The trivial vector bundle of rank r over m is the product m. A vector bundle for which a trivialization exists is called trivializable. M are vector spaces, the manifolds f in the local trivialization are vector spaces, and for each p. On a connected component of xall bers of a locally trivial vector bundle have the same. We say that a vector bundle is locally trivial, if for every x2xthere is an open neighborhood x2u. M, in other words, it assigns smoothly to each tangent space tpm a kdimensional subspace. 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. They are referred as rank 2 stable vector bundles on the complex projective space p3 8. Vector bundles in algebraic geometry enrique arrondo notes prepared for the first summer school on complex geometry villarrica, chile 79 december 2010 1. U r ntaking p b to b r by a vector space isomorphism for every b2u. A bundle over a manifold is trivial if it is simply the cartesian product of.
Intuitively this is the object we get by gluing at each point p. For this goal, we study the moduli space of 5tuples x,x,z,e. A group acting on it is found and it is shown that it acts in. Vector bundles and connections universiteit utrecht. We will call e the total space, m the base, 1p the fiber over p, and. U is called a local trivialization of e over u and u is a trivializing neighbour hood for. The vector bundle of rank rover xis said to be trivial if there exists a global bundle chart e x. Bring vector space back in picture cliff construct manifolds from given manifolds understand. M s1 which we claim is a non trivial real vector bundle of rank 1 over s1.
Im not sure if your comment is in response to mine. But there are many other interesting vector bundles. If the dimension of the vector space is mthen the bundle is often called an mplane bundle. Orientations on vector bundles universiteit utrecht. Vector bundles and charts if you open any book on di.
Fiber bundles, especially vector bundles, are ubiquitous in mathematics. Property t of vector bundles is in fact stronger than it appears to be. An example of vector bundle which is not trivial but nevertheless stably trivial is ts2 definition 10. Xbe a cp mapping between cp premanifolds with corners x. Wellknown examples of vector bundles are the tangent and cotangent bundles, tmand t m, as well as the tensor bundle tm x2m p. With these in mind, we see that the local triviality condition from the definition of vector bundles says that. Vector bundles in this section we recall some basics on vector bundles. B, onto b, is called a vector bundle of rank kover bif the following holds. Vectors v x2p x are thought of as arrows with their tail at x. 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. It is a real vector bundle of rank n dimm which in general is not trivial. Vector bundles and pullbacks nikolai nowaczyk november 2008 contents. Characteristic classes assign to the each vector bundle a cohomology class of the base space. E f is the tensor product bundle, which has rank rk erk f.
The topology of fiber bundles stanford mathematics. 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. M s1 which we claim is a non trivial real vector bundle of rank 1. Subbundles and quotient bundles motivation example n. Given a space x, we take a real or complex finite dimensional vector space v and. We have put a topology on fesuch that for each local trivialization. Applying this operation on each bre of a vector bundle e. E intersecting each ber of e in a vector subspace, such that the restriction pj e 0 is a. For any smooth manifold m, e m rk is a trivial bundle over m.
S2 of rank 2 for each k2z or a complex vector bundle of rank 1, since g 01. Linear algebra operations on vector bundles motivation. The tangent bundle tm is indeed a vector bundle, of rank dimm. One can recover the ber bundle from this principal bundle by a simple construction of. 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. A rank n vector bundle on a manifold m is a bration.
Its local coordinate charts are obtained by restricting the trivialization. This new vector bundle fe is called the associated vector bundle to e, given by the functor f. A simple example of a non trivial vector bundle is the in. In order to help motivate them, we rst discuss vector bundles, and how they. Cn yang, 1979 in this lecture we introduce the notion of a principal. U ck that is a biholomorphic map of complex manifolds.
E m is a choice of orientation for every fiber ex, such that for any local trivialization. Any example of manifold without global trivialization of. As one learns in an introductory smooth manifolds course, for any smooth man ifold m there is a naturally associated vector bundle. The notion of vector bundle is a basic extension to the geometric domain of the fundamental idea of a vector space. The euler class, the thom class and the chern classes are a few. 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.
A kvector bundle over mof rank kconsists of a bundle e m whose. A vector bundle of rank 1 is usually called a line bundle. 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. A vector bundle for which a trivialization exists is called trivializable accordingly, the local triviality condition in the definition of topological vector bundles def.
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. 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. The local trivialization shows that locally the map. Bundles, classifying spaces and characteristic classes contents. For the trivialization neighborhoods, only one is needed. For instance, to any vector space v we can associate its dual space v. Any example of manifold without global trivialization of tangent bundle. Rnbe a trivialization, and denote e i to be the ith vector in the standard basis of rn. In this case x is also a section of the vector bundle. This construction can also be applied brewise to vector bundles. We have put a topology on fesuch that for each local trivialization ej u u rn as. 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. 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.
Holomorphic vector bundles northwestern university. 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. The definition of a vector bundle shows that any vector bundle is locally trivial. Bis isomorphic to the pullback of a universal vector bundle e0. Often the vector bundles that one encounters do not arise right away as in the. Vector bundle examples trivial bundle moebius bundle tangent bundle cotangent bundle. By definition, a smooth kvector bundle of rank r over a smooth manifold m. We have seen that eis a vector sub bundle of a trivial bundle m rn over 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.
If the original data of clutching functions came from locally trivializations of a bundle. Now we put an inner product on rn, and take the ber f p of f at p2mto be the orthogonal complement of e. Induced bundles and vector bundles in the previous lecture, we have seen how principal gbundles correspond to equivalence classes of cocycles. 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. 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. Several other approaches of vector bundles can be found in 35, 15. Such a trivialization is called a holomorphic trivialization. We have seen that eis a vector subbundle of a trivial bundle m rn over m. 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. The open neighborhood u together with the homeomorphism is called a local trivialization of the vector bundle. Ebe a cp vector bundle map between cp vector bundles over a cp premanifold with corners x. Suppose we have a vector bundle on m, along with a trivialization over an open cover ui. Luckily, there is again a simpler criterion which ensures integrability.
Motivation let x,o be a cp premanifold with corners, 0. In the case there is no ambiguity about the base, we will denote a vector bundle by efor short. In mathematics, an orientation of a real vector bundle is a generalization of an orientation of a vector space. 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. Cohomology and vector bundles university of chicago. A trivialization of e is the choice of such an isomorphism.
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