What is Fiber bundle: Definition and 23 Discussions

In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is locally a product space, but globally may have a different topological structure. Specifically, the similarity between a space


{\displaystyle E}
and a product space


{\displaystyle B\times F}
is defined using a continuous surjective map,



{\displaystyle \pi \colon E\to B}
, that in small regions of E behaves just like a projection from corresponding regions of


{\displaystyle B\times F}


{\displaystyle B}
. The map


{\displaystyle \pi }
, called the projection or submersion of the bundle, is regarded as part of the structure of the bundle. The space


{\displaystyle E}
is known as the total space of the fiber bundle,


{\displaystyle B}
as the base space, and


{\displaystyle F}
the fiber.
In the trivial case,


{\displaystyle E}
is just


{\displaystyle B\times F}
, and the map π is just the projection from the product space to the first factor. This is called a trivial bundle. Examples of non-trivial fiber bundles include the Möbius strip and Klein bottle, as well as nontrivial covering spaces. Fiber bundles such as the tangent bundle of a manifold and more general vector bundles play an important role in differential geometry and differential topology, as do principal bundles.
Mappings between total spaces of fiber bundles that "commute" with the projection maps are known as bundle maps, and the class of fiber bundles forms a category with respect to such mappings. A bundle map from the base space itself (with the identity mapping as projection) to


{\displaystyle E}
is called a section of


{\displaystyle E}
. Fiber bundles can be specialized in a number of ways, the most common of which is requiring that the transition maps between the local trivial patches lie in a certain topological group, known as the structure group, acting on the fiber


{\displaystyle F}

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  1. cianfa72

    I Fiber bundle homeomorphism with the fiber

    Hi, in the definition of fiber bundle there is a continuous onto map ##\pi## from the total space ##E## into the base space ##B##. Then there are local trivialization maps ##\varphi: \pi^{-1}(U) \rightarrow U \times F## where the open set ##U## in the base space is the trivializing neighborhood...
  2. P

    A On the group notation of the 1975 Wu-Yang paper

    In the 1975 Wu–Yang paper on electromagnetism=fiber bundle theory Table 1: https://journals.aps.org/prd/pdf/10.1103/PhysRevD.12.3845 Wu & Yang use the notation ##\mathrm{U}_1(1)## for the bundle of electromagnetism and ##\mathrm{SU}_2## for the isospin gauge field. I am unfamiliar with this...
  3. V

    A How to derive Quantum Mechanics in curved physical space?

    I am following [this YouTube lecture by Schuller][1] where he finds the appropriate formalism for the quantum mechanics in the physical curved space. Everything makes sense to me but at the very end I see that we find the pull backed connection one-form on the base manifold. He says to the end...
  4. Thytanium

    A Special Relativity Fiber Bundle: R^3 & Lorentz Metric

    In the space-time of special relativity considered as fiber bundle, could it be stated that the base space is time and the fibers are space ##R^3## related to each other by the Lorentz metric as a connection and in this case would there be parallelism, and in this case: how would this fiber...
  5. O

    A On the relationship between Chern number and zeros of a section

    Greetings. I still struggle a little with the mathematics involved in the description of gauge theories in terms of fiber bundles, so please pardon and correct me if you find conceptual errors anywhere in this question. I would like to understand the connection (when it exists) between the...
  6. cianfa72

    I How Do Transition Functions Define a Möbius Strip in Fiber Bundles?

    Hi, starting from this (old) thread I'm a bit confused about the following: the transition function ##ϕ_{12}(b)## is defined just on the intersection ##U_1\cap U_2## and as said in that thread it actually amounts to the 'instructions' to glue together the two charts to obtain the Möbius...
  7. G

    I Visualizing the cotangent space to a sphere

    Is it correct to say that: the cotangent is given by the gradients (*) to all the curves passing through a point and it actually spans the same tangent space to a point of a sphere? If you visualize them as geometric planes (**), the cotangent and the tangent spaces are more than isomorphic...
  8. cianfa72

