# Normal bundle

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Concept in mathematics

For normal bundles in algebraic geometry, see [normal cone](/source/Normal_cone).

In [differential geometry](/source/Differential_geometry), a field of [mathematics](/source/Mathematics), a **normal bundle** is a particular kind of [vector bundle](/source/Vector_bundle), [complementary](/source/Complementary_angles) to the [tangent bundle](/source/Tangent_bundle), and coming from an [embedding](/source/Embedding) (or [immersion](/source/Immersion_(mathematics))).

## Definition

### Riemannian manifold

Let ( M , g ) {\displaystyle (M,g)} be a [Riemannian manifold](/source/Riemannian_manifold), and S ⊂ M {\displaystyle S\subset M} a [Riemannian submanifold](/source/Riemannian_submanifold). Define, for a given p ∈ S {\displaystyle p\in S} , a vector n ∈ T p M {\displaystyle n\in \mathrm {T} _{p}M} to be *[normal](/source/Normal_vector)* to S {\displaystyle S} whenever g ( n , v ) = 0 {\displaystyle g(n,v)=0} for all v ∈ T p S {\displaystyle v\in \mathrm {T} _{p}S} (so that n {\displaystyle n} is [orthogonal](/source/Orthogonal_complement) to T p S {\displaystyle \mathrm {T} _{p}S} ). The set N p S {\displaystyle \mathrm {N} _{p}S} of all such n {\displaystyle n} is then called the *normal space* to S {\displaystyle S} at p {\displaystyle p} .

Just as the total space of the [tangent bundle](/source/Tangent_bundle) to a manifold is constructed from all [tangent spaces](/source/Tangent_space) to the manifold, the total space of the **normal bundle**[1] N S {\displaystyle \mathrm {N} S} to S {\displaystyle S} is defined as

- N S := ∐ p ∈ S N p S {\displaystyle \mathrm {N} S:=\coprod _{p\in S}\mathrm {N} _{p}S} .

The **conormal bundle** is defined as the [dual bundle](/source/Dual_bundle) to the normal bundle. It can be realised naturally as a sub-bundle of the [cotangent bundle](/source/Cotangent_bundle).

### General definition

More abstractly, given an [immersion](/source/Immersion_(mathematics)) i : N → M {\displaystyle i:N\to M} (for instance an embedding), one can define a normal bundle of N {\displaystyle N} in M {\displaystyle M} , by at each point of N {\displaystyle N} , taking the [quotient space](/source/Quotient_space_(linear_algebra)) of the tangent space on M {\displaystyle M} by the tangent space on N {\displaystyle N} . For a Riemannian manifold one can identify this quotient with the orthogonal complement, but in general one cannot (such a choice is equivalent to a [section](/source/Section_(category_theory)) of the projection p : V → V / W {\displaystyle p:V\to V/W} ).

Thus the normal bundle is in general a *quotient* of the tangent bundle of the ambient space M {\displaystyle M} restricted to the subspace N {\displaystyle N} .

Formally, the **normal bundle**[2] to N {\displaystyle N} in M {\displaystyle M} is a quotient bundle of the tangent bundle on M {\displaystyle M} : one has the [short exact sequence](/source/Short_exact_sequence) of vector bundles on N {\displaystyle N} :

- 0 → T N → T M | i ( N ) → T M / N := T M | i ( N ) / T N → 0 {\displaystyle 0\to \mathrm {T} N\to \mathrm {T} M\vert _{i(N)}\to \mathrm {T} _{M/N}:=\mathrm {T} M\vert _{i(N)}/\mathrm {T} N\to 0}

where T M | i ( N ) {\displaystyle \mathrm {T} M\vert _{i(N)}} is the restriction of the tangent bundle on M {\displaystyle M} to N {\displaystyle N} (properly, the pullback i ∗ T M {\displaystyle i^{*}\mathrm {T} M} of the tangent bundle on M {\displaystyle M} to a vector bundle on N {\displaystyle N} via the map i {\displaystyle i} ). The fiber of the normal bundle T M / N ↠ π N {\displaystyle \mathrm {T} _{M/N}{\overset {\pi }{\twoheadrightarrow }}N} in p ∈ N {\displaystyle p\in N} is referred to as the **normal space at p {\displaystyle p}** (of N {\displaystyle N} in M {\displaystyle M} ).

