is the sheaf of smooth germs vanishing at and the isomorphism reduces to the definition of the tangent space in terms of germs of smooth functions on . 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 , 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 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, in the Grothendieck group. In case of an immersion in , the tangent bundle of the ambient space is trivial (since is contractible, hence parallelizable), so , and thus . 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 is embedded in to a symplectic manifold , such that the pullback of the symplectic form has constant rank on . Then one can define the symplectic normal bundle to X as the vector bundle over X with fibres where 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. By Darboux's theorem, the constant rank embedding is locally determined by . The isomorphism of symplectic vector bundles over implies that the symplectic normal bundle already determines the constant rank embedding locally. This feature is similar to the Riemannian case. References Algebraic geometry Differential geometry Differential topology Vector bundles