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).
be a Riemannian manifold
a Riemannian submanifold
. Define, for a given
, a vector
to be normal
). The set
of all such
is then called the normal space
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 to is defined as
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.
More abstractly, given an immersion
(for instance an embedding), one can define a normal bundle of N
, 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:
is the restriction of the tangent bundle on M
(properly, the pullback
of the tangent bundle on M
to a vector bundle on N
via the map
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
, 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,
in the Grothendieck group
In case of an immersion in
, the tangent bundle of the ambient space is trivial (since
is contractible, hence parallelizable
, and thus
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.