Normal bundle

Riemannian manifold

Let ( M , g ) be a Riemannian manifold, and S ⊂ M a Riemannian submanifold. Define, for a given p ∈ S , a vector n ∈ T p M to be normal to S whenever g ( n , v ) = 0 for all v ∈ T p S (so that n is orthogonal to T p S ). The set 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 N S to S 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 i : N → 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 → 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:

where T M | i ( N ) is the restriction of the tangent bundle on M to N (properly, the pullback i ∗ T M of the tangent bundle on M to a vector bundle on N via the map 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 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 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,

in the Grothendieck group. In case of an immersion in R N , the tangent bundle of the ambient space is trivial (since R N is contractible, hence parallelizable), so [ T N ] + [ T M / N ] = 0 , and thus [ T M / N ] = − [ T N ] .

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 , ω ) , 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

where i : X → 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 ∗ ( T M ) . The isomorphism

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.