Normal bundle

For normal bundles in algebraic geometry, see normal cone.

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 $v\in {\mathrm {T}}_{p}S$ (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 $i\colon N\to M$ (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 TM\vert _{{i(N)}}\to T_{{M/N}}:=TM\vert _{{i(N)}}/TN\to 0

where $TM\vert _{{i(N)}}$ 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 immersibility and embeddability of manifolds in Euclidean space.

For symplectic manifolds

Suppose a manifold $X$ is embedded in to a symplectic manifold $(M,\omega )$ , such that the pullback of the symplectic form has constant rank on $X$ . Then one can define the symplectic normal bundle to X as the vector bundle over X with fibres

(T_{{i(x)}}X)^{\omega }/(T_{{i(x)}}X\cap (T_{{i(x)}}X)^{\omega }),\quad x\in X,

where $i:X\rightarrow M$ 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.^[1]

By Darboux's theorem, the constant rank embedding is locally determined by $i*(TM)$ . The isomorphism

i^{*}(TM)\cong TX/\nu \oplus (TX)^{\omega }/\nu \oplus (\nu \oplus \nu ^{*}),\quad \nu =TX\cap (TX)^{\omega },

of symplectic vector bundles over $X$ implies that the symplectic normal bundle already determines the constant rank embedding locally. This feature is similar to the Riemannian case.

References

↑ Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X

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