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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 $S\subset M$ a Riemannian submanifold. Define, for a given $p\in S$ , a vector $n\in \mathrm {T} _{p}M$ to be normal to $S$ whenever $g(n,v)=0$ for all $v\in \mathrm {T} _{p}S$ (so that $n$ is orthogonal to $\mathrm {T} _{p}S$ ). The set $\mathrm {N} _{p}S$ 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] $\mathrm {N} S$ to $S$ is defined as

\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 $i:N\to M$ (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 $p:V\to V/W$ ).

Thus the normal bundle is in general a quotient of the tangent bundle of the ambient space $M$ restricted to the subspace $N$ .

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\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 $\mathrm {T} M\vert _{i(N)}$ is the restriction of the tangent bundle on $M$ to $N$ (properly, the pullback $i^{*}\mathrm {T} M$ of the tangent bundle on $M$ to a vector bundle on $N$ via the map $i$ ). The fiber of the normal bundle $\mathrm {T} _{M/N}{\overset {\pi }{\twoheadrightarrow }}N$ in $p\in N$ is referred to as the normal space at $p$ (of $N$ in $M$ ).

Conormal bundle

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

{\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 $x_{k+1},\dots ,x_{n}$ . Therefore we can define a non-degenerate pairing

(I_{Y}/I_{Y}^{\ 2})_{p}\times {\mathrm {T} _{X/Y}}_{p}\longrightarrow \mathbb {R}

that induces an isomorphism of sheaves $\mathrm {T} _{X/Y}\simeq (I_{Y}/I_{Y}^{\ 2})^{\vee }$ . We can rephrase this fact by introducing the conormal bundle $\mathrm {T} _{X/Y}^{*}$ defined via the conormal exact sequence

0\to \mathrm {T} _{X/Y}^{*}\rightarrowtail \Omega _{X}^{1}|_{Y}\twoheadrightarrow \Omega _{Y}^{1}\to 0

,

then $\mathrm {T} _{X/Y}^{*}\simeq (I_{Y}/I_{Y}^{\ 2})$ , viz. the sections of the conormal bundle are the cotangent vectors to $X$ vanishing on $\mathrm {T} Y$ .

When $Y=\lbrace p\rbrace$ 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 $X$

\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 $\mathbf {R} ^{N}$ , 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 $X$ , any two embeddings in $\mathbf {R} ^{N}$ 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 the integer ${N}$ 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,

[\mathrm {T} N]+[\mathrm {T} _{M/N}]=[\mathrm {T} M]

in the Grothendieck group. In case of an immersion in $\mathbf {R} ^{N}$ , the tangent bundle of the ambient space is trivial (since $\mathbf {R} ^{N}$ is contractible, hence parallelizable), so $[\mathrm {T} N]+[\mathrm {T} _{M/N}]=0$ , and thus $[\mathrm {T} _{M/N}]=-[\mathrm {T} N]$ .

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 $X$ 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

(\mathrm {T} _{i(x)}X)^{\omega }/(\mathrm {T} _{i(x)}X\cap (\mathrm {T} _{i(x)}X)^{\omega }),\quad x\in X,

where $i:X\rightarrow M$ denotes the embedding and $(\mathrm {T} X)^{\omega }$ is the symplectic orthogonal of $\mathrm {T} X$ in $\mathrm {T} M$ . 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.^[3]

By Darboux's theorem, the constant rank embedding is locally determined by $i^{*}(\mathrm {T} M)$ . The isomorphism

i^{*}(\mathrm {T} M)\cong \mathrm {T} X/\nu \oplus (\mathrm {T} X)^{\omega }/\nu \oplus (\nu \oplus \nu ^{*})

(where $\nu =\mathrm {T} X\cap (\mathrm {T} X)^{\omega }$ and $\nu ^{*}$ is the dual under $\omega$ ,) 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.

References

^ 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
^ Tammo tom Dieck, Algebraic Topology, (2010) EMS Textbooks in Mathematics ISBN 978-3-03719-048-7
^ Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X

[1] 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

[2] Tammo tom Dieck, Algebraic Topology, (2010) EMS Textbooks in Mathematics ISBN 978-3-03719-048-7

[3] Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X

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