Indifferential geometry, a field ofmathematics, anormal bundle is a particular kind ofvector bundle,complementary to thetangent bundle, and coming from anembedding (orimmersion).

Let${\displaystyle (M,g)}$ be aRiemannian manifold, and${\displaystyle S\subset M}$ aRiemannian submanifold. Define, for a given${\displaystyle p\in S}$, a vector${\displaystyle n\in {\mathrm{T}}_{p}M}$ to benormal to${\displaystyle S}$ whenever${\displaystyle g(n,v)=0}$ for all${\displaystyle v\in {\mathrm{T}}_{p}S}$ (so that${\displaystyle n}$ isorthogonal to${\displaystyle {\mathrm{T}}_{p}S}$). The set${\displaystyle {\mathrm{N}}_{p}S}$ of all such${\displaystyle n}$ is then called thenormal space to${\displaystyle S}$ at${\displaystyle p}$.

Just as the total space of thetangent bundle to a manifold is constructed from alltangent spaces to the manifold, the total space of thenormal bundle^[1]${\displaystyle \mathrm{N}S}$ to${\displaystyle S}$ is defined as

${\displaystyle \mathrm{N}S:=\coprod _{p\in S}{\mathrm{N}}_{p}S}$.

Theconormal bundle is defined as thedual bundle to the normal bundle. It can be realised naturally as a sub-bundle of thecotangent bundle.

More abstractly, given animmersion${\displaystyle i:N\to M}$ (for instance an embedding), one can define a normal bundle of${\displaystyle N}$ in${\displaystyle M}$, by at each point of${\displaystyle N}$, taking thequotient space of the tangent space on${\displaystyle M}$ by the tangent space on${\displaystyle N}$. For a Riemannian manifold one can identify this quotient with the orthogonal complement, but in general one cannot (such a choice is equivalent to asection of the projection${\displaystyle p:V\to V/W}$).

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

Formally, thenormal bundle^[2] to${\displaystyle N}$ in${\displaystyle M}$ is a quotient bundle of the tangent bundle on${\displaystyle M}$: one has theshort exact sequence of vector bundles on${\displaystyle N}$:

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

where${\displaystyle \mathrm{T}M{|}_{i(N)}}$ is the restriction of the tangent bundle on${\displaystyle M}$ to${\displaystyle N}$ (properly, the pullback${\displaystyle i^{\ast }\mathrm{T}M}$ of the tangent bundle on${\displaystyle M}$ to a vector bundle on${\displaystyle N}$ via the map${\displaystyle i}$). The fiber of the normal bundle${\displaystyle {\mathrm{T}}_{M/N}\stackrel{\pi }{\twoheadrightarrow }N}$ in${\displaystyle p\in N}$ is referred to as thenormal space at${\displaystyle p}$ (of${\displaystyle N}$ in${\displaystyle M}$).

If${\displaystyle Y\subseteq X}$ is a smooth submanifold of a manifold${\displaystyle X}$, we can pick local coordinates${\displaystyle (x_1,\dots ,x_n)}$ around${\displaystyle p\in Y}$ such that${\displaystyle Y}$ is locally defined by${\displaystyle x_{k+1}=\cdots =x_n=0}$; then with this choice of coordinates

and theideal sheaf is locally generated by${\displaystyle x_{k+1},\dots ,x_n}$. Therefore we can define a non-degenerate pairing

${\displaystyle (I_Y/I_Y^2{)}_p\times {{\mathrm{T}}_{X/Y}}_p⟶\mathbb{R}}$

that induces an isomorphism of sheaves${\displaystyle {\mathrm{T}}_{X/Y}\simeq (I_Y/I_Y^2{)}^{\vee }}$. We can rephrase this fact by introducing theconormal bundle${\displaystyle {\mathrm{T}}_{X/Y}^{\ast }}$ defined via theconormal exact sequence

${\displaystyle 0\to {\mathrm{T}}_{X/Y}^{\ast }\rightarrowtail {\mathrm{\Omega }}_X^1{|}_Y\twoheadrightarrow {\mathrm{\Omega }}_Y^1\to 0}$,

then${\displaystyle {\mathrm{T}}_{X/Y}^{\ast }\simeq (I_Y/I_Y^2)}$, viz. the sections of the conormal bundle are the cotangent vectors to${\displaystyle X}$ vanishing on${\displaystyle \mathrm{T}Y}$.

When${\displaystyle Y=\{p\}}$ is a point, then the ideal sheaf is the sheaf of smooth germs vanishing at${\displaystyle p}$ and the isomorphism reduces to thedefinition of the tangent space in terms of germs of smooth functions on${\displaystyle X}$

${\displaystyle {\mathrm{T}}_{X/\{p\}}^{\ast }\simeq ({\mathrm{T}}_{p}X{)}^{\vee }\simeq \frac{{\mathfrak{m}}_p}{{\mathfrak{m}}_p^2}}$.

Abstract manifolds have acanonical 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${\displaystyle {\mathbf{R}}^N}$, by theWhitney 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${\displaystyle X}$, any two embeddings in${\displaystyle {\mathbf{R}}^N}$ for sufficiently large${\displaystyle N}$ areregular homotopic, 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${\displaystyle N}$ could vary) is called thestable normal bundle.

The normal bundle is dual to the tangent bundle in the sense ofK-theory: by the above short exact sequence,

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

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

This is useful in the computation ofcharacteristic classes, and allows one to prove lower bounds on immersibility and embeddability of manifolds inEuclidean space.

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

${\displaystyle ({\mathrm{T}}_{i(x)}X{)}^{\omega }/({\mathrm{T}}_{i(x)}X\cap ({\mathrm{T}}_{i(x)}X{)}^{\omega }),x\in X,}$

where${\displaystyle i:X\to M}$ denotes the embedding and${\displaystyle (\mathrm{T}X{)}^{\omega }}$ is thesymplectic orthogonal of${\displaystyle \mathrm{T}X}$ in${\displaystyle \mathrm{T}M}$. 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]

ByDarboux's theorem, the constant rank embedding is locally determined by${\displaystyle i^{\ast }(\mathrm{T}M)}$. The isomorphism

${\displaystyle i^{\ast }(\mathrm{T}M)\cong \mathrm{T}X/\nu \oplus (\mathrm{T}X{)}^{\omega }/\nu \oplus (\nu \oplus {\nu }^{\ast })}$

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

• Algebraic geometry
• Differential geometry
• Differential topology
• Vector bundles

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