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 .
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 ). The fiber of the normal bundle in is referred to as the normal space at (of in ).
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,
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.
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.
- 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
- Tammo tom Dieck, Algebraic Topology, (2010) EMS Textbooks in Mathematics ISBN 978-3-03719-048-7
- Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X