Normal bundle

For normal bundles in algebraic geometry, see normal cone.

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 ${\displaystyle (M,g)}$ be a Riemannian manifold, and ${\displaystyle S\subset M}$ a Riemannian submanifold. Define, for a given ${\displaystyle p\in S}$, a vector ${\displaystyle n\in \mathrm {T} _{p}M}$ to be normal to ${\displaystyle S}$ whenever ${\displaystyle g(n,v)=0}$ for all ${\displaystyle v\in \mathrm {T} _{p}S}$ (so that ${\displaystyle n}$ is orthogonal to ${\displaystyle \mathrm {T} _{p}S}$). The set ${\displaystyle \mathrm {N} _{p}S}$ of all such ${\displaystyle n}$ is then called the normal space to ${\displaystyle S}$ at ${\displaystyle p}$.

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 ^[1] ${\displaystyle \mathrm {N} S}$ to ${\displaystyle S}$ is defined as

${\displaystyle \mathrm {N} S:=\coprod _{p\in S}\mathrm {N} _{p}S}$.

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 ${\displaystyle i:N\to M}$ (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 ${\displaystyle V\to V/W}$).

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:

${\displaystyle 0\to TN\to TM\vert _{i(N)}\to T_{M/N}:=TM\vert _{i(N)}/TN\to 0}$

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

Conormal bundle

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

${\displaystyle {\begin{aligned}T_{p}X&=\mathbb {R} {\Big \lbrace }{\frac {\partial }{\partial x_{1}}}|_{p},\dots ,{\frac {\partial }{\partial x_{n}}}|_{p}{\Big \rbrace }\\T_{p}Y&=\mathbb {R} {\Big \lbrace }{\frac {\partial }{\partial x_{1}}}|_{p},\dots ,{\frac {\partial }{\partial x_{k}}}|_{p}{\Big \rbrace }\\{T_{X/Y}}_{p}&=\mathbb {R} {\Big \lbrace }{\frac {\partial }{\partial x_{k+1}}}|_{p},\dots ,{\frac {\partial }{\partial x_{n}}}|_{p}{\Big \rbrace }\\\end{aligned}}}$

and the ideal sheaf is locally generated by ${\displaystyle x_{k+1},\dots ,x_{n}}$. Therefore we can define a non-degenerate pairing

${\displaystyle (I_{Y}/I_{Y}^{2})_{p}\times {T_{X/Y}}_{p}\longrightarrow \mathbb {R} }$

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

${\displaystyle 0\to T_{X/Y}^{*}\rightarrowtail \Omega _{X}^{1}|_{Y}\twoheadrightarrow \Omega _{Y}^{1}\to 0}$,

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

When ${\displaystyle Y=\lbrace p\rbrace }$ is a point, then the ideal sheaf is the sheaf of smooth germs vanishing at ${\displaystyle p}$ and the isomorphism reduces to the definition of the tangent space in terms of germs of smooth functions on ${\displaystyle X}$

${\displaystyle T_{X/\lbrace p\rbrace }^{*}\simeq (T_{p}X)^{\vee }\simeq {\frac {{\mathfrak {m}}_{p}}{{\mathfrak {m}}_{p}^{2}}}}$.

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 ${\displaystyle \mathbf {R} ^{N}}$, 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 M, any two embeddings in ${\displaystyle \mathbf {R} ^{N}}$ 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 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,

${\displaystyle [TN]+[T_{M/N}]=[TM]}$

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

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 ${\displaystyle X}$ is embedded in to a symplectic manifold ${\displaystyle (M,\omega )}$, such that the pullback of the symplectic form has constant rank on ${\displaystyle X}$. Then one can define the symplectic normal bundle to X as the vector bundle over X with fibres

${\displaystyle (T_{i(x)}X)^{\omega }/(T_{i(x)}X\cap (T_{i(x)}X)^{\omega }),\quad x\in X,}$

where ${\displaystyle i:X\rightarrow M}$ 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 ${\displaystyle i^{*}(TM)}$. The isomorphism

${\displaystyle i^{*}(TM)\cong TX/\nu \oplus (TX)^{\omega }/\nu \oplus (\nu \oplus \nu ^{*}),\quad \nu =TX\cap (TX)^{\omega },}$

of symplectic vector bundles over ${\displaystyle X}$ implies that the symplectic normal bundle already determines the constant rank embedding locally. This feature is similar to the Riemannian case.

References

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