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).


Riemannian manifold

Let (M,g) be a Riemannian manifold, and S subset M a Riemannian submanifold. Define, for a given p in S, a vector n in mathrm{T}_p M to be normal to S whenever g(n,v)=0 for all vin mathrm{T}_p S (so that n is orthogonal to mathrm{T}_p S). The set mathrm{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 mathrm{N} S to S is defined as

mathrm{N}S := coprod_{p in S} mathrm{N}_p S.

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 icolon N to 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 to 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:

0 to TN to TMvert_{i(N)} to T_{M/N} := TMvert_{i(N)} / TN to 0
where TMvert_{i(N)} is the restriction of the tangent bundle on M to N (properly, the pullback i^*TM 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 mathbf{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 mathbf{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,
[TN] + [T_{M/N}] = [TM]
in the Grothendieck group. In case of an immersion in mathbf{R}^N, the tangent bundle of the ambient space is trivial (since mathbf{R}^N is contractible, hence parallelizable), so [TN] + [T_{M/N}] = 0, and thus [T_{M/N}] = -[TN].

This is useful in the computation of characteristic classes, and allows one to prove lower bounds on immersability and embeddability of manifolds in Euclidean space.

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