# Questions tagged [principal-bundles]

In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product $X\times G$ of a space $X$ with a group $G$.

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### Hermitian metrics on the Associated Vector Bundle of a Principal $U(1)$-bundle

If a complex line bundle $L$ over some manifold $M$ has a Hermitian metric, then $M$ has a frame bundle, which is a principal $U(1)$-bundle over $M$. Now reverse this, if we have a principal $U(1)$-...
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### List all the circle bundles over the circle, sphere, projective plane, torus, and Klein bottle [closed]

List all the circle bundles over the circle, sphere, projective plane, torus, and Klein bottle. State the orientability of the total space, the base and the bundle (orientability of a circle bundle is ...
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### Generalizing a property of the two sphere

There are many circle bundles over the sphere $S^2$ (in fact infinitely many) but all of them are principal. Do there exist any other manifolds besides $S^2$ for which (nontrivial bundles exist ...
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### Parallel fundamenal vector fields on circle bundle

Since this question received no answer, let me go through a simpler case first. Let $P$ be a $U(1)$-bundle and suppose I have a metric on it that makes the fundamental vector field of the action ...
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### A simple example of a $G$-principal bundle that cannot be lifted to an $F$-principal bundle along an epimorphism $F\to G$.

I am trying to understand spin structures, in particular how they may fail to exist. To start with, I would like to see a most simple example of a $G$-principal bundle that cannot be lifted to an $F$-...
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### Moduli Space of Flat connection over Homology 3-Shpere

I'm trying to understand the space of flat connections over the trivial $SU(2)$ -bundle of a closed, oriented homology three-sphere (for the purpose of understanding the instanton Floer homology of it)...
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### Non existence of a preferred Horizontal subspace on a bundle. Why not ? (Basics)

If I choose a principal bundle, let us say $G\rightarrow P \rightarrow B$, with $G=U(1)$, $P=T^2$ (2-torus) and $B=S^1$. Can I choose to put my finger on the identity element of the group over a point ...
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### How to generate an associated connection

I know that we can generate an affine connection on a GL(n) bundle.If we have a fibre bundle whose structure group G is the subgroup of GL(n) with diagonalizable matrices M whose eigenvalues are +1 or ...
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### Equivalence between sections of the pullback bundle and lifts in the corresponding commutative diagram

Let $\pi: P \to B$ denote a principal $G$-bundle over base $B$, and let $f: B' \to B$ be a continuous map from another space $B'$ to $B$. I've been reading Stephen Mitchell's Notes on Principal ...
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### Tensor bundle as associated bundle

Given a smooth manifold $M$ of dimension $d$, consider the frame bundle $FM \overset{\pi_{FM}}\longrightarrow M$. We can construct the tangent bundle $TM \overset{\pi_{TM}}\longrightarrow M$ as an ...
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### Principal connections on $P$ and covariant derivatives on associated vector bundle $E=P\times_\rho V$

I would like to have a concrete proof or reference to the following fact: Let $P\rightarrow M$ be a principal $G$-bundle over an $n$-dimensional manifold $M$, and let $E=P\times_\rho V$ be an ...
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### Connection on tangent bundle induces connection on other fiber bundles (or possibly the reverse)?

As per Lee, every connection on a tangent bundle induces a connection on any tensor bundle in a natural fashion. Can this be extended to connections on more general fiber bundles, such as vector ...
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### Oriented Orthonormal Frame Bundle and Orientation-Preserving Isometry Group

Let $\mathbb{H}^3$ be the hyperbolic $3$-space. Let $F\mathbb{H}^3$ be the oriented orthonormal frame bundle, and let $\mathrm{Isom}^+(\mathbb{H}^3)$ be the orientation-preserving isometry group. It ...
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### Principal bundles with non-constant structure group

Take $P \rightarrow M$ to be a principal $G$-bundle. We modify the bundle so the structure group $G$ is not the same for all fibres of the bundle. Is there a name for such an object? Are there issues ...
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### Moduli space of connection on line bundle

I want to show that for a $U(1)$ bundle $P$ over a connected smooth 4-manifold $X$, the moduli space of Yang-Mills connection over $P$ is the torus $H^1(X,\mathbb{R})/H^1(X,\mathbb{Z})$. Now I reduce ...
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### Good exercices to understand vector, principal and associated bundles

I'm a new stuent of differential topology and in my course I came across bundles and I'm looking now for good exercices (books, links) to understand them well ! the 3 subjects that I need to really ...
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Suppose $P$ a principal bundle over connected manifold $B$ with correspondent Lie group $G=SU(2)$, and $A$ a connection on $P$. We say a map $\sigma \in Aut(P)$ a stabilizer of $A$ if $\sigma^*A=A$....