imported>MediaKyle: Adding local short description: "Concept in mathematics", overriding Wikidata description "vector bundle, complementary to the tangent bundle, associated to an embedding"

2025-05-03T15:21:48Z

Adding local short description: "Concept in mathematics", overriding Wikidata description "vector bundle, complementary to the tangent bundle, associated to an embedding"

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{{Short description|Concept in mathematics}}
{{for|normal bundles in algebraic geometry|normal cone}}
In [[differential geometry]], a field of [[mathematics]], a '''normal bundle''' is a particular kind of [[vector bundle]], [[complementary angles|complementary]] to the [[tangent bundle]], and coming from an [[embedding]] (or [[immersion (mathematics)|immersion]]).

==Definition==

===Riemannian manifold===
Let <math>(M,g)</math> be a [[Riemannian manifold]], and <math>S \subset M</math> a [[Riemannian submanifold]]. Define, for a given <math>p \in S</math>, a vector <math>n \in \mathrm{T}_p M</math> to be ''[[normal vector|normal]]'' to <math>S</math> whenever <math>g(n,v)=0</math> for all <math>v\in \mathrm{T}_p S</math> (so that <math>n</math> is [[orthogonal complement|orthogonal]] to <math>\mathrm{T}_p S</math>). The set <math>\mathrm{N}_p S</math> of all such <math>n</math> is then called the ''normal space'' to <math>S</math> at <math>p</math>.

Just as the total space of the [[tangent bundle]] to a manifold is constructed from all [[tangent space]]s to the manifold, the total space of the '''normal bundle'''<ref>John M. Lee, ''Riemannian Manifolds, An Introduction to Curvature'', (1997) Springer-Verlag New York, Graduate Texts in Mathematics 176 {{isbn|978-0-387-98271-7}}</ref> <math>\mathrm{N} S</math> to <math>S</math> is defined as
:<math>\mathrm{N}S := \coprod_{p \in S} \mathrm{N}_p S</math>.

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 (mathematics)|immersion]] <math>i: N \to M</math> (for instance an embedding), one can define a normal bundle of <math>N</math> in <math>M</math>, by at each point of <math>N</math>, taking the [[quotient space (linear algebra)|quotient space]] of the tangent space on <math>M</math> by the tangent space on <math>N</math>. 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 (category theory)|section]] of the projection <math>p:V \to V/W</math>).

Thus the normal bundle is in general a ''quotient'' of the tangent bundle of the ambient space <math>M</math> restricted to the subspace <math>N</math>.

Formally, the '''normal bundle'''<ref>[[Tammo tom Dieck]], ''Algebraic Topology'', (2010) EMS Textbooks in Mathematics {{isbn|978-3-03719-048-7}}</ref> to <math>N</math> in <math>M</math> is a quotient bundle of the tangent bundle on <math>M</math>: one has the [[short exact sequence]] of vector bundles on <math>N</math>:
:<math>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</math>

where <math>\mathrm{T}M\vert_{i(N)}</math> is the restriction of the tangent bundle on <math>M</math> to <math>N</math> (properly, the pullback <math>i^*\mathrm{T}M</math> of the tangent bundle on <math>M</math> to a vector bundle on <math>N</math> via the map <math>i</math>). The fiber of the normal bundle <math> \mathrm{T}_{M/N}\overset{\pi}{\twoheadrightarrow} N</math> in <math> p\in N</math> is referred to as the '''normal space at <math> p</math>''' (of <math>N</math> in <math>M</math>).

===Conormal bundle===
If <math>Y\subseteq X</math> is a smooth submanifold of a manifold <math>X</math>, we can pick local coordinates <math>(x_1,\dots,x_n)</math> around <math>p\in Y</math> such that <math> Y</math> is locally defined by <math>x_{k+1}=\dots=x_n=0</math>; then with this choice of coordinates

:<math>\begin{align}
\mathrm{T}_pX&=\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}_pY&=\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{align}</math>
and the [[ideal sheaf]] is locally generated by <math>x_{k+1},\dots,x_n</math>. Therefore we can define a non-degenerate pairing

