Normal Sheaf

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 $n \in \mathrm_p M$ to be '' normal'' to $S$ whenever $g(n,v)=0$ for all $v\in \mathrm_p S$ (so that $n$ is orthogonal to $\mathrm_p S$). The set $\mathrm_p S$ 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 $\mathrm S$ to $S$ is defined as
:$\mathrmS := \coprod_ \mathrm_p S$.

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 $V \to V/W$).

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 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'':
:$0 \to TN \to TM\vert_ \to T_ := TM\vert_ / TN \to 0$

where $TM\vert_$ is the restriction of the tangent bundle on ''M'' to ''N'' (properly, the pullback $i^*TM$ of the tangent bundle on ''M'' to a vector bundle on ''N'' via the map $i$). The fiber of the normal bundle $T_\overset N$ 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 $x_=\dots=x_n=0$; then with this choice of coordinates

:$\begin T_pX&=\mathbb\Big\lbrace\frac, _p,\dots, \frac, _p\Big\rbrace\\ T_pY&=\mathbb\Big\lbrace\frac, _p,\dots, \frac, _p\Big\rbrace\\ _p&=\mathbb\Big\lbrace\frac, _p,\dots, \frac, _p\Big\rbrace\\ \end$
and the ideal sheaf is locally generated by $x_,\dots,x_n$. Therefore we can define a non-degenerate pairing

:$(I_Y/I^2_Y)_p\times _p\longrightarrow \mathbb$
that induces an isomorphism of sheaves $T_\simeq(I_Y/I_Y^2)^\vee$. We can rephrase this fact by introducing the conormal bundle $T^*_$ defined via the conormal exact sequence

:$0\to T^*_\rightarrowtail \Omega^1_X, _Y\twoheadrightarrow \Omega^1_Y\to 0$,
then $T^*_\simeq (I_Y/I_Y^2)$, viz. the sections of the conormal bundle are the cotangent vectors to $X$ vanishing on $TY$.

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 $X$

:$T^*_\simeq (T_pX)^\vee\simeq\frac$.

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 $\mathbf^N$, 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 $\mathbf^N$ 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,
: N+ _= M/math>
in the Grothendieck group.
In case of an immersion in $\mathbf^N$, the tangent bundle of the ambient space is trivial (since $\mathbf^N$ is contractible, hence parallelizable), so N+ _= 0, and thus _= - 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 $X$ 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
:$(T_X)^\omega/(T_X\cap (T_X)^\omega), \quad x\in X,$
where $i:X\rightarrow M$ 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. Ralph Abraham and Jerrold E. Marsden, ''Foundations of Mechanics'', (1978) Benjamin-Cummings, London

By Darboux's theorem, the constant rank embedding is locally determined by $i^*(TM)$. The isomorphism
:$i^*(TM)\cong TX/\nu \oplus (TX)^\omega/\nu \oplus(\nu\oplus \nu^*), \quad \nu=TX\cap (TX)^\omega,$
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

{{DEFAULTSORT:Normal Bundle
Algebraic geometry
Differential geometry
Differential topology
Vector bundles