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 $(M,g)$ 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 $i\colon N \to M$ (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 normal bundle itself forms an N-dimensional manifold differentiably embedded in $\mathbf{R}^{2N}$ if the manifold itself is of dimension 2N.

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 compact 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.

Algebraic geometry

In algebraic geometry, the normal bundle NXY of a regular embedding i: XY, defined by some sheaf of ideals I is the vector bundle on X corresponding to the dual of the sheaf I/I2. The regularity of the embedding ensures that this sheaf is locally free and agrees with the normal cone CXY, which is defined as $Spec \oplus_{n \geq 0} I^n / I^{n+1}$.[1]

References

1. ^ Fulton, William (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] 2, Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4, MR 1644323, section B.7