Normal bundle explained
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
be a
Riemannian manifold, and
a
Riemannian submanifold. Define, for a given
, a vector
to be
normal to
whenever
for all
(so that
is
orthogonal to
). The set
of all such
is then called the
normal space to
at
.
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[1]
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.
General definition
(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
).
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[2] 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\toTN\toTM\verti(N)\toTM/N:=TM\verti(N)/TN\to0
where
is the restriction of the tangent bundle on
M to
N (properly, the pullback
of the tangent bundle on
M to a vector bundle on
N via the map
). The fiber of the normal bundle
TM/N\overset{\pi}{\twoheadrightarrow}N
in
is referred to as the
normal space at
(of
in
).
Conormal bundle
If
is a smooth submanifold of a manifold
, we can pick local coordinates
around
such that
is locally defined by
; then with this choice of coordinates
\begin{align}
T | |
| pX&=R\lbrace | \partial | \partialx1 |
|
|p,...,
|p\rbrace\\
T
| |
| pY&=R\lbrace | \partial | \partialx1 |
|
|p,...,
|p\rbrace\\
{TX/Y
}_p&=\mathbb\Big\lbrace\frac|_p,\dots, \frac|_p\Big\rbrace\\\endand the
ideal sheaf is locally generated by
. Therefore we can define a non-degenerate pairing
}_p\longrightarrow \mathbbthat induces an isomorphism of sheaves
. We can rephrase this fact by introducing the
conormal bundle
defined via the
conormal exact sequence0\to
→ tail
Y\twoheadrightarrow
0
,then
, viz. the sections of the conormal bundle are the cotangent vectors to
vanishing on
.
When
is a point, then the ideal sheaf is the sheaf of smooth germs vanishing at
and the isomorphism reduces to the definition of the tangent space in terms of germs of smooth functions on
.
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 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), so
, and thus
.
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
is embedded in to a
symplectic manifold
, such that the pullback of the symplectic form has constant rank on
. Then one can define the symplectic normal bundle to X as the vector bundle over X with fibres
(Ti(x)
X\cap(Ti(x)X)\omega), x\inX,
where
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.
[3] By Darboux's theorem, the constant rank embedding is locally determined by
. The isomorphism
i*(TM)\congTX/\nu ⊕ (TX)\omega/\nu ⊕ (\nu ⊕ \nu*), \nu=TX\cap(TX)\omega,
of symplectic vector bundles over
implies that the symplectic normal bundle already determines the constant rank embedding locally. This feature is similar to the Riemannian case.
Notes and References
- John M. Lee, Riemannian Manifolds, An Introduction to Curvature, (1997) Springer-Verlag New York, Graduate Texts in Mathematics 176
- [Tammo tom Dieck]
- [Ralph Abraham (mathematician)|Ralph Abraham]