Non trivial vector bundle pdf

As an application, we prove that a uniform vector bundle on a classical grassmannian of low rank is splitting. More explicitly, suppose x x denotes spacetime and f f denotes some space that one wants to map into. By peterweyl theorem, any gvector bundle over x is a summand of a trivial gbundle. Pdf splitting criteria for vector bundles induced by. The interest of the elliptic curve lies in the fact that it provides the first nontrivial case, grothendieck 6 having shown that for a rational curve every vector bundle is a direct sum of line. Pdf locally trivial quantum vector bundles and associated.

The narasimhanseshadri theorem tells us that every s. In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space x for example x could be a topological space, a manifold, or an algebraic variety. M s1 which we claim is a non trivial real vector bundle of rank 1 over s1. But for the generic semistable g c bundle on a torus, sequivalence is the same as isomorphism. Once you see the notion of vector bundle, next thing you want to see are examples of non trivial vector bundles. This does not happen for the annulus, which is an orientable vector bundle. A toric vector bundle on a complete toric variety is nef or ample if and. When the vector bundle in question is the tangent bundle, the unit sphere bundle is known as the unit tangent bundle. Let e be the space of tangent vectors to two dimensional sphere s2 embed. Langlands program, including its geometric reformulation, addressed primarily to physi. One requires that locally, on small enough patches of the base space x x, the variation of the fibers is constant up to isomorphism one says the vector bundle is locally trivial, but the key point of vector bundles is that there may be non trivial structure in how the collection of vector spaces globally glues together. Affine varieties with stably trivial algebraic vector bundles.

If you know the characteristic classes of your vector bundles, you can compute the characteristic classes of their direct sum, so if any of those dont vanish then the direct sum is nontrivial. Why is the tangent bundle to s1 and s3 trivial but not the. It is a real vector bundle of rank n dimm which in general is not trivial. A bundle over a manifold is trivial if it is simply the cartesian product of. You could look at the bundle of frames of the vector bundle. A gbundle is trivial if it is trivial as a vector bundle. The main point is that up to isomorphism, every real vector bundle over the circle is either trivial, or the whitney sum of a trivial bundle with the mobius bundle. Nontrivial torus equivariant vector bundles of rank three. M s1 which we claim is a non trivial real vector bundle of rank 1. In general you can prove vector bundles are nontrivial by the theory of characteristic classes. Suppose that decomposes into where is a line bundle and is an indecomposable vector bundle. If it were trivial, there would be a vector field with no zeros.

A simple example of a non trivial vector bundle is the in. Nontrivial torus equivariant vector bundles of rank three on p. S 3 and s 1 are lie groups, and any lie group has trivial tangent bundle. Every vector bundle on p splits as a sum of line bundles oa 1o a r, for some integers a 1a r. E the module of differentiable sections of a vector bundle e.

From integral curves to integral manifolds integral. If sis the trivial real vector bundle of rank one over m, then an order one partial differential operator is little more than a vector. You get the transition functions directly from the vector bundle. Example 2 304 let m be a differentiable manifold of. It must be emphasized that by far the deepest input in the preceding results is the whitney embedding theorem in the non compact case it is applied to the total space of vector bundles, which are never compact except possibly when bundle has rank 0. Pdf affine varieties with stably trivial algebraic vector. We define locally trivial quantum vector bundles qvb and qvb associated to locally trivial quantum principal fibre bundles. Eudml vector bundles with trivial determinant and second. If mis compact, then any vector bundle eover mis isomorphic to a subbundle of a trivial vector bundle over m. So a non zero section is vector field where the tangent vectors are all non zero.

But there are many other interesting vector bundles. Whenever possible examples were given to illustrate the role of vector bundles. Vector bundles are almost always required to be locally trivial, however, which means they are examples of fiber bundles. Vector bundles or at least, tangent bundles appear quite naturally when one tries to work with differential.

Tensor product of gvector bundle makes k gpxqinto a ring. The primary purpose of this paper is the study of algebraic vector bundles over an elliptic curve defined over an algebraically closed field k. Moreover, there are only few examples of such v ector bundles in general case see, 14, 10. O and the non trivial extension of o by o are sequivalent. The narasimhanseshadri theorem tells us that every sequivalence class contains up to isomorphism a unique representative that admits a. Vector bundles and connections universiteit utrecht.

In particular they always have a zerosection that maps every point in ghto the origin of the vector space that is the ber above that point. Sketch we can give a sketch of a proof of this as follows. We also prove that a vector bundle on a homogenous space is trivial if and only if the restriction of the vector bundle to every schubert line is trivial. In the case there is no ambiguity about the base, we will denote a vector bundle by efor short. Thanks for contributing an answer to mathematics stack exchange. A smooth map x eis said to be a smooth section of the bundle e,x. For example, if g is discrete then a principal gbundle with connected total space is the same thing as a regular. Pmhas no nontrivial nitedimensional representations. As an application, we prove that a vector bundle on a smooth toric variety x, whose anticanonical bundle has stable base locus of codimension at least three, is trivial when its restrictions to. To specify a locally trivial bundle it is not necessary to be. In general, all manifolds and groups will be assumed to be con nected.

Vector bundles of rank 1 is also called the line bundle. In class we discussed the notion of integral manifolds for e in m as well as maximal ones, essentially as a generalization of. This is just a statement of the famous \hairy ball theorem. O and the nontrivial extension of o by o are sequivalent. Nontrivial torus equivariant vector bundles of rank three on p2 37. Pdf one bundle is trivial, this phenomenon has been given a name. Definition for x x a topological space, then a topological vector bundle e x e \to x over a topological field k k is called trivial if its total space is the product topological space. A vector bundle of rank 1 is usually called a line bundle. The lie group structure on s 3 comes from identifying it with the unit quaternions the tangent bundle of s 2 is nontrivial by the hairy balls theorem. As another example, w e ha the dual bund le e t giv en b y the con tragredien t represen tation 1. E b, over a topological space, b, is a continuous map. In the case of s n and its tangent bundle t which is a rank n vector bundle, a section is a continuous map from s n into t that takes each point to a tangent vector of that point. There exist v ector bundles of an y degree and rank.

