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Concept
Normal bundle
Summary
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).
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.
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 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 ).
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 .
Official source
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