# 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 be a Riemannian manifold, and a Riemannian submanifold. Define, for a given , a vector to be *normal* to whenever for all (so that is orthogonal to ). The set of all such is then called the *normal space* to at .

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* to is defined as

- .

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 (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 ).

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*:

where is the restriction of the tangent bundle on *M* to *N* (properly, the pullback of the tangent bundle on *M* to a vector bundle on *N* via the map ).

## 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 , 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 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,

in the Grothendieck group. In case of an immersion in , the tangent bundle of the ambient space is trivial (since is contractible, hence parallelizable), so , and thus .

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 is embedded in to a symplectic manifold , such that the pullback of the symplectic form has constant rank on . Then one can define the symplectic normal bundle to X as the vector bundle over X with fibres

where 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 . The isomorphism

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

## References

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