normal bundle

Let X be an immersed submanifoldMathworldPlanetmath of M, with immersionMathworldPlanetmath i:XM. Then we can restrict the tangent bundleMathworldPlanetmath of M to N or more properly, take the pullback i*TM. This, as an vector bundleMathworldPlanetmath over X should contain a lot of information about the embedding of X into M. But there is a natural injection TXi*TM, and the subbundle which is the image of this only has information on the intrinsic properties of X, and thus is useless in obtaining information about the embedding of X into M. Instead, to get information on this, we take the quotient i*TM/TX=NX, the normal bundleMathworldPlanetmath of X. The normal bundle is very strongly dependent on the immersion i. If E is any vector bundle on X, then E is the normal bundle for the embedding of X into E as the zero sectionMathworldPlanetmath.

The normal bundle determines the local geometry of the embedding of X into M in the following sense: In M, there exists an open neighborhood UX which is diffeomorphicMathworldPlanetmath to NX by a diffeomorphism taking X to the zero section.

Title normal bundle
Canonical name NormalBundle
Date of creation 2013-03-22 13:59:06
Last modified on 2013-03-22 13:59:06
Owner bwebste (988)
Last modified by bwebste (988)
Numerical id 5
Author bwebste (988)
Entry type Definition
Classification msc 58A32