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 be a Riemannian manifold, and a Riemannian submanifold. Define, for a given , a vector to be normal to whenever for all (so that is orthogonal to ). The set of all such is then called the normal space to at . 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 to is defined as . 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 (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 ). 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: where is the restriction of the tangent bundle on M to N (properly, the pullback of the tangent bundle on M to a vector bundle on N via the map ). The fiber of the normal bundle in is referred to as the normal space at (of in ). Conormal bundle If is a smooth submanifold of a manifold , we can pick local coordinates around such that is locally defined by ; then with this choice of coordinates and the ideal sheaf is locally generated by . Therefore we can define a non-degenerate pairing that induces an isomorphism of sheaves . We can rephrase this fact by introducing the conormal bundle defined via the conormal exact sequence , then , viz. the sections of the conormal bundle are the cotangent vectors to vanishing on . When is a point, then the ideal sheaf