TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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Embedded submanifolds admit local defining submersions

Statement

Let $S \subseteq M^{m}$ be an embedded $k$ -submanifold and let $p \in S$ . Then $S$ admits a local defining map at $p$ .

Facts & Assumptions

Given: An embedded $k$ -submanifold $S \subseteq M^{m}$ and a point $p \in S$ .

[F1]

In a slice chart near $p$ , $S$ is the coordinate slice $R^{k} \times {0}$ (Embedded submanifolds and slice charts).

[F2]

A local defining map is a smooth submersion whose zero fibre is the local trace of the submanifold (Local defining maps for embedded submanifolds).

[L1]

Chart maps are diffeomorphisms onto open Euclidean sets (Chart maps are diffeomorphisms onto Euclidean open sets).

[L2]

Differentials satisfy the chain rule (The chain rule for differentials of smooth maps).

Proof

technique · direct

1.1

Choose a slice chart $φ : U \to Ω \subseteq R^{k} \times R^{m - k}$ at $p$ as in [F1], and let ${pr}_{2} : R^{k} \times R^{m - k} \to R^{m - k}$ be the second-factor projection. Define $Φ : = {pr}_{2} \circ φ : U \to R^{m - k}$ . Then $S \cap U = Φ^{- 1} (0)$ by the slice description.

F1F2givenconstruct

2.1

In the same coordinates, $Φ = {pr}_{2} \circ φ$ . By [L1], the differential $d φ_{q}$ is an isomorphism for every $q \in U$ . Applying [L2] gives $d Φ_{q} = d ({pr}_{2})_{φ (q)} \circ d φ_{q} .$ The Euclidean differential $d ({pr}_{2})_{φ (q)}$ is the coordinate projection onto the normal factor, hence surjective. Therefore $d Φ_{q}$ is surjective for every $q \in U$ , so $Φ$ is a submersion.

L1L2step 1.1

3.1

Therefore $Φ$ satisfies the definition in [F2] and is a local defining map for $S$ at $p$ .

F2step 1.1step 2.1∎

Depends on

Used by

The preimage theorem for submanifolds under submersions Theorem

Cited to discharge well-definedness by Local defining maps for embedded submanifolds.

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John M. Lee, Introduction to Smooth Manifolds, Embedded Submanifolds (standard reference, not scraped)