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

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 ${\displaystyle \mathrm {N} S}$ to ${\displaystyle S}$ is defined as

${\displaystyle \mathrm {N} S:=\coprod _{p\in S}\mathrm {N} _{p}S}$.

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 ${\displaystyle i\colon N\to M}$ (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 ${\displaystyle V\to V/W}$).

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:

${\displaystyle 0\to TN\to TM\vert _{i(N)}\to T_{M/N}:=TM\vert _{i(N)}/TN\to 0}$

where ${\displaystyle TM\vert _{i(N)}}$ is the restriction of the tangent bundle on M to N (properly, the pullback ${\displaystyle i^{*}TM}$ of the tangent bundle on M to a vector bundle on N via the map ${\displaystyle i}$).

## 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 ${\displaystyle \mathbf {R} ^{N}}$, 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 ${\displaystyle \mathbf {R} ^{N}}$ 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,

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

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

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

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

where ${\displaystyle i:X\rightarrow M}$ 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.[1]

By Darboux's theorem, the constant rank embedding is locally determined by ${\displaystyle i*(TM)}$. The isomorphism

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

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

## References

1. ^ Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X