Normal bundle

For normal bundles in algebraic geometry, see normal cone.

In differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding (or immersion).

Definition

Riemannian manifold

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

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 $\mathrm {N} S$ to $S$ is defined as

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

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.

General definition

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

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

Formally, the normal bundle to $N$ in $M$ is a quotient bundle of the tangent bundle on $M$: one has the short exact sequence of vector bundles on $N$:

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

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

Conormal bundle

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

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

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

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

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

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

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

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

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

Stable normal bundle

Abstract manifolds 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 $\mathbf {R} ^{N}$, 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 $X$, any two embeddings in $\mathbf {R} ^{N}$ for sufficiently large $N$ 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 ${N}$ could vary) is called the stable normal bundle.

Dual to tangent bundle

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

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

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

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.

For symplectic manifolds

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

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

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

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

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

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