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. Let for the data be an equivariant vector bundle of rank three on. There exists a differential structure on the associated vector bundle. If m is a smooth manifold and k is a nonnegative integer, then f. If each 1x is one dimensional vector space, it is called a line bundle. Pmhas no non trivial nitedimensional representations. Alternatively, you can think of it as a vector field on s n. Etale triviality of finite vector bundles over compact. Triviality of vector bundles and families of rational curves 2 3. The tangent bundle tm of a smooth manifold m has already been discussed in. If mis compact, then any vector bundle eover mis isomorphic to a sub bundle of a trivial vector bundle over m.

In the case a, more specific results can be obtained. Non trivial torus equivariant vector bundles of rank three on p2 37. Is every real vector bundle over the circle necessarily. Lectures on operator ktheory and the atiyahsinger index. A holomorphic vector bundle e over m would be called etale trivial if there is a finite connected etale covering y m such that the vector bundle. Homogeneous vector bundles raoul bott the annals of. Pdf vector bundles and their applications researchgate. A splitting criterion of vector bundles on homogenous spaces 9 5. Is every real vector bundle over the circle necessarily trivial. Pdf vector bundle of rank 2k, but a real vector bundle of rank 2kneed not in general admit a complex structure. A ricci soliton is a riemannian manifold m,g together with smooth vector. Vector bundles usc dana and david dornsife college of.

In particular if each 1x is a vector space which changes smoothly, it is called a vector bundle. But avoid asking for help, clarification, or responding to other answers. R, where lxg denotes the lie derivative of the metric g with respect to x. I2, thus concluding that the sum of two nontrivial bundles can be trivial. Chapter 7 vector bundles louisiana state university. The simplest line bundle is the cartesian product m r which assigns to each point a copy of the real numbers.

A real vector bundle over the circle is trivial if and only if it is orientable. If the dimension of the vector space is mthen the bundle is often called an mplane bundle. U ck that is a biholomorphic map of complex manifolds. The dimension of the vector spaces is called rank of the bundle and vector bundles of rank one are called line bundles. It is called the trivial rankk real vector bundle over m. It is called nontrivial if x is not a killing vector. The vector bundle of rank rover xis said to be trivial if there exists a global bundle chart e x. Thus the book contains sections on locally trivial bundles, and on the simplest. The answer is yes if m is compact, and the proof is similar to the proof of the \simple whitney embedding theorem in lecture 9. It must be emphasized that by far the deepest input in the preceding results is the whitney embedding theorem in the noncompact case it is applied to the total space of vector bundles, which are never compact except possibly when bundle has rank 0. How to prove that a vector bundle is trivial iff there are. A kvector bundle over mof rank kconsists of a bundle e m whose.

Non trivial vector bundle over nonparacompact contractible space. For a discrete structure group gauge group every bundle is flat, and in this case nontrivial bundles and nontrivial monodromy come down to essentially the same thing see also at local system. Then, x4 is a1contractible and has a nontrivial rank 2 vector bundle restrict a nontrivial bundle from q 4. Also, the vector spaces are usually required to be over the real or complex numbers, in which case the vector bundle is said to be a real or complex vector bundle respectively. This also deals with the nonparacompact nonhausdorff case, using a suitable wedge sum with the trivial bundle over a nonparacompact contractible space, so only the hausdorff nonparacompact case remains without an example. For x x a suitable space then a vector bundle over x x is called trivial if there is no twist in how the fibers vary over it. Every vector eld on the 2sphere, s2, has at least one zero. If e is a homogeneous vector bundle3 over x subject to a, i. The interest of the elliptic curve lies in the fact that it provides the first non trivial case, grothendieck 6 having shown that for a rational curve every vector bundle is a direct sum of line. How to prove that a vector bundle is trivial iff there are n global sections that form a basis on each fiber. If vis a vector space over a field k and if v c vis a subset, we denote.

This indicates the importance of introducing operators acting on the sections of nontrivial bundles. 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. In class we discussed the notion of integral manifolds for e in m as well as maximal ones, essentially as a generalization of the theory of integral curves for vector. How to prove that a vector bundle is trivial iff there are n.

O r 1 c l with a line bundle l has rank r and degree deg. This contradiction shows that the mobius band is not isomorphic to a trivial bundle. Apr 05, 2014 you could look at the bundle of frames of the vector bundle. The analogous degenerate conic bundles over p2 are the artinmumford. Then, x4 is a1contractible and has a non trivial rank 2 vector bundle restrict a non trivial bundle from q 4. Using this result and the theory of chern classes of vector bundles, we give a general criterion for a uniform vector bundle on a homogenous space to be splitting. The nontrivial part of this is showing that 1 is continuous. Aravind asok usc vector bundles and a 1 homotopy theory. How to prove a vector bundle is nontrivial using transition. Analele stiintifice ale universitatii ovidius constanta.

For any smooth manifold m, e m rk is a trivial bundle over m. In this case it is known that x is a projective algebraic variety lo, 3. On the differential geometry of homogeneous vector bundles. Here, i want to collect such examples with justification of one or two lines saying why this vector bundle is non trivial. Integral manifolds for trivial line bundles let m be a c. Triviality of vector bundles on homogenous spaces 6 4.