- Source: Normal bundle
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
)
{\displaystyle (M,g)}
be a Riemannian manifold, and
S
⊂
M
{\displaystyle S\subset M}
a Riemannian submanifold. Define, for a given
p
∈
S
{\displaystyle p\in S}
, a vector
n
∈
T
p
M
{\displaystyle n\in \mathrm {T} _{p}M}
to be normal to
S
{\displaystyle S}
whenever
g
(
n
,
v
)
=
0
{\displaystyle g(n,v)=0}
for all
v
∈
T
p
S
{\displaystyle v\in \mathrm {T} _{p}S}
(so that
n
{\displaystyle n}
is orthogonal to
T
p
S
{\displaystyle \mathrm {T} _{p}S}
). The set
N
p
S
{\displaystyle \mathrm {N} _{p}S}
of all such
n
{\displaystyle n}
is then called the normal space to
S
{\displaystyle S}
at
p
{\displaystyle 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
N
S
{\displaystyle \mathrm {N} S}
to
S
{\displaystyle S}
is defined as
N
S
:=
∐
p
∈
S
N
p
S
{\displaystyle \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
:
N
→
M
{\displaystyle i:N\to M}
(for instance an embedding), one can define a normal bundle of
N
{\displaystyle N}
in
M
{\displaystyle M}
, by at each point of
N
{\displaystyle N}
, taking the quotient space of the tangent space on
M
{\displaystyle M}
by the tangent space on
N
{\displaystyle 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
p
:
V
→
V
/
W
{\displaystyle p:V\to V/W}
).
Thus the normal bundle is in general a quotient of the tangent bundle of the ambient space
M
{\displaystyle M}
restricted to the subspace
N
{\displaystyle N}
.
Formally, the normal bundle to
N
{\displaystyle N}
in
M
{\displaystyle M}
is a quotient bundle of the tangent bundle on
M
{\displaystyle M}
: one has the short exact sequence of vector bundles on
N
{\displaystyle N}
:
0
→
T
N
→
T
M
|
i
(
N
)
→
T
M
/
N
:=
T
M
|
i
(
N
)
/
T
N
→
0
{\displaystyle 0\to \mathrm {T} N\to \mathrm {T} M\vert _{i(N)}\to \mathrm {T} _{M/N}:=\mathrm {T} M\vert _{i(N)}/\mathrm {T} N\to 0}
where
T
M
|
i
(
N
)
{\displaystyle \mathrm {T} M\vert _{i(N)}}
is the restriction of the tangent bundle on
M
{\displaystyle M}
to
N
{\displaystyle N}
(properly, the pullback
i
∗
T
M
{\displaystyle i^{*}\mathrm {T} M}
of the tangent bundle on
M
{\displaystyle M}
to a vector bundle on
N
{\displaystyle N}
via the map
i
{\displaystyle i}
). The fiber of the normal bundle
T
M
/
N
↠
π
N
{\displaystyle \mathrm {T} _{M/N}{\overset {\pi }{\twoheadrightarrow }}N}
in
p
∈
N
{\displaystyle p\in N}
is referred to as the normal space at
p
{\displaystyle p}
(of
N
{\displaystyle N}
in
M
{\displaystyle M}
).
= Conormal bundle
=If
Y
⊆
X
{\displaystyle Y\subseteq X}
is a smooth submanifold of a manifold
X
{\displaystyle X}
, we can pick local coordinates
(
x
1
,
…
,
x
n
)
{\displaystyle (x_{1},\dots ,x_{n})}
around
p
∈
Y
{\displaystyle p\in Y}
such that
Y
{\displaystyle Y}
is locally defined by
x
k
+
1
=
⋯
=
x
n
=
0
{\displaystyle x_{k+1}=\dots =x_{n}=0}
; then with this choice of coordinates
T
p
X
=
R
{
∂
∂
x
1
|
p
,
…
,
∂
∂
x
k
|
p
,
…
,
∂
∂
x
n
|
p
}
T
p
Y
=
R
{
∂
∂
x
1
|
p
,
…
,
∂
∂
x
k
|
p
}
T
X
/
Y
p
=
R
{
∂
∂
x
k
+
1
|
p
,
…
,
∂
∂
x
n
|
p
}
{\displaystyle {\begin{aligned}\mathrm {T} _{p}X&=\mathbb {R} {\Big \lbrace }{\frac {\partial }{\partial x_{1}}}{\Big |}_{p},\dots ,{\frac {\partial }{\partial x_{k}}}{\Big |}_{p},\dots ,{\frac {\partial }{\partial x_{n}}}{\Big |}_{p}{\Big \rbrace }\\\mathrm {T} _{p}Y&=\mathbb {R} {\Big \lbrace }{\frac {\partial }{\partial x_{1}}}{\Big |}_{p},\dots ,{\frac {\partial }{\partial x_{k}}}{\Big |}_{p}{\Big \rbrace }\\{\mathrm {T} _{X/Y}}_{p}&=\mathbb {R} {\Big \lbrace }{\frac {\partial }{\partial x_{k+1}}}{\Big |}_{p},\dots ,{\frac {\partial }{\partial x_{n}}}{\Big |}_{p}{\Big \rbrace }\\\end{aligned}}}
and the ideal sheaf is locally generated by
x
k
+
1
,
…
,
x
n
{\displaystyle x_{k+1},\dots ,x_{n}}
. Therefore we can define a non-degenerate pairing
(
I
Y
/
I
Y
2
)
p
×
T
X
/
Y
p
⟶
R
{\displaystyle (I_{Y}/I_{Y}^{\ 2})_{p}\times {\mathrm {T} _{X/Y}}_{p}\longrightarrow \mathbb {R} }
that induces an isomorphism of sheaves
T
X
/
Y
≃
(
I
Y
/
I
Y
2
)
∨
{\displaystyle \mathrm {T} _{X/Y}\simeq (I_{Y}/I_{Y}^{\ 2})^{\vee }}
. We can rephrase this fact by introducing the conormal bundle
T
X
/
Y
∗
{\displaystyle \mathrm {T} _{X/Y}^{*}}
defined via the conormal exact sequence
0
→
T
X
/
Y
∗
↣
Ω
X
1
|
Y
↠
Ω
Y
1
→
0
{\displaystyle 0\to \mathrm {T} _{X/Y}^{*}\rightarrowtail \Omega _{X}^{1}|_{Y}\twoheadrightarrow \Omega _{Y}^{1}\to 0}
,
then
T
X
/
Y
∗
≃
(
I
Y
/
I
Y
2
)
{\displaystyle \mathrm {T} _{X/Y}^{*}\simeq (I_{Y}/I_{Y}^{\ 2})}
, viz. the sections of the conormal bundle are the cotangent vectors to
X
{\displaystyle X}
vanishing on
T
Y
{\displaystyle \mathrm {T} Y}
.
