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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Pulling back Lie-valued forms to get connection and curvature?

According to Lemma 1 in https://en.wikipedia.org/wiki/Chern%E2%80%93Weil_homomorphism#Definition_of_the_homomorphism , if $\Omega$ is the curvature of a connection on a principal $G$-bundle $P\...
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Does $\{(f):ℙ↣ℙ\}$ contain any analytic members $f$?, and if so What is the simplest such injective $f$? Are all $f$ necessarily monotonic? [duplicate]

(Above, $ℙ≔\{\text{all primes}_ℤ\}$.) Are there any analytic functions that will give a unique prime output for every distinct prime input? Analyticity should preclude cheap reiterating upon a prime-...
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Levi Civita Connections vs. Ehresmann connection

Forgive me if I mess some of these concepts up or say something incorrect, I am still figuring out all the details of an Ehresmann connection in an associated vector bundle. So, how do we relate these ...
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Solutions of Yang-Mills equation in the case of $G=U_1$

If $P\to M$ is a principal $U_1$-bundle, and $A$ is a connection on $P$, then it's curvature $F_A$ is a $2$-form with coeficient in $P\times_G\mathfrak{u}_1$, where $\mathfrak{u}_1$ is the Lie ...
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What does it mean that a $G$-space embeds into its homotopy quotient as the fiber over the basepoint of the classifying space of $G$?

Let $G$ be a topological group and $M$ be a $G$-space. Let $EG \rightarrow BG$ be a universal $G$-bundle and let $M_G$ be the homotopy quotient $(EG \times M)/G$. What does it mean that $M$ embeds in $...
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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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Explicit formula of $S^1$-bundle over $S^2$ with euler number $n$

Suppose $M \to S^2$ is an orientable circle bundle with Euler number $n$. Regarding $S^2$ as $D^2_1 \cup_{\text{id}} D^2_2$, since $M$ is trivial over $D^2_i$ ($i=1,2$), we have $M=(S^1\times D^2_1)\...
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Tensor fields defining $G$-structure are parallel

Suppose $G \leq GL_n(\mathbb{R})$ is the stabilizer of some tensors $T^0_1, ..., T^0_k$, let $P$ be a $G$-structure on a manifold, i.e. a principal $G$ subbundle of the frame bundle of $M$ and let $...
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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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"Tightening up" a map in the fibre of a principal torus bundle

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 $...
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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 ...
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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 ...
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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 $\mathscr{G}$ ...
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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 ...
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$SU(2)$-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 ...
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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 $P$, ...
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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 ...
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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 ...
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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 ...
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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 $G$...
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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 ...
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2 votes
1 answer
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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) ...
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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 ...
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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 ...
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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 ...
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3 votes
1 answer
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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 ...
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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 ...
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2 votes
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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 ...
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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 ...
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3 votes
2 answers
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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 $P$ ...
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1 answer
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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$(G)$...
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2 votes
1 answer
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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 ...
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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^\...
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1 answer
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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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3 votes
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Associate bundles, equivariant sections and tangent elements

Consider a principal $G$-bundle over a manifold $X$, and consider yet another manifold $B$ endowed with a $G$-action. Everything is assumed to be smooth. The associated bundle $$ \mathcal{B}=P\times_G ...
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Fibers of a principal $G$- bundle are diffeomorphic to $G$.

Definition: A principal bundle $\pi:P \rightarrow M$ with structure group $G$ is a fiber bundle $P$ with a right action of the Lie group $G$ on the fibers, such that $$\pi(pg)= \pi(p), \quad p \in P , ...
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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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3 votes
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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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1 vote
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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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stabilizer of connection on SU(2) bundle

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