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

be a Riemannian manifold a Riemannian submanifold to be normal is orthogonal is then called the normal space.

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 is defined as

1. .

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, taking the quotient space. 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).

Thus the normal bundle is in general a quotient.

Formally, the normal bundle: one has the short exact sequence:

is referred to as the ).

Conormal bundle

; then with this choice of coordinates

and the ideal sheaf. Therefore we can define a non-degenerate pairing

. We can rephrase this fact by introducing the conormal bundle defined via the conormal exact sequence

1. ,

.

and the isomorphism reduces to the definition of the tangent space

1. .

Stable normal bundle

Abstract manifolds have a canonical, by the Whitney embedding theorem, every manifold admits a normal bundle, given such an embedding.

are regular homotopic 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 is contractible, hence parallelizable.

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

is embedded in to a symplectic manifold with fibres

is the symplectic orthogonal. 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 isomorphism

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

References