DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-31

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Embedded submanifolds and slice charts

Definition

Let $M^{m}$ be a smooth manifold, let $k \in N$ satisfy $0 \leq k \leq m$ , and let $S \subseteq M$ . One says that $S$ is an embedded $k$ -dimensional submanifold of $M$ when for every $p \in S$ there is a smooth chart $φ : U \to φ (U) \subseteq R^{m}$ with $p \in U$ such that

$φ (S \cap U) = φ (U) \cap (R^{k} \times {0}) .$

Such a chart is a slice chart for $S$ at $p$ . The topology on $S$ is the subspace topology inherited from $M$ (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).

The next lemma proves that the restricted slice charts define a smooth $k$ -manifold structure on $S$ .

Depends on

Used by

A discrete embedded submanifold is locally closed and countable Corollary
Codimension and hypersurfaces Definition
Local defining maps for embedded submanifolds Definition
An embedded submanifold need not be open in the ambient manifold False statement
The image of every immersion need not be an embedded submanifold False statement
The intrinsic topology of an immersed submanifold need not be the subspace topology False statement
Slice-chart restrictions form a smooth atlas Lemma
Smoothness into an embedded submanifold is an initial property Proposition
Smoothness of a map on an embedded submanifold is local in the ambient space Proposition
The diagonal is an embedded submanifold Proposition
The image of a smooth embedding is an embedded submanifold Proposition
The smooth structure of an embedded submanifold is unique Proposition
A regular level set is an embedded submanifold Theorem
Embedded submanifolds admit local defining submersions Theorem

Dependency tree · two levels

8 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)
Will J. Merry, Differential Geometry, Definition 6.6 (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds, Definition 12 (standard reference, not scraped)