Riemannian manifold edit

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 edit

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 edit

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 ${\displaystyle X}$ 

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

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.

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.

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.