Normal bundle

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

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

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

The conormal bundle is defined as the dual bundle to the normal bundle. It can be realised naturally as a sub-bundle of the cotangent bundle.

More abstractly, given an immersion ${\displaystyle i:N\to M}$ (for instance an embedding), one can define a normal bundle of ${\displaystyle N}$ in ${\displaystyle M}$, by at each point of ${\displaystyle N}$, taking the quotient 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 a section of the projection ${\displaystyle p:V\to V/W}$).

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

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

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

where ${\displaystyle \mathrm {T} M\vert _{i(N)}}$ is the restriction of the tangent bundle on ${\displaystyle M}$ to ${\displaystyle N}$ (properly, the pullback ${\displaystyle i^{*}\mathrm {T} M}$ 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 ${\displaystyle \mathrm {T} _{M/N}{\overset {\pi }{\twoheadrightarrow }}N}$ in ${\displaystyle p\in N}$ is referred to as the normal 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 ${\displaystyle x_{k+1}=\dots =x_{n}=0}$; then with this choice of coordinates

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

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

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

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 the conormal bundle ${\displaystyle \mathrm {T} _{X/Y}^{*}}$ defined via the conormal exact sequence

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

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

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

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

Abstract manifolds have a canonical 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 the 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 ${\displaystyle X}$, any two embeddings in ${\displaystyle \mathbf {R} ^{N}}$ for sufficiently large ${\displaystyle N}$ are regular 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 the stable normal bundle.

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

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

in the Grothendieck 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, hence parallelizable), 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 of characteristic classes, and allows one to prove lower bounds on immersibility and embeddability of manifolds in Euclidean space.

Suppose a manifold ${\displaystyle X}$ is embedded in to a symplectic 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

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

where ${\displaystyle i:X\rightarrow M}$ denotes the embedding and ${\displaystyle (\mathrm {T} X)^{\omega }}$ is the symplectic 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]

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

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

(where ${\displaystyle \nu =\mathrm {T} X\cap (\mathrm {T} X)^{\omega }}$ and ${\displaystyle \nu ^{*}}$ 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.