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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 $(M,g)$ 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 $p$.### 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 $V\; to\; V/W$).## 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.## 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,

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.

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:

- $0\; to\; TN\; to\; TMvert\_\{i(N)\}\; to\; T\_\{M/N\}\; :=\; TMvert\_\{i(N)\}\; /\; TN\; to\; 0$

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.

- $[TN]\; +\; [T\_\{M/N\}]\; =\; [TM]$

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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Last updated on Sunday January 20, 2008 at 05:46:43 PST (GMT -0800)

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This article is licensed under the GNU Free Documentation License.

Last updated on Sunday January 20, 2008 at 05:46:43 PST (GMT -0800)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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