What is Bundles: Definition and 57 Discussions

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): to every point x of the space X we associate (or "attach") a vector space V(x) in such a way that these vector spaces fit together to form another space of the same kind as X (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over X.
The simplest example is the case that the family of vector spaces is constant, i.e., there is a fixed vector space V such that V(x) = V for all x in X: in this case there is a copy of V for each x in X and these copies fit together to form the vector bundle X × V over X. Such vector bundles are said to be trivial. A more complicated (and prototypical) class of examples are the tangent bundles of smooth (or differentiable) manifolds: to every point of such a manifold we attach the tangent space to the manifold at that point. Tangent bundles are not, in general, trivial bundles. For example, the tangent bundle of the sphere is non-trivial by the hairy ball theorem. In general, a manifold is said to be parallelizable if, and only if, its tangent bundle is trivial.
Vector bundles are almost always required to be locally trivial, however, which means they are examples of fiber bundles. 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). Complex vector bundles can be viewed as real vector bundles with additional structure. In the following, we focus on real vector bundles in the category of topological spaces.

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  1. Euge

    POTW What is the Relationship Between Fiber Bundles of Spheres?

    Prove that if there is a fiber bundle ##S^k \to S^m \to S^n##, then ##k = n-1## and ##m = 2n-1##.
  2. A

    I Tangent Bundle of Product is diffeomorphic to Product of Tangent Bundles

    My apologies if this question is trivial. I have searched the forum and haven't found an existing answer to this question. I've been working through differential geometry problem sets I found online (associated with MATH 481 at UIUC) and am struggling to show that T(MxN) is diffeomorphic to TM...
  3. K

    I Gauge Theory and Fiber Bundles

    Hopefully, I am in the right forum. I am trying to get an intuitive understanding of how fiber bundles can describe gauge theories. Gauge fields transform in the adjoint representation and can be decomposed as: Wμ = Wμata Gauge field = Gauge group x generators in the adjoint...
  4. K

    I Connections (principal bundles) ....

    I read about connections on principal bundles. I don't have the knowledge nor the time to learn about principal bundles in the first place. Never the less this makes me wonder if such connections are the same as those talked about in the context of tangent vector spaces. Are they the same thing?
  5. Abhishek11235

    A Penrose paragraph on Bundle Cross-section?

    I am reading "Road to Reality" by Rogen Penrose. In chapter 15, Fibre and Gauge Connection ,while going through Clifford Bundle, he says: .""""...Of course, this in itself does not tell us why the Clifford bundle has no continuous cross-sections. To understand this it will be helpful to look at...
  6. Q

    Yang Mills & Fiber Bundles Resource

    Hi everyone, Does anyone know of a good intuitive resource for learning Yang-Mills theory and Fiber Bundles? Ultimately my goal is to gain a geometric understanding of gauge theory generally. I have been studying differential forms and exterior calculus. Thanks!
  7. S

    I Connections on principal bundles

    Hello! I am reading about connections on principal bundles and the book I read introduces the connection one form as ##\omega \in \mathfrak{g} \otimes T^*P##, where ## \mathfrak{g}## is the Lie Algebra associated with the principle bundle P. I am a bit confused about what does this mean...
  8. J

    A Is the Berry connection compatible with the metric?

    Hello, Is the Berry connection compatible with the metric(covariant derivative of metric vanishes) in the same way that the Levi-Civita connection is compatible with the metric(as in Riemannanian Geometry and General Relativity)? Also, does it have torsion? It must either have torsion or not be...
  9. S

    I Understanding Fibre Bundles: A Layman's Guide

    Hello! I am having some troubles understanding fibre bundles and I would be really grateful if someone can explain them to me in layman terms (at least how to visualize them). To begin with, I am not sure what is the fibre bundle, is it the projection function, or the total space (or something...
  10. L

    A Integration along a loop in the base space of U(1) bundles

    Let ##P## be a ##U(1)## principal bundle over base space ##M##. In physics there are phenomenons related to a loop integration in ##M##, such as the Berry's phase ##\gamma = \oint_C A ## where ##C(t)## is a loop in ##M##, and ##A## is the gauge potential (pull back of connection one-form of...
  11. I

