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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.

In mathematics, 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.

Specifically, the similarity between a space $E$ and a product space $B × F$ is defined using a continuous surjective map: $\pi \colon E \to B$ that in small regions of $E$ behaves just like a projection from corresponding regions of $B × F$ to $B$. The map $π$, called the projection or submersion of the bundle, is regarded as part of the structure of the bundle. The space $E$ is known as the total space of the fiber bundle, $B$ as the base space, and $F$ the fiber.

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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 relation of order $k$ is a subset of the $k$th…
hase_olaf
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Elementary proof of the fact that any orientable 3-manifold is parallelizable

A parallelizable manifold $M$ is a smooth manifold such that there exist smooth vector fields $V_1,...,V_n$ where $n$ is the dimension of $M$, such that at any point $p\in M$, the tangent vectors $V_1(p),...,V_n(p)$ provide a basis for the tangent…
Pandora
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Why is the tangent bundle orientable?

Let $M$ be a smooth manifold. How do I show that the tangent bundle $TM$ of $M$ is orientable?
Lonely Penguin
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When is a fibration a fiber bundle?

In this question I am using Wiki's definitions for fibration and fiber bundle. I want to be general in asking my question, but I am mostly interested in smooth compact manifolds and smooth fibrations and bundle projection between them. Under some…
Lor
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Equivalence of Definitions of Principal $G$-bundle

I've finally gotten around to learning about principal $G$-bundles. In the literature, I've encountered (more than) four different definitions. Since I'm still a beginner, it's unclear to me whether these definitions are equivalent or not. I would…
Jesse Madnick
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What are the differences between a fiber bundle and a sheaf?

They are similar. Both contain a projection map and one can define sections, moreover the fiber of the fiber bundle is just like the stalk of the sheaf. But what are the differences between them? Maybe a sheaf is more abstract and can break down,…
Strongart
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What is the relation between connections on principal bundles and connections on vector bundles?

I'm reading Kobayashi / Nomizu 's vol. I. I am reading about connections in principal G-bundles. After that chapter there's a chapter on (linear) connections on Vector bundles. Since we can associate to every principal bundle a vector bundle (via…
Sak
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Local triviality of principal bundles

Suppose I define a principal $G$-bundle as a map $\pi: P \to M$ with a smooth right action of $G$ on $P$ that acts freely and transitively on the fibers of $\pi$. Does it follow that $P$ is locally isomorphic to $M \times G$ with the obvious right…
Eric O. Korman
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What does "locally trivial" do for us?

For the following we will work in the smooth category. (But examples in the topological category is also welcome.) The usual definition of a fibre bundle is Def A fibre bundle is the quadruple $(E,B,\pi,F)$ where $E,B,F$ are differentiable…
Willie Wong
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An intuitive vision of fiber bundles

In my mind it is clear the formal definition of a fiber bundle but I can not have a geometric image of it. Roughly speaking, given three topological spaces $X, B, F$ with a continuous surjection $\pi: X\rightarrow B$, we "attach" to every point $b$…
Dubious
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Does curvature zero mean the bundle is trivial?

Let $P\to M$ be some Bundle over $M$. I know that, if $P$ is a trivial bundle it must have curvature zero. Say I have the converse, my curvature is zero. Does this imply that the bundle ist trivial? If not, what can actually be said about the…
FMN
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Projection of fiber bundle is a submersion

I'm just wondering about my proof for the following fact. I get the feeling it is almost trivial but I am still getting a feel for geometry and so it doesn't seem 'obvious' to me just yet. The projection $\pi$ of the fiber bundle $(E,\pi,M,F)$ is a…
beedge89
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Do Hopf bundles give all relations between these "composition factors"?

Write a fiber bundle $F\to E\to B$ in short as $E=B\ltimes F$ (in analogy with groups). (This is not necessary, but: given another bundle $X\to B\to Y$, we can write $E=(Y\ltimes X)\ltimes F$, but we may also compose $E\to B$ with $B\to Y$ to get…
anon
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Fiber bundle is compact if base and fiber are

I want to show that the total space $E$ is compact if the fiber $F$ and the base space $B$ are compact. Let $\pi$ denote the fiber projection. Since every point in $B$ has an open neighborhood $U$ whose preimage $\pi^{-1}(U)$ is homeomorphic to…
Stefan Hamcke
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An example of a triple $(E,\pi,M)$ which is not a vector bundle

What is an example of a pair of finite dimensional $C^{\infty}$ manifolds $E$ and $M$, and a smooth function $\pi:E\rightarrow M$ such that $\pi^{-1}(p)$ has a vector space structure for each $p\in M\ $ (all of them with same dimension), but it is…
MBL
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