    I Galilean spacetime as a fiber bundle

    Hi, reading the book "The Road to Reality" by Roger Penrose I was a bit confused about the notion of Galilean spacetime as fiber bundle (section 17.2). As explained there, each fiber over absolute time ##t## is a copy of ##\mathbf E^3## (an instance of it over each ##t##), there exist no...
  9. cianfa72

    I Product Space vs Fiber Bundle: Understanding the Difference

    Hi, I'm not a really mathematician...I've a doubt about the difference between a trivial example of fiber bundle and the cartesian product space. Consider the product space ## B \times F ## : from sources I read it is an example of trivial fiber bundle with ##B## as base space and ##F## the...
  10. Geofleur

    A Pathological example of a Principle Fiber Bundle?

    In Hassani's Mathematical Physics, a principal fiber bundle is defined as shown below. I wanted to see if there is a way to view a tangent bundle as a PFB, even if the resulting structure would have to be globally trivial, so I came up with this idea: Let ##P = {\rm I\!R} \times {\rm l\!R}##...
  11. 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...
  12. G

    A Question About Elementary Fiber Bundle Example

    My question regards the classic example of trivial vs non-trivial fiber bundles, the 'cylinder' and the 'Mobius band.' I'm using Nakahara's 'Geometry, Topology, and Physics', specifically Example 9.1. Here's the text: Then he says (and this is the bit where he loses me): My questions are: i)...
  13. 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!
  14. L

    A Can I find a smooth vector field on the patches of a torus?

    I am looks at problems that use the line integrals ##\frac{i}{{2\pi }}\oint_C A ## over a closed loop to evaluate the Chern number ##\frac{i}{{2\pi }}\int_T F ## of a U(1) bundle on a torus . I am looking at two literatures, in the first one the torus is divided like this then the Chern number...
  15. 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...
  16. I

    How to calculate attenuation for optical fiber bundle?

    Hi, If I understood it correctly, the coefficient of attenuation for a single optical fiber, is alpha = (10/L)*log(P(0)/P(L)). Assuming if I knew the properties of the optical fiber and the amount of optical fiber in the bundle, then the total attenuation for the bundle is approximately number...
  17. O

    A Tangent Bundle: Why It's Important

    Good Morning All: I am now understanding a bit -- just a bit: still struggling - about the tangent bundle. But I have no idea WHY this is important. As I understand, at every point on a manifold (or, more appropriately: at the coordinates placed on a manifold by a mapping), we study the union...
  18. Padrepapp

    Coupling Xe Arc Lamp into Fiber Bundle

    Hey, we are trying to couple the light of a 75 W Xe Arc Lamp (Hamamatsu L2194) into a 800um(0,8mm) diameter fiber bundle (7 fibers). Now we have 2 plano convex lenses (25mm diameter, 30mm EFL, edmund serial #45-364), the first for collimating the second for focusing onto the fiber. We are...
  19. Matterwave

    Understanding the Inverse of a Fiber Bundle Projection Map

    Hey guys, I've often seen in the definition of a Fiber bundle a projection map \pi: E\rightarrow B where E is the fiber bundle and B is the base manifold. This projection is used to project each individual fiber to its base point on the base manifold. I then see a lot of references to...
  20. K

    On transition functions of fiber bundle

    I don't understand why the constructed fiber bundle E have g_{\alpha \beta} as its transition function. The problem is in the pdf file,thank you!
  21. Rasalhague

    Fiber Bundle Basics: An Overview of Penrose's Twisted Bundles

    In The Road to Reality, § 15.2, Roger Penrose introduces the concept of a fibre bundle. Here I'll modify his notation to that of Wikipedia and other sources, so that B stands for base space (rather than the bundle), F for fibre, and E for the total space, which I think Penrose calls the (fibre)...
  22. P

    Transmitting Images through a Coherent Fiber Bundle

    A coherent fiber bundle is very useful in transmitting images. Let's say we have a gaussian beam from a laser as our source. Now the acceptance angle(2*alpha) is determined from the location of the beam waist, right? The light ray will then go down the fiber core with the use of a converging...
  23. J

    Foliation, fibration, fiber bundle?

    what's the difference among those three objects? Any body can give me some examples? Thanks