A kvector bundle over mof rank kconsists of a bundle e m whose. The open neighborhood u together with the homeomorphism is called a local trivialization of the vector bundle. Wellknown examples of vector bundles are the tangent and cotangent bundles, tmand t m, as well as the tensor bundle tm x2m p. The local trivialization shows that locally the map. We have seen that eis a vector sub bundle of a trivial bundle m rn over m. Bundles, classifying spaces and characteristic classes contents. E f is the tensor product bundle, which has rank rk erk f. Subbundles and quotient bundles motivation example n. One can recover the ber bundle from this principal bundle by a simple construction of. We have put a topology on fesuch that for each local trivialization ej u u rn as. U r ntaking p b to b r by a vector space isomorphism for every b2u. Such a trivialization is called a holomorphic trivialization. M is a complex vector bundle together with the structure of a complex manifold on e, such that for any x2mthere exists u3x in mand trivialization. We can also consider the category of all vector bundles over a fixed base space x.
The trivial vector bundle of rank r over m is the product m. On a connected component of xall bers of a locally trivial vector bundle have the same dimension, which is called the rank of this vector bundle. Given a space x, we take a real or complex finite dimensional vector space v and. Applying this operation on each bre of a vector bundle e. B, onto b, is called a vector bundle of rank kover bif the following holds. A vector bundle of rank 1 is usually called a line bundle. Luckily, there is again a simpler criterion which ensures integrability.
Intuitively this is the object we get by gluing at each point p. A trivialization of e is the choice of such an isomorphism. Cn yang, 1979 in this lecture we introduce the notion of a principal. The tangent bundle tm is indeed a vector bundle, of rank dimm. From this perspective the answer is ambiguous, unless e happens to. The notion of vector bundle is a basic extension to the geometric domain of the fundamental idea of a vector space. As we have seen in the previous section, holomorphic vector bundles are complex vector bundles for which the local trivialization has been chosen in such a way that the transition functions are holomorphic. But there are many other interesting vector bundles. Property t of vector bundles is in fact stronger than it appears to be. Mequipped with a connection such that there exists a trivialization in which the covariant derivative has the form above over each open set in the trivialization. Holomorphic vector bundles northwestern university. Induced bundles and vector bundles in the previous lecture, we have seen how principal gbundles correspond to equivalence classes of cocycles.
Bis isomorphic to the pullback of a universal vector bundle e0. This new vector bundle fe is called the associated vector bundle to e, given by the functor f. For any smooth manifold m, e m rk is a trivial bundle over m. Vector bundles and connections universiteit utrecht. 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. In this study, we illustrate an application of vector bundles by prolonging isometric actions from a manifold base manifold to another by using a vector bundle structure. On a connected component of xall bers of a locally trivial vector bundle have the same. A vector bundle for which a trivialization exists is called trivializable. Orientations on vector bundles universiteit utrecht. Motivation let x,o be a cp premanifold with corners, 0. Several other approaches of vector bundles can be found in 35, 15. The euler class, the thom class and the chern classes are a few. Vector bundles and structure groups a vector bundle over a topological space m or with base space m is, essentially, family of vector spaces continuously parametrized by m. The most important example is the tangent bundle, a real vector bundle.
Any example of manifold without global trivialization of. The elements of the fiber over x in a vector bundle can be viewed as abstract vectors in an internal space, with the local trivialization. Vector bundles in algebraic geometry enrique arrondo notes prepared for the first summer school on complex geometry villarrica, chile 79 december 2010 1. Characteristic classes assign to the each vector bundle a cohomology class of the base space.
A vector bundle is said to be stably trivial if it becomes trivial after taking the direct sum with a vector bundle isomorphic to the trivial bundle. Vector bundles and charts if you open any book on di. We have put a topology on fesuch that for each local trivialization. For the trivialization neighborhoods, only one is needed.
In order to help motivate them, we rst discuss vector bundles, and how they. We will call e the total space, m the base, 1p the fiber over p, and. With these in mind, we see that the local triviality condition from the definition of vector bundles says that. Often the vector bundles that one encounters do not arise right away as in the. Xbe a cp mapping between cp premanifolds with corners x. Linear algebra operations on vector bundles motivation. Prolongations of isometric actions to vector bundles. While we accomplish this in a certain sense by showing that any vector bundle e. A vector bundle for which a trivialization exists is called trivializable accordingly, the local triviality condition in the definition of topological vector bundles def.
