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 $N$:

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

where $\mathrm{T}M{|}_{i(N)}$ 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 ${\mathrm{T}}_{M/N}\stackrel{\pi }{\twoheadrightarrow }N$ 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

$\begin{array}{rl}{\mathrm{T}}_{p}X& =\mathbb{ℝ}\{\frac{\partial }{\partial x_1}{|}_p,\dots ,\frac{\partial }{\partial x_k}{|}_p,\dots ,\frac{\partial }{\partial x_n}{|}_p\}\\ {\mathrm{T}}_{p}Y& =\mathbb{ℝ}\{\frac{\partial }{\partial x_1}{|}_p,\dots ,\frac{\partial }{\partial x_k}{|}_p\}\\ {{\mathrm{T}}_{X/Y}}_p& =\mathbb{ℝ}\{\frac{\partial }{\partial x_{k+1}}{|}_p,\dots ,\frac{\partial }{\partial x_n}{|}_p\}\\ \end{array}$

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

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

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

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

then $$\begin{gathered} {\mathrm{T}}_{X/Y}^{*}\simeq (I_Y/I_Y^{ \\ 2}) \end{gathered}$$, viz. the sections of the conormal bundle are the cotangent vectors to $X$ vanishing on $\mathrm{T}Y$. When $Y=\{p\}$ 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 $X$

$$\begin{gathered} {\mathrm{T}}_{X/\{p\}}^{*}\simeq ({\mathrm{T}}_{p}X{)}^{\vee }\simeq \frac{{\mathfrak{𝔪}}_p}{{\mathfrak{𝔪}}_p^{ \\ 2}} \end{gathered}$$.

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

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

where $i:X\to M$ 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.