Let be an immersed submanifold of , with immersion . Then we can restrict the tangent bundle of to or more properly, take the pullback . This, as an vector bundle over should contain a lot of information about the embedding of into . But there is a natural injection , and the subbundle which is the image of this only has information on the intrinsic properties of , and thus is useless in obtaining information about the embedding of into . Instead, to get information on this, we take the quotient , the normal bundle of . The normal bundle is very strongly dependent on the immersion . If is any vector bundle on , then is the normal bundle for the embedding of into as the zero section.
The normal bundle determines the local geometry of the embedding of into in the following sense: In , there exists an open neighborhood which is diffeomorphic to by a diffeomorphism taking to the zero section.
|Date of creation||2013-03-22 13:59:06|
|Last modified on||2013-03-22 13:59:06|
|Last modified by||bwebste (988)|