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An embedding into Euclidean space gives a rank-(n-m) stable normal inverse

Statement

Assume countable choice ${A C}_{ω}$ . Let $i : M^{m} ↪ R^{n}$ be a smooth embedding of a closed smooth $m$ -manifold with $n > m$ , and let $ν_{i} = i^{*} T R^{n} / d i (T M)$ be its normal quotient, identified with the orthogonal complement of $d i (T M)$ by a Euclidean metric (Normal and conormal bundles of an embedded submanifold, Assuming countable choice, an ambient metric identifies the two normal bundles). Then $ν_{i}$ is a smooth real bundle of rank $n - m$ , and the orthogonal splitting together with the canonical trivialization $i^{*} T R^{n} ≅ ε^{n}$ gives a smooth bundle isomorphism $φ : T M \oplus ν_{i} \to ε^{n}$ . Hence $(ν_{i}, φ)$ is a rank- $(n - m)$ stable normal inverse of $M$ in the sense of Stable normal inverse of the tangent bundle. Consequently every closed smooth $m$ -manifold admits a stable normal inverse: apply Every smooth manifold embeds in some finite-dimensional Euclidean space to obtain an embedding into some $R^{N}$ . The countable-choice hypothesis is exactly the one inherited from the metric and tubular identifications of the published embedding normal-bundle definition; no further choice is made.

Facts & Assumptions

Given: A smooth embedding $i : M^{m} ↪ R^{n}$ of a closed smooth $m$ -manifold with $n > m$ , and countable choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

The normal-bundle set of the embedded submanifold $M \subseteq R^{n}$ is the fibrewise quotient $ν = ∐_{p \in M} T_{p} R^{n} / T_{p} M$ , with the smooth vector-bundle structure supplied for such quotients; the defining quotient of the pullback, $ν_{i} = i^{*} T R^{n} / d i (T M)$ , is the same bundle under the canonical identification of $i^{*} T R^{n}$ with $T R^{n} ∣_{M}$ (Normal and conormal bundles of an embedded submanifold).

[F2]

Assume ${A C}_{ω}$ ; for an embedded submanifold and a Riemannian metric $g$ on the ambient manifold, the quotient map restricts to a smooth bundle isomorphism $T S^{⊥} \to T M ∣_{S} / T S$ ; for $S = M \subseteq R^{n}$ with the Euclidean metric this identifies $ν_{i}$ with the orthogonal complement $d i (T M)^{⊥}$ (Assuming countable choice, an ambient metric identifies the two normal bundles).

[F3]

For a compact (in particular closed) smooth $M$ and a smooth embedding $i : M ↪ R^{N}$ with $N \geq m$ , the published normal-bundle definition gives a smooth real bundle $ν_{i}$ of rank $N - m$ with $T M \oplus ν_{i} ≅ ε^{N}$ ; the only choice used is the inherited ${A C}_{ω}$ of the metric and tubular identifications (Stable normal bundle of a compact smooth manifold, the rank of the quotient).

[F4]

Under ${A C}_{ω}$ the identity chart of $R^{n}$ is a global smooth chart, so its induced tangent-bundle chart trivializes the Euclidean tangent bundle, $T R^{n} ≅ ε_{R^{n}}^{n}$ ; pulling this trivialization back along the smooth map $i$ and applying the choice-free product-pullback lemma gives the canonical trivialization $i^{*} T R^{n} ≅ i^{*} ε_{R^{n}}^{n} ≅ ε^{n}$ (Assuming countable choice, the tangent bundle has a canonical smooth 2n-manifold structure, The induced tangent bundle chart, The pullback of a trivial smooth vector bundle is canonically trivial).

[F5]

Under ${A C}_{ω}$ every smooth $m$ -manifold embeds smoothly into some finite-dimensional Euclidean space (Every smooth manifold embeds in some finite-dimensional Euclidean space).

