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

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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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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
A smooth map transverse to an embedded submanifold Definition
Codimension and hypersurfaces Definition
Dead-end components Definition
Euclidean hypersurface normals, shape operators and curvature Definition
Induced connection and second fundamental form Definition
Local defining maps for embedded submanifolds Definition
Local transversals to a regular foliation Definition
Normal and conormal bundles of an embedded submanifold Definition
Reeb components of a codimension-one foliation Definition
Seifert smoothing and Seifert circles of an oriented link diagram Definition
Self-transverse immersions and the double point locus Definition
Taut codimension-one foliations Definition
The flowout of an embedded submanifold by a vector field Definition
Transverse complementary-dimensional intersection sets Definition
Whitney circle for a pair of intersection points Definition
The double point dimension count for surfaces in four- and five-space Example
An embedded submanifold need not be open in the ambient manifold False statement
Every leaf of a regular foliation is an embedded submanifold False statement
The cut locus of a point is always a smooth hypersurface 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
A compact C¹ foliation leaf is an embedded hypersurface Lemma
A double point has two disjoint embedded sheet disks meeting transversely Lemma
A non-closed leaf of a codimension-one foliation meets a closed transversal Lemma
A vector field along an embedded submanifold extends to a neighbourhood and globally when the submanifold is closed Lemma
Arcs joining two points of a connected submanifold avoiding finitely many points Lemma
Closedness of compact leaves diffeomorphic to a finite-fundamental-group leaf Lemma
Compact transverse complementary intersections are finite Lemma
Compactness gives a compactly supported time-dependent velocity field Lemma
Embedded bands joining two framed spheres exist Lemma
Finite holonomy acts on a small transverse disk Lemma
Normal bundle of the zero locus of a transverse section Lemma
Normal push-off zeros are the self-intersection points Lemma
Restriction of a foliation transverse to the boundary Lemma
Slice-chart restrictions form a smooth atlas Lemma
Smooth Euclidean hypersurface graphs and compact localization Lemma
Smooth relative isotopy extension for disk arcs with puncture endpoints Lemma
Smooth relative isotopy extension for finite disk arc systems Lemma
Sphere graph charts, surface density, and a finite partition Lemma

…and 21 more results.

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)