Normal bundle

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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

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

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

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

), so ${\displaystyle [TN]+[T_{M/N}]=0}$, and thus ${\displaystyle [T_{M/N}]=-[TN]}$.

This is useful in the computation of

.

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