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).

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 $$V \to V/W$$).

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:
 * $$0 \to TN \to TM\vert_{i(N)} \to T_{M/N} := TM\vert_{i(N)} / TN \to 0$$

where $$TM\vert_{i(N)}$$ is the restriction of the tangent bundle on M to N (properly, the pullback $$i^*TM$$ of the tangent bundle on M to a vector bundle on N via the map $$i$$). The fiber of the normal bundle $$ T_{M/N}\overset{\pi}{\twoheadrightarrow} N$$ 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{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}$$ and the ideal sheaf is locally generated by $$x_{k+1},\dots,x_n$$. Therefore we can define a non-degenerate pairing


 * $$(I_Y/I^2_Y)_p\times {T_{X/Y}}_p\longrightarrow \mathbb{R}$$

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


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

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

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 $$ X$$


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

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 M, 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 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,
 * $$[TN] + [T_{M/N}] = [TM]$$

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 $$[TN] + [T_{M/N}] = 0$$, and thus $$[T_{M/N}] = -[TN]$$.

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
 * $$ (T_{i(x)}X)^\omega/(T_{i(x)}X\cap (T_{i(x)}X)^\omega), \quad x\in X,$$

where $$i:X\rightarrow M$$ 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 $$i^*(TM)$$. The isomorphism
 * $$ i^*(TM)\cong TX/\nu \oplus (TX)^\omega/\nu \oplus(\nu\oplus \nu^*), \quad \nu=TX\cap (TX)^\omega,$$

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.