# 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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### A topological Ehresmann's theorem

A proper local homeomorphism is a covering space (with some mild conditions on the spaces involved). I want to know about the following generalization, which I believe is false but cannot come up with ...
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### Pullback of a covering map along the covering map.

Consider $H$ as a subgroup of $G$, we can push them to the level of classifying spaces $p: BH \rightarrow BG$. This is a covering space with fiber $[G:H]$. What is the pullback of this covering map ...
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### Fiber of continuous surjection from higher to lower dimensional unit balls

Lets $B_k=\{x\in\mathbb{R}^k|\Vert{x}\Vert\le1\}$ denotes the k-dimension closed unit ball. Suppose $f:B_n\rightarrow{B_m}$ is a continuous surjective map where m<n. For an interior point y of $B_m$...
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### Bundle homomorphisms and sections of Hom-bundle

I reading "Complex Topological K-theory" by Efton Park and came across exercise 1.4. Let $V,W$ be vector bundles over a compact Hausdorff space $X$. a) Show that the collection $Hom(V,W)$ of ...
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### Riemann curvature tensor "as a" vector in the total space?

I have been reading through Wald's book on General Relativity, often using other sources to help gain a deeper understanding of the mathematics. Through this supplemental learning, I encountered ...
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### How to show that any section here will give rise to family of functions and how to specify the compatibility conditions

This question was asked in my exercises of differential geometry and I am completely struck on the problem. So, I am posting it here in hope I will get some leads: Question: Let $M$ be a manifold of ...
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### Erlangen-Style approach to Homogneoues Fiber Bundles

In connection with a philosophy of physics project I have recently been looking at fiber bundles from an Erlangen-inspired perspective. My inspiration is the following well-known result: Every smooth ...
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### Bott Tu local constant presheaf

I´m reading Bott Tu Differential forms in algebraic topology and I have a question. In the page 169 we wnat calculate cohomology of a fiber bundle $\pi:E\to M$ with fiber $F$, and we define the ...
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### Product in the category of smooth vector bundles

Is the product in the category of smooth vector bundles just the direct product of smooth vector bundles? More precisely, if $\pi:E \rightarrow B$ and $\pi': E' \rightarrow B'$ are two smooth vector ...
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### Connection on pullback principal bundle

For a Lie group $G$, let $\pi: E \rightarrow M$ be a principal $G$-bundle over a smooth manifold $M$. Let $\omega: TE \rightarrow Lie(G)$ be a connection 1-form on $E$. Let $f:N \rightarrow M$ be ...
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### Descomposition of Tangent bundled in sum of line bundles

Let $P \rightarrow M$ a G-structure where $G\subset GL(n)$ is a group of diagonal matrix. Prove that tangent bundle is isomorphic at $TM=\xi_1 \oplus \xi_2 \oplus \dots \oplus \xi_n$, where $\xi_j$ ...
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### "Cohomology Operations" - Steenrod-Epstein: frame fields on $S^{n-1}$ as sections of a fibre bundle

In the book "Cohomology Operations" by Steenrod and Epstein, at page 56, the author says that the existence of a field of $k$-frames (i.e. a function that maps a point of the sphere to a $k$-...
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### Principal Bundle Definition Question Action restriction

Let $G$ be a topological group and $\pi: P\rightarrow M$ a fiber bundle with fiber $G$ (working in continuous category). We say that $\pi$ is a principal G-bundle if (1) $G$ acts freely on the right ...
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### Intuition of curvature

The only physical intuition of curvature that I know: parallel transport along a closed loop doesn't close (e.g. parallel transport of a tangent vector on a sphere). The Ambrose-Singer theorem says &...
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### Are all $\mathbf{P^1}$-fibrations projective bundles? [duplicate]

Maybe there is a very simple answer to my question or a counterexample - I don't see it: Let $\mathcal{X}\rightarrow C$ be a flat, smooth family with fibres $\cong \mathbf{P^1}$ of dim $2$ over a ...
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### When is a principal bundle with group $G \times H$ the product of a principal bundle with group $G$ and one with group $H$?

In general, it is not true that a fiber bundle with a product fiber is the product of two fiber bundles with the factors as fibers (think of vector bundles). However, I read somewhere that every 2-...
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### Trivialization of the orthonormal frame bundle

Given an $n$-dimensional manifold $M$ with Riemannian metric $\eta$, the orthonormal frame bundle over $M$ is defined as the set of all orthonormal frames over all points in $M$. Denote such bundle by ...
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### Scalar fields with definite weight under conformal maps as sections of some bundle

Let $(M,g)$ be a smooth Riemannian manifold. A scalar field in $(M,g)$ is merely a map $\phi:M\to \mathbb{R}$. Under a diffeomorphism $f:(M,g)\to (M,g)$ it transforms to $\phi' = \phi\circ f$. We then ...
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### An orientable line bundle is always trivial?

In general, there exist non-trivial orientable bundles. But if we only consider line bundles, then orientable bundles are always trivial bundles. (Similarly, non-trivial bundles are always non-...
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