We say it is the topological vector bundle glued from the transition functions. P k 0 kv yields a vector bundle in fact, bundle of real. Notice that e m rk is a vector bundle of rank k, with the obvious projection map and local trivialization. E m denote a vector bundle, and choose a local trivialization. For this goal, we study the moduli space of 5tuples x,x,z,e. For a eld f r or c and a positive integer k, a rank kfvector bundle. M, 1pep has the structure of kdimensional vector space, each p. U is called a local trivialization of eover uand uis a trivializing neighbourhood for e. Finally, the quotient coprojections constitute a local trivialization of this vector bundle over the given open cover. E m with typical fibre rn and local trivializations.
A morphism of vector bundles is a ber bundle morphism which restricts to a. A rank 1 bundle is often referred to as a line bundle. Vector bundles and connections the exposition of vector. Motivation we want to study the bundle analogues of subspaces and quotients of nitedimensional vector spaces. E 2 taking each ber p 1 1 b to the ber p 1 2 b by a linear isomorphism. Sections of 2vector bundles urs schreiber january 4, 2007 higher categories and their applications, fields institute workshop, 2007, jan. Prolongations of isometric actions to vector bundles. Thanks for contributing an answer to mathematics stack exchange. Motivation let x,o be a cp premanifold with corners, 0.
Canonical trivializations functions of a principal bundle for some purposes it is useful to have local trivialization functions. Fare vector bundles then e f is called the direct or \whitney sum, and has. C, an arbitrary semistable vector bundle fon c of rank r and a. A kvector bundle over mof rank kconsists of a bundle e m whose. It is called the trivial rankkcomplex vector bundle. But there are many other interesting vector bundles. We show how the amplitude of holonomies on a vector bundle can be controlled by the integral of the curvature of the connection on a surface enclosed by the curve.
With these in mind, we see that the local triviality condition from the definition of vector bundles says that. If the dimension of the vector space is mthen the bundle is often called an mplane bundle. As a riemannian metric on m is an inner product on the vector bundle tm, theorem 3. Hence, e has a unique structure of vector bundle over mwith projection. The vector bundle e m is called trivial it is isomorphic to the trivial rankk bundle. Xbe a smooth ber oriented vector bundle over a compact smooth oriented manifold x. A vector bundle of rank 1 is usually called a line bundle. By definition, a smooth kvector bundle of rank r over a smooth manifold m. But avoid asking for help, clarification, or responding to other answers.
Assume that the rank ber dimension of eis the same as the dimension nof x. U in the definition is called a local trivialization of. Vector bundles of rank 1 is also called the line bundle. The notion of vector bundle is a basic extension to the geometric domain of the fundamental idea of a vector space. Vector bundles of rank one are also called line bundles. If mis a smooth manifold and kis a nonnegative integer, then. The vector bundle of rank rover xis said to be trivial if. Introduction let e be a vector bundle over the manifold m with. Any example of manifold without global trivialization of. Cohomology and vector bundles university of chicago.
Thus, a vector bundle represents linearization of nonlinear structure of manifolds. A local system is a bundle with locally constant sheaf of sections. In particular, the restriction of a vector bundle to a submanifold of the base is a vector bundle over the submanifold. The triple must satisfy the condition that there be a local trivialization f. A vector bundle has the local appearance of a product, but may have nontrivial global topology. In the other direction, for a vector bundle eover x, one takes the frame bundle or the trivialization bundle. If r 1 then e is called a real or complex line bundle over m. In this chapter, we introduce an important generalization of tangent bundles. The most important example is the tangent bundle, a real vector bundle. The tangent bundle tm is indeed a vector bundle, of rank dim m. In mathematics, and particularly topology, a fiber bundle or, in british english, fibre bundle is a space that is locally a product space, but globally may have a different topological structure. The open neighborhood u together with the homeomorphism is called a local trivialization of the vector bundle. R on rk, the resulting bundle e is a vector bundle of rank k over m. The local trivialization shows that locally the map.
Similarly, taking fv to be the whole exterior algebra v. On the chern correspondence for principal fibre bundles with. The following gives a functorial correspondence between vector bundles and g gl n bundles. Vector bundles and connections universiteit utrecht. Recall that a choice of local coordinates xon m determines a choice of local trivial ization of tm using the basis vector. A connection on a manifold mis a connection on its tangent bundle tm.
This new vector bundle fe is called the \associated vector bundle to e, given by the functor f. A complex vector bundle of rank kis also a real vector bundle of rank 2k, but a real vector bundle of rank 2kneed not in general admit a complex structure. Therefore e c x ec \to x is a topological vector bundle def. U for a principal bundle which satisfy certain additional. Each local trivial ization u, for x2u, yields such an isomorphism uj e.
We then go on to discuss local and global sections of vector bundles. M for which the local trivialization has been chosen in such a way that the transition functions are locally constant. Given a space x, we take a real or complex finite dimensional vector space v and. A section of eis a smooth choice of a vector in each. E m is a choice of orientation for every fiber ex, such that for any local trivialization. A bundle over a manifold is trivial if it is simply the cartesian product of. For an isolated zero p2x of a smooth section sof ede ne the local degree deg p s of sat pby deg p s. Bis called the base and ethe total space of this vector bundle. The notion of global or local section of a ber bundle e.
The set of global sections will be denoted by sece and the set of local sections over an open subset u. U is called a local trivialization of e over u and u is a trivializing neighbour hood for e. One of the main objects associated to vector bundles are their local sections. Chapter 7 vector bundles louisiana state university. Estimates of the amplitude of holonomies by the curvature. Vector bundles in this section we recall some basics on vector bundles. M are vector spaces, the manifolds f in the local trivialization are vector spaces, and for each p. A smooth real vector bundle of rank k over the base manifold m is a manifold e called the total space, together with a smooth surjection e. X such that for every x 2 x, there is an open neighborhood u of x and a homeomorphism also called a local trivialization. I have seen several equivalent characterizations bundles acted upon nicely by the fundamental group of the base, bundles admitting flat connections, etc, and from this i can construct examples of vector bundles that are provably not local systems e. We have developed the notion of a cp vector bundle over xas a certain kind of cp mapping e xthat is roughly speaking a cp varying family of. Bis said to be a real or complex vector bundle corresponding to the typical bre being a real or complex. Fiber bundles are denoted by various combination of components and. We shall require the definition of parallel transport in fiber bundles to satisfy two.