Normal bundle

Summary

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[1] 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[2] 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 ). The fiber of the normal bundle in is referred to as the normal space at (of in ).

Conormal bundle

If is a smooth submanifold of a manifold , we can pick local coordinates around such that is locally defined by ; 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 , viz. the sections of the conormal bundle are the cotangent vectors to vanishing on .

When is a point, then the ideal sheaf is the sheaf of smooth germs vanishing at and the isomorphism reduces to the definition of the tangent space in terms of germs of smooth functions on

.

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 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.[3]

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

  1. ^ John M. Lee, Riemannian Manifolds, An Introduction to Curvature, (1997) Springer-Verlag New York, Graduate Texts in Mathematics 176 ISBN 978-0-387-98271-7
  2. ^ Tammo tom Dieck, Algebraic Topology, (2010) EMS Textbooks in Mathematics ISBN 978-3-03719-048-7
  3. ^ Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X