    Quantum QFT: groups, effective action, fiber bundles, anomalies, EFT

    Hi, I am looking for textbooks in QFT. I studied QFT using Peskin And Schroeder + two year master's degree QFT programme. I want to know about the next items: 1) Lorentz group and Lie group (precise adjectives, group representation and connection between fields and spins from the standpoint of...
  12. N

    A Gauge Theory: Principal G Bundles

    I've been studying TQFT and gauge theory. Dijkgraaf-Witten theory in particular. One learns that a topological field theory applied to a manifold outputs the number of principal G bundles of a manifold. My question is for the Physicists in the room, why do you want to know the number of...
  13. sehrish shakir

    Pressure drop in rod bundles without spacer grid

    Hello! I am studying the mentioned topic I am confused in calculating the average value of friction factor in sub channels ... please help ?? thank you in advance
  14. fresh_42

    Does anybody know an introduction to Lie Algebra Bundles

    A textbook or even better some openly available pdf would be preferred. I have no idea where to start a search and I normally prefer recommendations over just picking any book from some publisher. Thank you.
  15. 1

    Two questions for vector bundles

    1 Let A → N, B → N be two vector bundles over a manifold N. How to show that there is a vector bundle Hom(A, B) whose fiber above x ∈ N is Hom(A, B)x := Hom(Ax, Bx)? 2 Let A → N, B → N be two vector bundles over a manifold N. Let C∞(A, B) denote the space of maps of vector bundles from A to B...
  16. W

    Restricting Base of Bundle. Which Properties are Preserved?

    Hi All, Please let me set things up before the actual question: I have a Lefschetz fibration ## f: W^4 \rightarrow S^2 ## , where ## W^4## is a closed, oriented manifold, and ## S^2## is the 2-disk. This f is a smooth map with finitely-many singularities {## x_1, x_2,..,x_n ##}, so that we have...
  17. camilus

    Multiplicity free fibers in maps between vector bundles

    For a map between vector bundles (which commute with a certain Lie groups like Sl2R or GL2R), what does it mean exactly for a fiber to be multiplicity free? Eplanations would be good, but examples would be even better. Thanks in advance, Gauss bless you! CM
  18. JonnyMaddox

    What is the Geometric Interpretation of Principal Bundles with Lie Group Fibres?

    I'm trying to learn how to think about principal bundles where the fibre is a lie group with local trivialization ϕ^{-1}_i:π(U_i)→U_i×G . For example ϕ^{-1}_i:π(S^2)→S^2×U(1) (if that makes sense) . But I don't know how to think of this (and other products with lie groups like that)...
  19. ChrisVer

    Which Book is Best for Studying Fiber Bundles in Yang-Mills Gauge Theories?

    I would like a good source from which I can study fiber bundles (mainly their application in Yang-Mills gauge theories, but also in differential geometry)... I tried to study them from the advanced differential geometry (note)book of 1 of my professors but it was a mess and it confused me even...
  20. C

    Is Fibre Bundles Cartan's Generalization of Klein's Erlagen Program?

    As I understand it, Felix Klein sought to classify geometries with respect to what groups G that respected the structure of the given space X. Lately i read in an article on "the history of connections" by Freeman Kamielle that Cartan wished to generalize this notion. Is it correct to think of...
  21. L

    Arahnov - Bohm effect and U(1) bundles

    I read in a differential geometry paper that Maxwell's equations can be formulated in terms of a connection on a Hermitian line bundle on Minkowski space. I understand the derivation of the field strength 2 form,the proof that Maxwell's equations say that its exterior derivative is zero and its...
  22. L

    On local trivializations and transition functions of fibre bundles

    Hi everyone! I would like to ask you some clarifications on an explicit example of local trivializations and transition functions of fibre bundles: namely on the [-1,1]\hookrightarrow E\rightarrow S^1 bundle (which I guess is the simplest possible example). Following Nakahara (chapter 9...
  23. Spinnor

    Fiber bundles and the fiber for Electromagnetism.

