$ pointwise. If $f (p) = g (p) = c$ at $p \in S$ , then, using the inclusion $T_{p} C \subset T_{p} M$ , their differentials satisfy $d (F_{f})_{p} = d (F_{g})_{p} = d (ϕ_{c})_{p} ∣_{T_{p} C}$ . Consequently their images have the same tangent hyperplane at $p$ and the same exterior tangent side. The common differential restricts to the identity on $T_{p} S$ . No equality of $d f_{p}$ and $d g_{p}$ is required.

Facts & Assumptions

[F1]

A smooth vector field on the boundaryless manifold $U$ has a unique maximal smooth flow on an open domain; its time curves are its unique maximal integral curves. The fundamental theorem on flows

[F2]

An embedded smooth submanifold with boundary carries a manifold-with-boundary smooth structure whose inclusion is a smooth embedding. Embedded smooth submanifolds with boundary

[F3]

A smooth embedding has injective differential on the full tangent space, including at boundary points, and is a homeomorphism onto its image. Immersions and embeddings for manifolds with boundary

[F4]

In a boundary chart the full tangent space has inward, outward and boundary-tangent vectors distinguished by the sign of the last coordinate; this distinction is chart independent. Inward, outward, and boundary-tangent vectors

[F5]

Smooth half-space maps obey the chain rule for their full derivatives at boundary points, as computed from smooth local extensions. Chain rule for smooth half-space maps

[F6]

For the maximal flow on $U$ , $U_{t} = {x \in U : (t, x) \in D}$ is open and $ϕ_{t} : U_{t} \to U_{- t}$ is a diffeomorphism with inverse $ϕ_{- t}$ . Time-t flow maps are diffeomorphisms between open domains

[F7]

The collar assumption uses a smooth boundary collar: a smooth embedding $c : S \times [0, ε) \to C$ with $c (p, 0) = p$ and image an open neighborhood of $S = \partial C$ in $C$ ; the width may first be a positive smooth function of $p$ . math:def-smooth-collar-of-a-manifold-boundary

[F8]

For the boundary inclusion $i : S = \partial C ↪ C$ , $d i_{p}$ identifies $T_{p} S$ with the boundary-tangent hyperplane of $T_{p} C$ . The boundary tangent space is the boundary-tangent hyperplane

[F9]

The differential of a diffeomorphism $F : N \to P$ is a linear isomorphism $d F_{p} : T_{p} N \to T_{F (p)} P$ at every point. The differential of a diffeomorphism is an isomorphism

Proof

Given: The assumptions in the Statement.

Goal: Prove boundary fixation, equality of the full tangent maps at equal heights, and equality of their inward image half-spaces.

1.1F1F2F5F8given

For $p \in S$ , the constant curve at $p$ is an integral curve of $K$ for every real time, since $K (p) = 0$ . Uniqueness in [F1] identifies it with the maximal integral curve through $p$ . Thus $ϕ (t, p) = p$ for all $t \in R$ , and $F_{f} ∣_{S} = F_{g} ∣_{S} = {id}_{S}$ . In particular, the fixed-time flow restricts to the identity on $S$ , so its differential on $T_{p} S$ , identified with the boundary-tangent hyperplane by [F8], is the identity.

2.1F1F2F3F5givenstep 1.1algebra

Fix $p \in S$ and $h \in {f, g}$ . The map $x \mapsto (h (x), x)$ takes values in the open set $D$ ; near $(h (p), p)$ this openness permits the usual smooth coordinate differentiation, including a smooth extension across the boundary of $C$ . Apply the half-space chain rule to its composition with $ϕ$ . For every $v \in T_{p} C$ , using the differential of the inclusion into $U$ , this gives $d (F_{h})_{p} (v) = d (ϕ_{h (p)})_{p} (v) + K_{ϕ (h (p), p)} d h_{p} (v)$ . Here $\partial_{t} ϕ (t, x) = K_{ϕ (t, x)}$ follows from the integral-curve property of [F1]. Step 1.1 and $K (p) = 0$ therefore give $d (F_{h})_{p} (v) = d (ϕ_{h (p)})_{p} (v)$ on the full tangent space, without any condition on $d h_{p}$ .

3.1F2F3F6F9givenstep 2.1

If $f (p) = g (p) = c$ , step 2.1 gives equality of the two differentials with the restriction of $d (ϕ_{c})_{p}$ . Since $(c, p) \in D$ , [F6] applies on the ambient open set $U_{c}$ ; by [F9], this ambient differential is an invertible map $T_{p} U \to T_{p} U$ . Its restriction to $T_{p} C$ is injective. The embedding hypotheses and [F2]–[F3] identify $T_{p} F_{h} (C)$ with $d (F_{h})_{p} (T_{p} C)$ , so the two image tangent hyperplanes agree.

4.1F3F4F8step 1.1step 2.1step 3.1∎

By [F4] and [F8], the closed half-space $H_{p}^{+} C$ in the Statement is intrinsic to the collar interior. The equal linear maps in step 3.1 carry this same half-space to equal half-spaces in the common tangent hyperplane, precisely the asserted common exterior tangent side. Their restriction to $T_{p} S$ is the identity by step 1.1. Together with the pointwise boundary fixation in step 1.1, this proves all the conclusions without requiring equality of the height derivatives.

Depends on

Used by · 3

Dependency tree · 2 levels

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