# Difficulty with the pullback bundle

by xaos
Tags: bundle, difficulty, pullback
 Sci Advisor P: 1,722 Difficulty with the pullback bundle The induced bundle is a natural way to create a new bundle from an old one. One could ask the question which vector bundles are induced and which are not. The remarkable answer is that they all are and they are all pullbacks of a single vector bundle over a single space. This universal space is called the Grassmann manifold of n-planes in R$^{∞}$ and the bundle is called the universal n-plane bundle. This space is similar to an infinite dimensional projective space. So in some sense the study of bundles over a space is the study of maps into this Grassman manifold. Further if two such maps are homotopic then the induced bundles are isomorphic so one is really studying homotopy classes of maps into the Grassmann manifold. Because of this, the cohomology of the Grassman manifold is extremely important since it's pullback under the map reflects the bundle itself. the study of this cohomology and it's pullbacks is one approach to the study of Characteristic classes. These classes contain information about the structure of the bundle e.g. whether the bundle has a non-zero section. For smooth manifolds there characteristic classes tell you things about the tangent bundle of the manifold. Further the universal bundle has a universal connection and all connections on all smooth vector bundles are pullbacks of the universal connection. this means that all Rimannian geometries are pull backs of a single universal geometry on the universal bundle. A famous theorem related wedge products of the curvature 2 form matrix to the characterisitc classes of the bundle so there are universal geometric invariants of smooth manifolds (and also all smooth bundles over a smooth manifold.) One example for oriented bundles with a Riemannain metric is the real Euler class whose integral over the manifold is its Euler characteristic. Here are a couple of examples for you to think about. - Suppose you have a smooth surface - like a torus - embedded in 3 space. The surface is orientable (proof?) and so has a well defined unit normal. The unit normal to each point on the surface assigns a point on the sphere of radius 1 centered at the origin. So it defines a mapping of the surface into the unit sphere. It is called the Gauss map. (You should convince yourself that the mapping is smooth.) Is the tangent bundle the induced bundle under this map? What bundle map do you get if you parallel translate each tangent vector to the surface to the tangent plane to the sphere at the point determined by the unit normal? Note that this map differs in general from the differential of the Gauss map which may have singularities or may distort the tangent plane. - Suppose a differentiable map is a local diffeomorphism. Is the induced bundle of the tangent bundle the tangent bundle? - Suppose an oriented vector bundle has a non-zero section i.e. a non-zero vector field. Do all bundles induced from it also have a non-zero section? - What is the difference between the Hopf fibration and the tangent circle bundle of the 2 sphere? Does the Euler class distinguish them?