    The circle is the fiber over Minkowski spacetime for electromagnetism? I want to make connection to the classical vector potential via some " picture" involving this circle. Does the following come close? Can I consider a 3 dimensional surface in C_1XMinkowski space that at a given slice of...
  24. B

    Tangent Bundles, T(MxN) is Diffeomorphic to TM x TN

    Homework Statement If M and N are smooth manifolds, then T(MxN) is diffeomorphic to TM x TN Homework Equations The Attempt at a Solution So I'm here let ((p,q),v) \in T(MxN) then p \in M and q \in N and v \in T(p,q)(MxN). so T(p,q)(MxN) v = \sum_{i=1}^{m+n}...
  25. L

    A Question about circle bundles

    This question asks whether every circle bundle comes from a 2 plane bundle. Paracompact space please - preferably a manifold. By circle bundle I mean the usual thing, a fiber bundle with fiber, a circle, that is locally a product bundle. The transition functions lie in some group of...
  26. Jim Kata

    Twistor question the link holomorphic vector bundles and anti self dual guage

    Hi, I was working through a Twistor paper and it was explaining the link between holomorphic vector bundles and anti self dual gauges and it had an equation like this, for electro-magnetism. \lambda^a \lambda^b(\frac{\partial A_{b\dot{b}}}{\partial x^{a\dot{a}}}-\frac{\partial...
  27. D

    Why Can Some Line Bundles Have Nonzero Sections While Others Cannot?

    Hey guys, I am confused about the concept of sections of vector bundles. Mobius band is nontrivial line bundle over circle so we can not find any nonzero section from circle to mobius strip. However a plane which is twisted once is a trivial line bundle over line. That means there is a nonzero...
  28. Matterwave

    Exploring Frame Bundles on Manifolds

    Ok, so I don't have much of an intuition for frame bundles, so I have some basic questions. A frame bundle over a manifold M is a principle bundle who's fibers are the sets of ordered bases for the vector fields on M right. 1) This means that any point in the fiber (say, over a point m in M)...
  29. H

    Principal bundles with connections

    Suppose (P,M,\pi,G) is a G-principal bundle. With this I mean a locally trivial fibration (G acts freely on P) over M=P/G with total space P and typical fibre G, as well as a differentiable surjective submersion \pi\colon P\to M. In this case M is nearly a manifold, but may be non-Hausdorff...
  30. L

    Fibre bundles for describing gauge invariance

    Hello all ! My question: Does fibre bundles are necessary for describing gauge invariance in electromagnetic case? Or fibre bundles uses only for describing gauge invariance in cases of weak, electroweak and strong interactions? Thanks
  31. K

    Methods for computing partial-costs of product bundles?

    This might go into stats, I'm not sure. But I'll throw it out there. You are at the grocery store and they have two product bundles: Four bananas and three limes for $10. Two grapefruits and five limes for $12. You want to come up with a way to compute the average cost of a lime, the...
  32. B

    Def./Examples of Spin Structure Re Trivial Bundles?

    Hi, All: I am reading a paper in which , if I understood well, a spin structure in a manifold M is equivalent to M admitting a trivialization of the tg.bundle over the 1-skeleton of M ( I guess M is assumed to be "nice-enough" so that it is a simplicial complex ) so that...
  33. L

    Invariant Polynomials on complexified bundles with connection

    I would like to know if the following correct. Suppose I start with a connection on a real vector bundle and extend it to the complexification of the bundle. The curvature forms of the complexification seem to be the same as curvature 2 forms of the real bundle. From this it seems that the...
  34. L

    Vectyor bundles with all zero characteristic classes

    Is there an example of a real vector bundle over a compact smooth manifold with all zero characteristic classes (Euler class,Stiefel-Whitney classes and Pontryagin classes) that is non-trivial?
  35. T

    How Does Light Behave Before Passing Through a Slit in Diffraction Experiments?

    I have a question regarding light bundles and the diffraction of waves. I've been trying to wrap my head around the processes that govern how diffraction works and it all seems to make sense to me regarding water waves and sound. If I just apply Huygens' principle that every point in a wave is...
  36. C

    Conductor Bundles: Reduce Impedance Per Meter

    http://www.kabculus.com/capacitance-and-inductance-matrices/node7.html this short page describes conductor bundles, which are power transmission lines hung parallel to each other. i think the page is trying to explain why, when they are hung that way that they reduce impedance per meter, but i...
  37. B

