# Questions tagged [fiber-bundles]

For questions about fiber bundle, which is a space that is locally a product space, but globally may have a different topological structure.

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### Why is a PDE a submanifold (and not just a subset)?

I struggle a bit with understanding the idea behind the definition of a PDE on a fibred manifold. Let $\pi: E \to M$ be a smooth locally trivial fibre bundle. In Gromovs words a partial differential ...
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### Connection in fibre bundle from discontinuous group action

I am trying to understand connections in fibre bundles. I thought of the following problem: Let $\Gamma$ be the discrete group generated by \begin{pmatrix} 1 & 3 & 0 \\ 0 & 1 & 0 \\ ...
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### How to classify principal bundles over a 2 dimensional surface?

I just want to how much people know about this at the moment? I thought this is elementary and may be of execrise level, but a quick google search showed serious papers written on this subject (like ...
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### Does a left group action on a principal bundle induce an action on associated vector bundles?

Let $G\hookrightarrow P\xrightarrow{\pi}M$ be a principal $G$-bundle with right action $\cdot$ and suppose we are also given a left action $\rho: U\times P\rightarrow P$ of some group $U$ on $P$. ...
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### Metric on total space, connection and Hopf fibration

As is well known that the three sphere $S^3$ may be viewed as a $S^1$ bundle over base space $S^2$. And it is more interesting, but likewise confusing to me that the metric on $S^3$ may be written as: ...
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### Sections of endomorphisms of a vector bundle

The following could be rather silly question but I haven'd found it stated explicitly; from the other side, it seems to me, that this fact is used often without comments. The problem is the ...
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### Covariant derivative on the base space

The basic definition of a covariant derivative for a Lie algebra valued n-form $\alpha \in \Omega^n(P)\otimes T_eG$ with $P$ a principle bundle with base space a manifold $M$, and $T_eG$ the Lie ...
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### Are most physics books wrong about the covariant derivative and connection?

I have always read in many physics books that a valid way of intuitively introducing the covariant derivative and the connection was the following: (example in GR but same thing for gauge theories) ...
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### Gysin sequence for the sphere bundle $B[O_a \times O_B]^+ \to BO_a \times BO_b$?

I have the sphere bundle $S^0 \hookrightarrow B[O_a \times O_b]^+ \stackrel{p}{\to} BO_a \times BO_b$ that can be thought of like: $B[O_a \times O_b]^+$ as the set of tuples of vector spaces $E^a$ ...
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### What is the geometric meaning of powers of the first Stiefel-Whitney class?

If $M$ is an $n$-dimensional manifold, I know $\\w_1^n(M)$ is a Stiefel-Whitney number of the manifold, but what does its vanishing geometrically mean? More generally, does the Stiefel-Whitney height ...
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### Identifying sheaves of sections of fiber bundles

Given a topological space $X$, there's a fundamental category equivalence between local homeomorphisms to $X$ and sheaves of sets over $X$. One direction takes a local homeomorphism to its sheaf of ...
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### Problem with equivalent definition of a integrable $G$-structure

I'm reading Kobayashi's book Transformation Groups in Differential Geometry and I don't understand a thing at page 2. It this proposition: My problem is that I don't understand the converse of this ...
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### Mapping Tori and Monodromy

I have a question regarding the right setup for mapping tori. Let me give the definitions that I use first. Let $I$ denote the interval $[0,2\pi]$ and let $I^*$ denote the quotient of $I$ by the ...
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### Why is a Dirac operator involutive only if the curvature and torsion are in the image of the kernel of the symbol under exterior multiplication?

I am trying to understand the classic paper by Atiyah, Hitchin, and Singer https://www.jstor.org/stable/79638 and I'm getting stuck on part of the proof of proposition 3.1. The proposition is ...
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### Roots of a canonical line bundle on a compact Riemann surface

Suppose we have a compact Riemann surface $X$ of genus $g$. Let $K$ denote the canonical line bundle on $X$, it's well known that $deg\ K=2g-2$. A square root of $K$ by definition is a holomorphic ...
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### Computing Stiefel Whitney classes

I am computing the cohomology of $BO(2) \times B0(3)$ and I would like to identify the Stiefiel Whitney classes of this space. For instance, I know $H^*(BO(2);\mathbb{Z}/2)\cong \mathbb{Z}/2[w_1,w_2]$...
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I would like to understand better this point. In generalized complex geometry the generalized bundle $E$ is defined as a non-trivial fibration of the cotangent bundle $T^*M$ over the tangent bundle $... 0answers 247 views ### Correspondence between line bundles and$U(1)$-bundles: a mistake from the physicists? I am reading a paper written by physicists and they say the following: Let$(L,h,\nabla)$be an holomorphic line bundle equipped with a Hermitian metric$h$and Chern connection$\nabla$. If$e^{f_{...
Let $P\to M$ a principal fibre bundle with fibre $G$, and let $A\in \Omega^{1}(P)\otimes\mathfrak{g}$ be a connection on $P$, where $\mathfrak{g}$ is the Lie algebra of $G$. Associated to every ...