PropositionStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-31

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Assuming countable choice, an ambient metric identifies the two normal bundles

Statement

Assume ${A C}_{ω}$ . Let $S \subseteq M$ be an embedded submanifold and let $g$ be a Riemannian metric on $M$ . If $q : T M ∣_{S} \to ν (S) = T M ∣_{S} / T S$ is the quotient map, then

$q ∣_{T S^{⊥}} : T S^{⊥} ⟶ ν (S)$

is a smooth vector-bundle isomorphism. Thus the fixed metric $g$ canonically identifies the quotient normal bundle with its $g$ -orthogonal realization.

Facts & Assumptions

Given: The axiom ${A C}_{ω}$ , an embedded submanifold $S \subseteq M$ , and an ambient Riemannian metric $g$ on $M$ .

[L0]

Under ${A C}_{ω}$ , $T M$ has a smooth manifold structure for which the induced tangent-bundle charts form a smooth atlas (Assuming countable choice, the tangent bundle has a canonical smooth 2n-manifold structure).

[L4]

An induced tangent-bundle chart sends $v = \sum_{i} v^{i} \partial_{x^{i}} ∣_{p}$ to $(x (p), (v^{1}, \dots, v^{n}))$ (The induced tangent bundle chart).

[L5]

A smooth vector bundle is locally trivialized by fibrewise linear charts (Smooth vector bundles, rank, fibres, and trivial bundles).

[L6]

The inclusion $i : S ↪ M$ is smooth (The inclusion of an embedded submanifold is a smooth embedding).

[L7]

Pullback along a smooth map carries a smooth vector bundle to a smooth vector bundle (The pullback fibre product is a smooth vector bundle).

[L8]

In a slice chart, $S$ is a coordinate slice (Embedded submanifolds and slice charts).

[L9]

A smooth subbundle is locally spanned by part of a smooth ambient frame (Vector subbundles).

[F0]

A smooth bundle metric is a fibrewise inner product whose pairing of any two smooth local sections is smooth (Smooth bundle metrics).

[L1]

The orthogonal complement of a smooth subbundle is a smooth subbundle (Orthogonal complements of subbundles are smooth subbundles).

[L2]

The quotient map $T M ∣_{S} \to ν (S)$ is a smooth bundle map (The canonical map to a quotient bundle is a smooth bundle map).

[L3]

A fibrewise bijective smooth bundle map over the identity is a bundle isomorphism (A fibrewise bijective smooth bundle map over a diffeomorphism is a bundle isomorphism).

Proof

technique · direct

1.1F0L0L1L4L5L6L7L8L9given

By [L0], [L4], and [L5], the induced charts make $T M \to M$ a smooth vector bundle. By [L6] and [L7], its pullback along $i$ is the smooth vector bundle $T M ∣_{S} \to S$ . In a slice chart from [L8], its coordinate frame is $\partial_{x_{1}}, \dots, \partial_{x_{k}}, \partial_{y_{1}}, \dots, \partial_{y_{c}}$ and $T S$ is spanned by the first $k$ vectors, so [L9] makes $T S$ a smooth subbundle. In that pulled-back frame, the coefficients of $g ∣_{S}$ are the smooth coefficient functions of $g$ composed with the smooth inclusion $i$ ; hence [F0] makes $g ∣_{S}$ a smooth bundle metric on $T M ∣_{S}$ . Applying [L1], the orthogonal complements $T S_{p}^{⊥}$ form a smooth subbundle and fibrewise $T_{p} M = T_{p} S \oplus T_{p} S^{⊥}$ .

2.1L2L3step 1.1algebra∎

Restrict the quotient map of [L2] to $T S^{⊥}$ . On each fibre this is the usual linear isomorphism from a chosen complement onto the quotient by $T_{p} S$ . Hence the restricted map $T S^{⊥} \to ν (S)$ is fibrewise bijective, so [L3] shows that it is a smooth bundle isomorphism.

Depends on

Used by

Vanishing stable characteristic classes do not make two embeddings isotopic Counterexample
Framings of a normal bundle Definition
Normal bundle of a formal immersion Definition
Stable normal bundle of a compact smooth manifold Definition
Tangential and normal projections along a Riemannian submanifold Definition
The normal addition map for a Euclidean submanifold Definition
Assuming countable choice, the normal bundle of the sphere is trivial Example
The normal line of an oriented hypersurface is trivial Example
The orthogonal normal bundle of a submanifold is defined without a metric False statement
An embedding into Euclidean space gives a rank-(n-m) stable normal inverse Lemma
Formal immersion gives the tangent normal-bundle identity Lemma
The tubular neighbourhood theorem in a smooth ambient manifold Theorem

Dependency tree · two levels

46 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 (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)