### Conormal bundle

If Y ⊆ X {\displaystyle Y\subseteq X} is a smooth submanifold of a manifold X {\displaystyle X} , we can pick local coordinates ( x 1 , … , x n ) {\displaystyle (x_{1},\dots ,x_{n})} around p ∈ Y {\displaystyle p\in Y} such that Y {\displaystyle Y} is locally defined by x k + 1 = ⋯ = x n = 0 {\displaystyle x_{k+1}=\dots =x_{n}=0} ; then with this choice of coordinates

- T p X = R { ∂ ∂ x 1 | p , … , ∂ ∂ x k | p , … , ∂ ∂ x n | p } T p Y = R { ∂ ∂ x 1 | p , … , ∂ ∂ x k | p } T X / Y p = R { ∂ ∂ x k + 1 | p , … , ∂ ∂ x n | p } {\displaystyle {\begin{aligned}\mathrm {T} _{p}X&=\mathbb {R} {\Big \lbrace }{\frac {\partial }{\partial x_{1}}}{\Big |}_{p},\dots ,{\frac {\partial }{\partial x_{k}}}{\Big |}_{p},\dots ,{\frac {\partial }{\partial x_{n}}}{\Big |}_{p}{\Big \rbrace }\\\mathrm {T} _{p}Y&=\mathbb {R} {\Big \lbrace }{\frac {\partial }{\partial x_{1}}}{\Big |}_{p},\dots ,{\frac {\partial }{\partial x_{k}}}{\Big |}_{p}{\Big \rbrace }\\{\mathrm {T} _{X/Y}}_{p}&=\mathbb {R} {\Big \lbrace }{\frac {\partial }{\partial x_{k+1}}}{\Big |}_{p},\dots ,{\frac {\partial }{\partial x_{n}}}{\Big |}_{p}{\Big \rbrace }\\\end{aligned}}}

and the [ideal sheaf](/source/Ideal_sheaf) is locally generated by x k + 1 , … , x n {\displaystyle x_{k+1},\dots ,x_{n}} . Therefore we can define a non-degenerate pairing

- ( I Y / I Y 2 ) p × T X / Y p ⟶ R {\displaystyle (I_{Y}/I_{Y}^{\ 2})_{p}\times {\mathrm {T} _{X/Y}}_{p}\longrightarrow \mathbb {R} }

that induces an isomorphism of sheaves T X / Y ≃ ( I Y / I Y 2 ) ∨ {\displaystyle \mathrm {T} _{X/Y}\simeq (I_{Y}/I_{Y}^{\ 2})^{\vee }} . We can rephrase this fact by introducing the **conormal bundle** T X / Y ∗ {\displaystyle \mathrm {T} _{X/Y}^{*}} defined via the **conormal exact sequence**

- 0 → T X / Y ∗ ↣ Ω X 1 | Y ↠ Ω Y 1 → 0 {\displaystyle 0\to \mathrm {T} _{X/Y}^{*}\rightarrowtail \Omega _{X}^{1}|_{Y}\twoheadrightarrow \Omega _{Y}^{1}\to 0} ,

then T X / Y ∗ ≃ ( I Y / I Y 2 ) {\displaystyle \mathrm {T} _{X/Y}^{*}\simeq (I_{Y}/I_{Y}^{\ 2})} , viz. the sections of the conormal bundle are the cotangent vectors to X {\displaystyle X} vanishing on T Y {\displaystyle \mathrm {T} Y} .

When Y = { p } {\displaystyle Y=\lbrace p\rbrace } is a point, then the ideal sheaf is the sheaf of smooth germs vanishing at p {\displaystyle p} and the isomorphism reduces to the [definition of the tangent space](/source/Tangent_space#Definition_via_cotangent_spaces) in terms of germs of smooth functions on X {\displaystyle X}

- T X / { p } ∗ ≃ ( T p X ) ∨ ≃ m p m p 2 {\displaystyle \mathrm {T} _{X/\lbrace p\rbrace }^{*}\simeq (\mathrm {T} _{p}X)^{\vee }\simeq {\frac {{\mathfrak {m}}_{p}}{{\mathfrak {m}}_{p}^{\ 2}}}} .

## Stable normal bundle

[Abstract manifolds](/source/Abstract_manifold) have a [canonical](/source/Canonical_form) tangent bundle, but do not have a normal bundle: only an embedding (or immersion) of a manifold in another yields a normal bundle. However, since every manifold can be embedded in R N {\displaystyle \mathbf {R} ^{N}} , by the [Whitney embedding theorem](/source/Whitney_embedding_theorem), every manifold admits a normal bundle, given such an embedding.

There is in general no natural choice of embedding, but for a given manifold X {\displaystyle X} , any two embeddings in R N {\displaystyle \mathbf {R} ^{N}} for sufficiently large N {\displaystyle N} are [regular homotopic](/source/Regular_homotopy), and hence induce the same normal bundle. The resulting class of normal bundles (it is a class of bundles and not a specific bundle because the integer N {\displaystyle {N}} could vary) is called the [stable normal bundle](/source/Stable_normal_bundle).

