Normal bundle

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

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 [math]\displaystyle{ \mathrm{N} S }[/math] to [math]\displaystyle{ S }[/math] is defined as

[math]\displaystyle{ \mathrm{N}S := \coprod_{p \in S} \mathrm{N}_p S }[/math].

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 [math]\displaystyle{ i: N \to M }[/math] (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 [math]\displaystyle{ V \to V/W }[/math]).

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

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:

[math]\displaystyle{ 0 \to TN \to TM\vert_{i(N)} \to T_{M/N} := TM\vert_{i(N)} / TN \to 0 }[/math]

where [math]\displaystyle{ TM\vert_{i(N)} }[/math] is the restriction of the tangent bundle on M to N (properly, the pullback [math]\displaystyle{ i^*TM }[/math] of the tangent bundle on M to a vector bundle on N via the map [math]\displaystyle{ i }[/math]). The fiber of the normal bundle [math]\displaystyle{ T_{M/N}\overset{\pi}{\twoheadrightarrow} N }[/math] in [math]\displaystyle{ p\in N }[/math] is referred to as the normal space at [math]\displaystyle{ p }[/math] (of [math]\displaystyle{ N }[/math] in [math]\displaystyle{ M }[/math]).

Conormal bundle

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

[math]\displaystyle{ \begin{align} T_pX&=\mathbb{R}\Big\lbrace\frac{\partial}{\partial x_1}|_p,\dots, \frac{\partial}{\partial x_n}|_p\Big\rbrace\\ T_pY&=\mathbb{R}\Big\lbrace\frac{\partial}{\partial x_1}|_p,\dots, \frac{\partial}{\partial x_k}|_p\Big\rbrace\\ {T_{X/Y}}_p&=\mathbb{R}\Big\lbrace\frac{\partial}{\partial x_{k+1}}|_p,\dots, \frac{\partial}{\partial x_n}|_p\Big\rbrace\\ \end{align} }[/math]

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

[math]\displaystyle{ (I_Y/I^2_Y)_p\times {T_{X/Y}}_p\longrightarrow \mathbb{R} }[/math]

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

[math]\displaystyle{ 0\to T^*_{X/Y}\rightarrowtail \Omega^1_X|_Y\twoheadrightarrow \Omega^1_Y\to 0 }[/math],

then [math]\displaystyle{ T^*_{X/Y}\simeq (I_Y/I_Y^2) }[/math], viz. the sections of the conormal bundle are the cotangent vectors to [math]\displaystyle{ X }[/math] vanishing on [math]\displaystyle{ TY }[/math].

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

[math]\displaystyle{ T^*_{X/\lbrace p\rbrace}\simeq (T_pX)^\vee\simeq\frac{\mathfrak{m}_p}{\mathfrak{m}_p^2} }[/math].

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 [math]\displaystyle{ \mathbf{R}^N }[/math], 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 M, any two embeddings in [math]\displaystyle{ \mathbf{R}^N }[/math] 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 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,

[math]\displaystyle{ [TN] + [T_{M/N}] = [TM] }[/math]

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

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

[math]\displaystyle{ (T_{i(x)}X)^\omega/(T_{i(x)}X\cap (T_{i(x)}X)^\omega), \quad x\in X, }[/math]

where [math]\displaystyle{ i:X\rightarrow M }[/math] denotes the embedding. 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 [math]\displaystyle{ i^*(TM) }[/math]. The isomorphism

[math]\displaystyle{ i^*(TM)\cong TX/\nu \oplus (TX)^\omega/\nu \oplus(\nu\oplus \nu^*), \quad \nu=TX\cap (TX)^\omega, }[/math]

of symplectic vector bundles over [math]\displaystyle{ X }[/math] implies that the symplectic normal bundle already determines the constant rank embedding locally. This feature is similar to the Riemannian case.