:<math>(I_Y/I_Y^{\ 2})_p\times {\mathrm{T}_{X/Y}}_p\longrightarrow \mathbb{R}</math>
that induces an isomorphism of sheaves <math>\mathrm{T}_{X/Y}\simeq(I_Y/I_Y^{\ 2})^\vee</math>. We can rephrase this fact by introducing the '''conormal bundle''' <math>\mathrm{T}^*_{X/Y}</math> defined via the '''conormal exact sequence'''

:<math>0\to \mathrm{T}^*_{X/Y}\rightarrowtail \Omega^1_X|_Y\twoheadrightarrow \Omega^1_Y\to 0</math>,
then <math>\mathrm{T}^*_{X/Y}\simeq (I_Y/I_Y^{\ 2})</math>, viz. the sections of the conormal bundle are the cotangent vectors to <math>X</math> vanishing on <math>\mathrm{T}Y</math>.

When <math>Y=\lbrace p\rbrace</math> is a point, then the ideal sheaf is the sheaf of smooth germs vanishing at <math>p</math> and the isomorphism reduces to the [[Tangent_space#Definition_via_cotangent_spaces|definition of the tangent space]] in terms of germs of smooth functions on <math> X</math>

:<math> \mathrm{T}^*_{X/\lbrace p\rbrace}\simeq (\mathrm{T}_pX)^\vee\simeq\frac{\mathfrak{m}_p}{\mathfrak{m}_p^{\ 2}}</math>.

==Stable normal bundle==
[[Abstract manifold|Abstract manifolds]] have a [[canonical form|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 <math>\mathbf{R}^{N}</math>, 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 <math>X</math>, any two embeddings in <math>\mathbf{R}^N</math> for sufficiently large <math>N</math> are [[regular homotopy|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 <math>{N}</math> 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,
:<math>[\mathrm{T}N] + [\mathrm{T}_{M/N}] = [\mathrm{T}M]</math>
in the [[Grothendieck group]].
In case of an immersion in <math>\mathbf{R}^N</math>, the tangent bundle of the ambient space is trivial (since <math>\mathbf{R}^N</math> is contractible, hence [[parallelizable]]), so <math>[\mathrm{T}N] + [\mathrm{T}_{M/N}] = 0</math>, and thus <math>[\mathrm{T}_{M/N}] = -[\mathrm{T}N]</math>.

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 <math>X</math> is embedded in to a [[symplectic manifold]] <math>(M,\omega)</math>, such that the pullback of the symplectic form has constant rank on <math>X</math>. Then one can define the symplectic normal bundle to <math>X</math> as the vector bundle over <math>X</math> with fibres
:<math> (\mathrm{T}_{i(x)}X)^\omega/(\mathrm{T}_{i(x)}X\cap (\mathrm{T}_{i(x)}X)^\omega), \quad x\in X,</math>
where <math>i:X\rightarrow M</math> denotes the embedding and <math>(\mathrm{T}X)^\omega</math> is the [[symplectic vector space#subspace|symplectic orthogonal]] of <math>\mathrm{T}X</math> in <math>\mathrm{T}M</math>. 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.<ref>[[Ralph Abraham (mathematician)|Ralph Abraham]] and [[Jerrold E. Marsden]], ''Foundations of Mechanics'', (1978) Benjamin-Cummings, London {{isbn|0-8053-0102-X}}</ref>

By [[Darboux's theorem]], the constant rank embedding is locally determined by <math>i^*(\mathrm{T}M)</math>. The isomorphism
:<math> i^*(\mathrm{T}M)\cong \mathrm{T}X/\nu \oplus (\mathrm{T}X)^\omega/\nu \oplus(\nu\oplus \nu^*)</math>
(where <math> \nu=\mathrm{T}X\cap (\mathrm{T}X)^\omega</math> and <math>\nu^*</math> is the dual under <math>\omega</math>,)
of symplectic vector bundles over <math>X</math> implies that the symplectic normal bundle already determines the constant rank embedding locally. This feature is similar to the Riemannian case.

==References==

{{reflist}}

{{Manifolds}}

{{DEFAULTSORT:Normal Bundle}}
[[Category:Algebraic geometry]]
[[Category:Differential geometry]]
[[Category:Differential topology]]
[[Category:Vector bundles]]

Normal bundle - Revision history

imported>MediaKyle: Adding local short description: "Concept in mathematics", overriding Wikidata description "vector bundle, complementary to the tangent bundle, associated to an embedding"