# 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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Suppose $p_1:E_1\to B$ and $p_2: E_2\to B$ are two compact principal $\mathbb{T}^d$-bundles over the same base $B$. Suppose there exists a fibre-preserving map $F:E_1\to E_2$ that covers the identity $... 2 votes 0 answers 40 views ### Relation between G-connection and second fundamental form when embedding is in principal G-bundle I'm pretty familiar with intrinsic geometry utilized in say General relativity for instance, and I understand the intrinsic curvature$\Omega$2-form of a connection$A$on a manifold$M$of dimension ... 2 votes 0 answers 145 views ### Fiber bundle orientability vs manifold orientability I read this question about vector bundles Bundle orientability vs manifold orientability In the answer to this question the last sentence states the following (I think fairly well known) result about ... 1 vote 0 answers 26 views ### Standard metric on adjoint bundle Consider we have a principal$G$-bundle$P$over a closed manifold$V$. Denote$\mathfrak{g}_P$by the associated bundle$P\times_G \mathfrak{g}$where$G$acts by adjoint action. Denote$\mathscr{G}$... 0 votes 0 answers 30 views ### homotopy equivalent of mapping space I'm reading the book Geometry of Four Manifolds by Donaldson. In page 179, he mentions that: There exist a homotopy equivalent$\Omega^n(BSU(2))\cong \Omega^{n-1}(SU(2))$. Here$\Omega^m(X)$means the ... 3 votes 2 answers 53 views ###$SU(2)$-bundles over four-manifolds are determined by top cells Suppose we have a compact oriented simply-connected four manifold$X$and a$SU(2)$bundle$P$with$2$nd Chern class$=k$over$X$. We know$X$have a top cell$e^4$, if we collapse all lower cells ... 0 votes 1 answer 54 views ### The definition of principal$G$bundle Here is the definition of principal$G$-bundle in "differential geometry" written by Taubes. Fix a smooth manifold$M$, and a Lie group$G$. A principal$G$-bundle is a smooth manifold$P$, ... 5 votes 0 answers 69 views ### Connections on Principal bundles & Covariant derivatives on Vector bundles Nowadays I'm reading "Differential geometry" written by Taubes. I have some problems and I guess that there may be some typos or I must get something wrong. Suppose vector bundle$E$is ... 0 votes 0 answers 59 views ### Parallel fundamental vector fields Suppose I have a principal bundle$P$relative to the group$G$. Suppose I have a torsionless connection on$TP$for which the fundamental vector fields relative to the$G$action are parallel. Can I ... 2 votes 2 answers 71 views ### Second order cohomology and$U(1)$-principal bundles In Homotopy Moment Map the following is stated If the symplectic form represents an integral cohomology class, then it corresponds to the curvature of a principal$U(1)$-bundle equipped with a ... 3 votes 1 answer 59 views ### Why$g^{-1}$in the definition of associated vector bundles? In the definition of the associated vector bundle$E$to a principal bundle$\pi:P\rightarrow M$, the equivalence relation is $$(p,v)\sim(pg,g^{-1}v)$$ where$p\in P$,$v\in V$,$g\in G$, Lie group$G$... 0 votes 1 answer 75 views ### Exterior covariant derivative on associated bundle Let$(P,\pi,M;G)$be a principal fibre bundle over$M$with connection$1$-form$A:TP\rightarrow \mathfrak{g}$. Let$\rho:G\rightarrow V$be a representation. The connection$A$now induces a ... 2 votes 1 answer 71 views ### Triviality of Sp(TM) Let M be a symplectic manifold of dimension$2n$and$TM$denote its tangent bundle. Let Sp(TM) denote the bundle over M whose fibers are linear maps preserving symplectic structure on M. Is Sp(TM) ... 0 votes 0 answers 40 views ### Why is the frame bundle of the Möbius strip the Z2 bundle? In the wiki figure it says "The frame bundle$\mathcal{F}(E)$of the Möbius strip$E$is a non-trivial principal$\mathbb {Z} /2\mathbb{Z}$-bundle over the circle." Shouldn't the frame ... 2 votes 0 answers 53 views ### Derivation of Yang-Mills functional I was trying to calculate the critical points of Yang-Mills functional. And I failed to show that$F_{A+ta}=F_{A}+t\nabla_Aa+t^2a\wedge a$. Here is my attempt: Suppose all calculation is in a local ... 1 vote 0 answers 55 views ### Characteristic class of principal bundle I've been told that principal$G$-bundles$E \to M$are classified by specifying a characteristic class$c(E) \in H^2(M,\pi_1(G)) ≈ \pi_1(G)$I have a few questions Given a bundle$E$how do we get a ... 3 votes 1 answer 72 views ### What is adjoint bundle for trivial bundle? Let$P =M\times G\to M$be a principal$G$-bundle on$M$(first coordinate projection) What is$ad(P)$? Here$ad(E) = E\times_{Ad}g$is a vector bundle on$M$[where$g$= Lie$(G)$,$E$is any ... 0 votes 0 answers 34 views ### What should be the definition of holomorphic principal bundle? As written in the title, What should be the definition of a holomorphic principal$G$bundle where$G$is a complex Lie group on a complex manifold$M$? For smooth vector bundles$E \to M$I know ... 2 votes 0 answers 36 views ### Confusion regarding definition of energy operator/harmonic metric I will start by laying out the context. Let$\pi : E\to M$be a principal$G$-bundle. For subgroup$H$of$G$a "reduction of structure group" is defined to be a section of the$G/H$-fibre ... 3 votes 1 answer 47 views ### Question regarding reduction of structure group Let$\pi:P \to M$be a principal$G$-bundle. Given a subgroup$H$of$G$one can consider the fibre bundle$P_H :=P\times_{G}G/H$whose fibres are the coset space$G/H$. We define a reduction of ... 3 votes 2 answers 83 views ### How do we go from a covariant derivative on a principal bundle to a covariant derivative on an associated bundle Let$M$be a smooth manifold and$\pi:P\to M$a principal$G$bundle over$M$. Suppose that$P$is equipped with a connection one form$\omega$. We can define an exterior covariant derivative on$P$... 1 vote 1 answer 64 views ### Confusion regarding definition of gauge transformation Let$E \to M$be a principal$G$-bundle. The gauge group is the group of$G$-bundle automorphisms of$E$. A connection on$E$can be thought of as a global$g$-valued 1-form on$E$where$g$= Lie$(G)$... 2 votes 1 answer 56 views ### Connection on a principal$S^1$bundle Let$\pi:M\to B$be a principal$S^1$-bundle over a symplectic manifold$(B,\omega)$. Is it always possible to construct a vector field$R\in \mathfrak{X}(M)$such that the$S^1$action on$M$is ... 1 vote 1 answer 64 views ### What is the correspondence between gauge field terminology and bundle terminology in electromagnetism? In electromagnetism, the electromagnetic field tensor can be expressed as $$F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu.$$ If we let$A= A_\mu dx^\mu$, since$F= \frac{1}{2} F_{\mu \nu} dx^\... 37 views

### 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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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 $\sigma^*A=A$....