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.

References