We prefer to think of sections as \vector valued maps on m, which can be added and multiplied by scalars, and wed like to think of the directional derivative dsxx as something which respects this linear structure. We say that a vector bundle is locally trivial, if for every x2xthere is an open neighborhood x2u. S2 of rank 2 for each k2z or a complex vector bundle of rank 1, since g 01. Ebe a cp vector bundle map between cp vector bundles over a cp premanifold with corners x. A smooth real vector bundle of rank k over the base manifold m is a manifold e. For this goal, we study the moduli space of 5tuples x,x,z,e. Vector bundle examples trivial bundle moebius bundle tangent bundle cotangent bundle.
Vector bundles and pullbacks nikolai nowaczyk november 2008 contents. Cohomology and vector bundles university of chicago. A bundle over a manifold is trivial if it is simply the cartesian product of. For instance, to any vector space v we can associate its dual space v. In this case x is also a section of the vector bundle. Im using the letter m to denote the base space of the vector bundle as a concession to the fact that in most of the applications well be. M s1 which we claim is a non trivial real vector bundle of rank 1 over s1. If mis compact, then for any vector bundle eover m, there exists a vector bundle f over mso that e f is a trivial bundle over m. If the original data of clutching functions came from locally trivializations of a bundle. Fiber bundles, especially vector bundles, are ubiquitous in mathematics. M, in other words, it assigns smoothly to each tangent space tpm a kdimensional subspace. A rank n vector bundle on a manifold m is a bration.
Now we put an inner product on rn, and take the ber f p of f at p2mto be the orthogonal complement of e. This construction can also be applied brewise to vector bundles. Vector bundles in this section we recall some basics on vector bundles. U ck that is a biholomorphic map of complex manifolds. Vectors v x2p x are thought of as arrows with their tail at x. In the case there is no ambiguity about the base, we will denote a vector bundle by efor short. U is called a local trivialization of e over u and u is a trivializing neighbour hood for. E m is a choice of orientation for every fiber ex, such that for any local trivialization.
Cohomology and vector bundles corrin clarkson reu 2008 september 28, 2008 abstract vector bundles are a generalization of the cross product of a topological space with a vector space. Aug 11, 2019 an isomorphism of a rank k vector bundle e over x with the trivial bundle of rank k over x is called a trivialization of e, and e is then said to be trivial or trivializable. An isomorphism of a rank k vector bundle e over x with the trivial bundle of rank k over x is called a trivialization of e, and e is then said to be trivial or trivializable. Its local coordinate charts are obtained by restricting the trivialization. It is a real vector bundle of rank n dimm which in general is not trivial. 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. This is a 2 dimensional vector space with the point xas its zero vector 0 x. By definition, a smooth kvector bundle of rank r over a smooth manifold m. Im not sure if your comment is in response to mine. If the dimension of the vector space is mthen the bundle is often called an mplane bundle. A group acting on it is found and it is shown that it acts in. Any example of manifold without global trivialization of tangent bundle.
Vector bundles of rank 1 is also called the line bundle. The topology of fiber bundles stanford mathematics. Using a partition of unity argument assuming youre working over a sufficiently nice space, or your vector bundle is a pullback of a universal one look up numerable cover for more on this but note that all the answers to this question tacitly assume this, any finite dimensional vector bundle admits a continuous choice of inner product. The vector bundle of rank rover xis said to be trivial if there exists a global bundle chart e x. An example of vector bundle which is not trivial but nevertheless stably trivial is ts2 definition 10. Suppose we have a vector bundle on m, along with a trivialization over an open cover ui. M s1 which we claim is a non trivial real vector bundle of rank 1. A simple example of a non trivial vector bundle is the in. M are vector spaces, the manifolds f in the local trivialization are vector spaces, and for each p. In mathematics, an orientation of a real vector bundle is a generalization of an orientation of a vector space.