## Dual to tangent bundle

The normal bundle is dual to the tangent bundle in the sense of [K-theory](/source/K-theory): by the above short exact sequence,

- [ T N ] + [ T M / N ] = [ T M ] {\displaystyle [\mathrm {T} N]+[\mathrm {T} _{M/N}]=[\mathrm {T} M]}

in the [Grothendieck group](/source/Grothendieck_group). In case of an immersion in R N {\displaystyle \mathbf {R} ^{N}} , the tangent bundle of the ambient space is trivial (since R N {\displaystyle \mathbf {R} ^{N}} is contractible, hence [parallelizable](/source/Parallelizable)), so [ T N ] + [ T M / N ] = 0 {\displaystyle [\mathrm {T} N]+[\mathrm {T} _{M/N}]=0} , and thus [ T M / N ] = − [ T N ] {\displaystyle [\mathrm {T} _{M/N}]=-[\mathrm {T} N]} .

This is useful in the computation of [characteristic classes](/source/Characteristic_classes), and allows one to prove lower bounds on immersibility and embeddability of manifolds in [Euclidean space](/source/Euclidean_space).

## For symplectic manifolds

Suppose a manifold X {\displaystyle X} is embedded in to a [symplectic manifold](/source/Symplectic_manifold) ( M , ω ) {\displaystyle (M,\omega )} , such that the pullback of the symplectic form has constant rank on X {\displaystyle X} . Then one can define the symplectic normal bundle to X {\displaystyle X} as the vector bundle over X {\displaystyle X} with fibres

- ( T i ( x ) X ) ω / ( T i ( x ) X ∩ ( T i ( x ) X ) ω ) , x ∈ X , {\displaystyle (\mathrm {T} _{i(x)}X)^{\omega }/(\mathrm {T} _{i(x)}X\cap (\mathrm {T} _{i(x)}X)^{\omega }),\quad x\in X,}

where i : X → M {\displaystyle i:X\rightarrow M} denotes the embedding and ( T X ) ω {\displaystyle (\mathrm {T} X)^{\omega }} is the [symplectic orthogonal](/source/Symplectic_vector_space#subspace) of T X {\displaystyle \mathrm {T} X} in T M {\displaystyle \mathrm {T} M} . Notice that the constant rank condition ensures that these normal spaces fit together to form a bundle. Furthermore, any fibre inherits the structure of a symplectic vector space.[3]

By [Darboux's theorem](/source/Darboux's_theorem), the constant rank embedding is locally determined by i ∗ ( T M ) {\displaystyle i^{*}(\mathrm {T} M)} . The isomorphism

- i ∗ ( T M ) ≅ T X / ν ⊕ ( T X ) ω / ν ⊕ ( ν ⊕ ν ∗ ) {\displaystyle i^{*}(\mathrm {T} M)\cong \mathrm {T} X/\nu \oplus (\mathrm {T} X)^{\omega }/\nu \oplus (\nu \oplus \nu ^{*})}

(where ν = T X ∩ ( T X ) ω {\displaystyle \nu =\mathrm {T} X\cap (\mathrm {T} X)^{\omega }} and ν ∗ {\displaystyle \nu ^{*}} is the dual under ω {\displaystyle \omega } ,) of symplectic vector bundles over X {\displaystyle X} implies that the symplectic normal bundle already determines the constant rank embedding locally. This feature is similar to the Riemannian case.

## References

1. **[^](#cite_ref-1)** John M. Lee, *Riemannian Manifolds, An Introduction to Curvature*, (1997) Springer-Verlag New York, Graduate Texts in Mathematics 176 [ISBN](/source/ISBN_(identifier)) [978-0-387-98271-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-98271-7)

1. **[^](#cite_ref-2)** [Tammo tom Dieck](/source/Tammo_tom_Dieck), *Algebraic Topology*, (2010) EMS Textbooks in Mathematics [ISBN](/source/ISBN_(identifier)) [978-3-03719-048-7](https://en.wikipedia.org/wiki/Special:BookSources/978-3-03719-048-7)

1. **[^](#cite_ref-3)** [Ralph Abraham](/source/Ralph_Abraham_(mathematician)) and [Jerrold E. Marsden](/source/Jerrold_E._Marsden), *Foundations of Mechanics*, (1978) Benjamin-Cummings, London [ISBN](/source/ISBN_(identifier)) [0-8053-0102-X](https://en.wikipedia.org/wiki/Special:BookSources/0-8053-0102-X)

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Adapted from the Wikipedia article [Normal bundle](https://en.wikipedia.org/wiki/Normal_bundle) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Normal_bundle?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.