When
Y
=
{
p
}
{\displaystyle Y=\lbrace p\rbrace }
is a point, then the ideal sheaf is the sheaf of smooth germs vanishing at
p
{\displaystyle p}
and the isomorphism reduces to the definition of the tangent space in terms of germs of smooth functions on
X
{\displaystyle X}
T
X
/
{
p
}
∗
≃
(
T
p
X
)
∨
≃
m
p
m
p
2
{\displaystyle \mathrm {T} _{X/\lbrace p\rbrace }^{*}\simeq (\mathrm {T} _{p}X)^{\vee }\simeq {\frac {{\mathfrak {m}}_{p}}{{\mathfrak {m}}_{p}^{\ 2}}}}
.
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
R
N
{\displaystyle \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 manifold
X
{\displaystyle X}
, any two embeddings in
R
N
{\displaystyle \mathbf {R} ^{N}}
for sufficiently large
N
{\displaystyle 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 the integer
N
{\displaystyle {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,
[
T
N
]
+
[
T
M
/
N
]
=
[
T
M
]
{\displaystyle [\mathrm {T} N]+[\mathrm {T} _{M/N}]=[\mathrm {T} M]}
in the Grothendieck group.
In case of an immersion in
R
N
{\displaystyle \mathbf {R} ^{N}}
, the tangent bundle of the ambient space is trivial (since
R
N
{\displaystyle \mathbf {R} ^{N}}
is contractible, hence parallelizable), so
[
T
N
]
+
[
T
M
/
N
]
=
0
{\displaystyle [\mathrm {T} N]+[\mathrm {T} _{M/N}]=0}
, and thus
[
T
M
/
N
]
=
−
[
T
N
]
{\displaystyle [\mathrm {T} _{M/N}]=-[\mathrm {T} N]}
.
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
{\displaystyle X}
is embedded in to a symplectic manifold
(
M
,
ω
)
{\displaystyle (M,\omega )}
, such that the pullback of the symplectic form has constant rank on
X
{\displaystyle X}
. Then one can define the symplectic normal bundle to
X
{\displaystyle X}
as the vector bundle over
X
{\displaystyle X}
with fibres
(
T
i
(
x
)
X
)
ω
/
(
T
i
(
x
)
X
∩
(
T
i
(
x
)
X
)
ω
)
,
x
∈
X
,
{\displaystyle (\mathrm {T} _{i(x)}X)^{\omega }/(\mathrm {T} _{i(x)}X\cap (\mathrm {T} _{i(x)}X)^{\omega }),\quad x\in X,}
where
i
:
X
→
M
{\displaystyle i:X\rightarrow M}
denotes the embedding and
(
T
X
)
ω
{\displaystyle (\mathrm {T} X)^{\omega }}
is the symplectic orthogonal of
T
X
{\displaystyle \mathrm {T} X}
in
T
M
{\displaystyle \mathrm {T} M}
. 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.
By Darboux's theorem, the constant rank embedding is locally determined by
i
∗
(
T
M
)
{\displaystyle i^{*}(\mathrm {T} M)}
. The isomorphism
i
∗
(
T
M
)
≅
T
X
/
ν
⊕
(
T
X
)
ω
/
ν
⊕
(
ν
⊕
ν
∗
)
{\displaystyle i^{*}(\mathrm {T} M)\cong \mathrm {T} X/\nu \oplus (\mathrm {T} X)^{\omega }/\nu \oplus (\nu \oplus \nu ^{*})}
(where
ν
=
T
X
∩
(
T
X
)
ω
{\displaystyle \nu =\mathrm {T} X\cap (\mathrm {T} X)^{\omega }}
and
ν
∗
{\displaystyle \nu ^{*}}
is the dual under
ω
{\displaystyle \omega }
,)
of symplectic vector bundles over
X
{\displaystyle X}
implies that the symplectic normal bundle already determines the constant rank embedding locally. This feature is similar to the Riemannian case.
References
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- Normal bundle
- Stable normal bundle
- Normal cone
- Normal
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- Immersion (mathematics)
- Normal (geometry)
- Right bundle branch block
- Canonical bundle
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