Definitions

# 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).

## Definition

### Riemannian manifold

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

$mathrm\left\{N\right\}S := coprod_\left\{p in S\right\} mathrm\left\{N\right\}_p S$.

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:

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

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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