    Classifying Bundles; Fixed Base and Fiber

    Hi, everyone: I am trying to find a result for the number of bundles (up to bundle iso.) over a fixed base and fixed fiber. For example, for B=S<sup>1</sup> , and fiber I=[0,1] I think that there are two; the cylinder and the Mobius strip. I think that the reason there are...
  38. M

    Global Section of Fiber Bundles with Contractible Fiber

    Is it true in complete generality that every fiber bundle with contractible fiber have a global section? Or do some sort of restrictions on the bundle need to be made? I ran across a mention of this fact in Guillemin and Sternberg's "Supersymmetry..." and I'm not sure how to prove it.
  39. M

    Exploring Gauge Bundle Breaking in Yang-Mills Theory

    Hello, suppose you start with Yang Mills theory with some gauge group G, for example SU(5). Then you turn on a gauge bundle, say a U(1) bundle, and the group breaks down. I know that from hearsay but I wonder how would you describe that explicitly in formulas? meha
  40. B

    Which Book on Fiber Bundles is Best for Beginners with an Intuitive Approach?

    Becuase of geometric phase,I'm looking for a good book on fiber bundles, with a minimum of prerequistes and that takes a more intutive rather than formal approach.I am reading a book called modern differential geometry for physicists. It is a good book but sometimes abstract. I know about...
  41. M

    Trying to get my head around tangent bundles

    Hello, Say you have a function f on the domain R^n, and an integral transform P which integrates f over all possible straight lines in R^n. I am lead to believe that the range of this is R^(2n), or a tangent bundle, which I am having MASSIVE problems visualising! Am I right in saying the...
  42. W

    SO(n) actions on vector bundles

    An oriented surface with a Riemannian metric has a natural action of the unit circle on its tangent bundle. Rotate the tangent vector through the angle theta in the positively direction. Is there a natural action of SO(n) on the tangent bundle of an oriented Riemannian n-manifold? Same...
  43. B

    How Can a Function Be a Submersion on Manifolds Without Forming a Fiber Bundle?

    How would one go about to construct a function on (smooth) manifolds that is a submersion without being (the projection map of) a fiber bundle?
  44. E

    What mistake did I make in proving that all tangent bundles are trivial?

    In trying to understand why not all tangent bundles are trivial, I've attempted to prove that they are all trivial and see where things go wrong. Unfortunately, I finished the proof and cannot find my mistake. Here it is: Let M be an n-manifold with coordinate charts (U_\alpha...
  45. Z

    Classification of Vector Bundles over Spheres

    This is a question from Hirsch's Differential Topology book: show that there is a bijective correspondence between K^k(S^n) \leftrightarrow \pi_{n-1}(GL(k)) , where K^k(S^n) denotes the isomorphism classes of rank k vector bundles over the sphere. The basic idea is that any vector bundle...
  46. W

    Bundles and global sections, triviality.

    Hi: I am trying to understand more geometrically the relation between triviality of bundles and existence of global sections. This is what I have for now. Please comment/critique: Let p:E-->B be a fiber bundle : consider E embedded in B as the 0 section. Then...
  47. haushofer

    Exterior derivatives on fibre bundles

    Hi, I have a small question about exterior derivatives d on defined on principal bundles P. We have the Ehresmann connection on a principal bundle P, represented by a Lie-algebra valued one-form omega. We can use the section sigma to pull this one-form back to our basemanifold, where the...
  48. haushofer

    Elementary questions about fibre bundles

    Hi, I'm a little stuck on Nakahara's treatment about fibre bundles. I hope someone can give me a clear answer on this; they are quite elementary questions, I guess. We have: * A principal bundle P(M,G) * A fibre G_{p} at p= \pi(u) Then the vertical subspace V_{u}P is defined as a...
  49. T

    Calculate the coefficient of friction between the two bundles of candy

    Homework Statement Mrs. K has just received a large shipment of candy at the local Blockbuster. Wanting to get out of the store as fast as possible (so that she can return to her loving husband) she uses the shrink wrap machine to bundle the candy into two blocks, one of 8 kg and the other...
  50. M

    Are vector bundles principal bundles?

    As far as I know, principal bundle is a fiber bundle with a fiber beeing a principal homogeneous space (or a topological group). According this definition vector bundle is a special principal bundle, because vector space with vector addition as group operation is a topological group. But I feel...