[F6]

A stable normal inverse of $M$ is a pair $(ν, φ)$ with $ν \to M$ a smooth real bundle of finite rank $k$ and $φ : T M \oplus ν \to ε^{m + k}$ a smooth bundle isomorphism; a rank- $k$ stable normal inverse is one with $rank ν = k$ (Stable normal inverse of the tangent bundle, Smooth vector bundles, rank, fibres, and trivial bundles).

Proof

1.1F1F3F6

Regard $i$ as an embedding of $M$ as an embedded submanifold $i (M) \subseteq R^{n}$ and let $ν_{i}$ be its normal quotient as in [F1]. By [F3] the quotient carries a smooth real vector-bundle structure of rank $n - m$ ; the rank is the difference of the ranks of the ambient tangent bundle of $R^{n}$ and of $d i (T M)$ , computed fibrewise, and equals $n - m$ because $d i$ is fibrewise injective.

1.2F2F3F4

By [F2] the Euclidean metric identifies the quotient $ν_{i}$ with the orthogonal complement $d i (T M)^{⊥}$ , which is a smooth subbundle of $i^{*} T R^{n}$ ; the orthogonal decomposition of the Euclidean bundle gives $i^{*} T R^{n} = d i (T M) \oplus d i (T M)^{⊥} ≅ T M \oplus ν_{i}$ , where the first summand is identified with $T M$ through the isomorphism $d i : T M \to d i (T M)$ . Composing this isomorphism with the canonical trivialization $i^{*} T R^{n} ≅ ε^{n}$ of [F4], which exists because the identity chart of $R^{n}$ trivializes $T R^{n}$ and the product-pullback lemma trivializes its pullback, gives a smooth bundle isomorphism $φ : T M \oplus ν_{i} ⟶ ε^{n} .$

2.1F2F4F6step 1.1step 1.2

Since $ν_{i}$ has rank $n - m$ by step 1.1 and $φ$ is a smooth bundle isomorphism onto $ε^{n} = ε^{m + (n - m)}$ , the pair $(ν_{i}, φ)$ is a rank- $(n - m)$ stable normal inverse of $M$ in the sense of [F6].

3.1F3F4F5F6step 1.1step 1.2step 2.1∎

For existence, let $M$ be any closed smooth $m$ -manifold. By [F5] there is a smooth embedding $j : M ↪ R^{N}$ into some finite-dimensional Euclidean space; the construction above applies to $j$ provided $N > m$ . If $N \leq m$ for the particular embedding produced, compose with the inclusion $R^{N} ↪ R^{N + 1} ↪ \dots ↪ R^{m + 1}$ (each an embedding of a linear subspace as a closed subset, hence a smooth embedding with $d i$ injective) to obtain an embedding into some $R^{n}$ with $n > m$ ; replacing the ambient metric by the standard Euclidean one leaves the argument unchanged. Applying steps 1.1–2.1 to that embedding produces a stable normal inverse of $M$ . The only choice principle used is the ${A C}_{ω}$ inherited from [F2] and [F5]; the trivialization [F4] is canonical, and no embedding, metric or complement is selected beyond the given ones.

Depends on

Used by

Embedding obstructions include all immersion normal-class obstructions Corollary
Top normal classes vanish for Euclidean embeddings Corollary
Vanishing stable characteristic classes do not make two embeddings isotopic Counterexample
Normal Stiefel-Whitney and Pontryagin classes of a closed manifold Definition
The normal line of an oriented hypersurface is trivial Example

Cited to discharge well-definedness by Stable normal inverse of the tangent bundle.

Dependency tree · two levels

63 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 W. Milnor and James D. Stasheff, Characteristic Classes (Annals of Mathematics Studies 74, Princeton University Press; complete text) (standard reference, not scraped)
Ralph L. Cohen, Immersions of Manifolds and Homotopy Theory (lecture notes, 30 June 2022; complete 46-page text) (standard reference, not scraped)
Ralph L. Cohen, Bundles, Manifolds, and Homotopy (author draft, complete 568-page text) (standard reference, not scraped)