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

(M,g)

be a Riemannian manifold, and

S\subsetM

a Riemannian submanifold. Define, for a given

p\inS

, a vector

n\inTpM

to be normal to

S

whenever

g(n,v)=0

for all

v\inTpS

(so that

n

is orthogonal to

TpS

). The set

NpS

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[1]

NS

to

S

is defined as

NS:=\coprodpNpS

.

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

i:N\toM

(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\toV/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[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

TM\verti(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

). The fiber of the normal bundle

TM/N\overset{\pi}{\twoheadrightarrow}N

in

p\inN

is referred to as the normal space at

p

(of

N

in

M

).

Conormal bundle

If

Y\subseteqX

is a smooth submanifold of a manifold

X

, we can pick local coordinates

(x1,...,xn)

around

p\inY

such that

Y

is locally defined by

xk+1=...=xn=0

; then with this choice of coordinates
\begin{align} T
pX&=R\lbrace\partial
\partialx1

|p,...,

\partial
\partialxn

|p\rbrace\\ T

pY&=R\lbrace\partial
\partialx1

|p,...,

\partial
\partialxk

|p\rbrace\\ {TX/Y

}_p&=\mathbb\Big\lbrace\frac|_p,\dots, \frac|_p\Big\rbrace\\\endand the ideal sheaf is locally generated by

xk+1,...,xn

. Therefore we can define a non-degenerate pairing
2
(I
Y)

p x {TX/Y

}_p\longrightarrow \mathbbthat induces an isomorphism of sheaves

TX/Y\simeq(IY/I

2)
Y

\vee

. We can rephrase this fact by introducing the conormal bundle
*
T
X/Y
defined via the conormal exact sequence

0\to

*
T
X/Y

tail

1
\Omega
X|

Y\twoheadrightarrow

1
\Omega
Y\to

0

,then
*
T
X/Y

\simeq(IY/I

2)
Y
, viz. the sections of the conormal bundle are the cotangent vectors to

X

vanishing on

TY

.

When

Y=\lbracep\rbrace

is a point, then the ideal sheaf is the sheaf of smooth germs vanishing at

p

and the isomorphism reduces to the definition of the tangent space in terms of germs of smooth functions on

X

*
T
X/\lbracep\rbrace

\simeq

2}
(T
p
.

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

RN

, 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

RN

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]+[TM/N]=[TM]

in the Grothendieck group.In case of an immersion in

RN

, the tangent bundle of the ambient space is trivial (since

RN

is contractible, hence parallelizable), so

[TN]+[TM/N]=0

, and thus

[TM/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

(Ti(x)

\omega/(T
X)
i(x)

X\cap(Ti(x)X)\omega),x\inX,

where

i:XM

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

i*(TM)

. The isomorphism

i*(TM)\congTX/\nu(TX)\omega/\nu(\nu\nu*),\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.

Notes and References

  1. John M. Lee, Riemannian Manifolds, An Introduction to Curvature, (1997) Springer-Verlag New York, Graduate Texts in Mathematics 176
  2. [Tammo tom Dieck]
  3. [Ralph Abraham (mathematician)|Ralph Abraham]