Normal bundle

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 \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 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{align} \mathrm{T}_pX&=\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}_pY&=\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{align} 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^1_X|_Y\twoheadrightarrow \Omega^1_Y\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}_pX)^\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. 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==