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).

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[1]

It can be realised naturally as a sub-bundle of the cotangent bundle.

(for instance an embedding), one can define a normal bundle of

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

: one has the short exact sequence of vector bundles on

is a smooth submanifold of a manifold

; then with this choice of coordinates and the ideal sheaf is locally generated by

Therefore we can define a non-degenerate pairing that induces an isomorphism of sheaves

We can rephrase this fact by introducing the conormal bundle

defined via the conormal exact sequence then

the sections of the conormal bundle are the cotangent vectors to

and the isomorphism reduces to the definition of the tangent space in terms of germs of smooth functions on

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.

, 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 manifold

are regular homotopic, and hence induce the same normal bundle.

could vary) is called the stable normal 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.

, the tangent bundle of the ambient space is trivial (since

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.

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

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.

[3] By Darboux's theorem, the constant rank embedding is locally determined by

implies that the symplectic normal bundle already determines the constant rank embedding locally.

This feature is similar to the Riemannian case.