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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Principal bundles over affine schemes

I came up with the following question. Let us suppose that we have an affine scheme $X=\text{Spec}(A)$. It is well known that vector bundles on $X$ are equivalent to $A$-modules. In particular, if ...
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Existence of smooth sections in principal bundles.

Consider a smooth principal bundle $(P, M, \pi, G)$. For every open $U \subset M$ (except $M$ itself) in the base space $M$ is there a local section? I.e, a smooth map $s: U \rightarrow P$ that ...
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Equivalence between fiber bundles with fiber F, and principal Aut(F)-bundles

First, let's fix some definitions. Definition 1: Let $G$ be a topological group. A principal G-bundle is a triple $\xi = (E,\pi,B)$, where $E$ is a right $G$-space and the action of $G$ on $E$ is free ...
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On characterisation of smooth $G$ equivariant morphisms between Product manifolds with $G$ action

In particular I am interested in the following! Let $M$ be a smooth manifold and $G$ be a Lie Group. Let $\rho: (M \times G)\times G \rightarrow M \times G$ be the smooth action of $G$ on $M \times G$ ...
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Tangent space of a Principal bundle

Suppose we have a Principal bundle with a Lie group G in fiber. It is known that through the trivializations it can locally be expressed as a product of the base space M and the group in the fiber. ...
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Isomorphism between a principal bundle and a pullback bundle.

I have seen in many texts on the classification of main bundles that, given two homotopically equivalent X and Y spaces, this equivalence being the function $f: Y \rightarrow X$, given a group G, if $...
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Formula for covariant derivatives on principal bundles

I'm reading the notes on gauge theory by José Figueroa-O'Farrill and got stuck on an exercise. To state it, let me first explain my notation. Let $G$ be a Lie group, $P\to M$ a principal $G$-bundle, $...
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Isomorphism between basic forms on principal bundle and forms on the associated vector bundle?

$P$ is a principal $G$-bundle, $ρ:G→GL(V)$ is a representation, and $E$ is the associated vector bundle of $P$, so $E=(P×V)/G$ with the right action $(p,v)⋅g=(p⋅g,ρ(g^{−1})v)$. I have read that ...
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Modifying trivializations in principal bundles

I am trying to figure out when modifying the trivializing neighborhoods of a principal (or just locally trivial) bundle alters the global bundle structure. As a practical example in which I ...
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When is a functor the associated bundle construction?

Let $G$ be a group and $X$ be a space. I am principally interested in these objects in the category of schemes over some base $S$, but an answer in another geometric category would be welcome. ...
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Under which conditions every 2-form is a curvature form?

Let $M$ be a smooth manifold, $G$ a Lie group and $P\rightarrow M$ a smooth principal $G$-bundle. Let $\Omega^1_{eq} (P;\mathfrak{g})$ denote the space of $G$-equivariant $\mathfrak{g}$-valued 1-forms ...
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Under surjective submersion, pushforward of a $k$-form is smooth

Let $\pi:P\to M$ be a principal $G$-bundle with some connection $\omega\in\Omega^1(P,\mathfrak g)$. Let $\psi\in \Omega^k(P)$ such that $\psi_{p\cdot g}\big(r_{g*}(v_1),...,r_{g*}(v_k)\big)=\psi_p(v_1,...
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Definition of principal bundle

I would like to ask about the relative definition of a principal bundle. Let $G$ an algebraic group acting trivially on a scheme $S$. A principal $G$-bundle over $S$ is a a $G$-fibration $P\...
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Principal bundle over another principal bundle again principal?

I want to expand upon an earlier (partially answered) question: Is a principal bundle of a principal bundle still principal? As in said question, let $M, P_1, P_2$ be manifolds, let $P_1 \overset{\...
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Curvature of orthogonal distribution in a Riemannian principal space

Let $P \to M$ be a principal $G$-bundle and $\langle\cdot,\cdot\rangle$ be a $G$-invariant Riemannian metric on $P$. We have an Ehresmann connection for this bundle defined by $${\rm Hor}_p(P) \doteq {...
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Visualizing $|\mathcal{B}\mathbb{Z}| \simeq S^1$.

The classifying space of the integer group $\mathbb{Z}$ can be defined as the geometric realization of the underlying groupoid $\mathcal{B}\mathbb{Z}$. To unwind, $\mathcal{B}\mathbb{Z}$ is simply ...
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$\mathbb{R}$-bundle

Why a principal $\mathbb{R}$-bundle is always trivial? I know that a principal bundle of the form $(E,B,\pi,G)$ is trivial if and only if it admits a global section $f:B\to E$. So which section ...
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Examples of principal bundles

I´m actually working with principal bundles and I´m looking for particular examples. For instance, a principal bundle of structure group $Gl(n)$ is just a vector bundle. I´m wondering what happens if ...
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Induced Projective map of a vector $X$

I've been going through connections on fibre bundles in Nakahara's Topology, Geometry and Physics from 2003 and I wondered if someone could answer this question for me (from page 395, exercise 10.1 a.)...
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Yang-Mills Field Strength Compatibility Function

I've been going through Nakahara's Topology, Geometry and Physics from 2003 and I'm struggling to fully understand this derivation from page 410: Say there exists two different fields, $\mathcal{F}_i ...
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Principal $G$-bundle induces to $P/G=M$

For a bundle $(P,G,\pi,M)$, the principal $G$-bundle satisfies $(1)$ The induced action $$P_x\times G \rightarrow P_x$$ is free and transitive and $G$ preserves the fibre of $p$ $(2)$ The bundle ...
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Are vertical bundles naturally “principal”?

Suppose I have a principle vector bundle $\pi:E\rightarrow M$ with a group $G$. By definition, the action of $G$ preserves fibers, so $g\circ\pi=\pi\circ g$. This means that the differential maps ...
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Connection 1-form on associated principal bundle

Let $Q$ be a principal $H$-bundle for some (Lie) group $H$, and $\omega_Q\in \Omega^1(Q,\mathfrak{h})$ a connection 1-form on $Q$. Given an $H$-representation it is straightforward to construct a ...
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A foliations as a G-stucture

According to this Wikipedia entry, a foliation is a particular G-structure whose structure group reduction induced by block matrices. I tried to find more details of this approach to foliations, but ...
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Tangent and cotangent bundle as an associated bundle

In a book I read the isomorphisms below were mentioned without any explanation. Is there any intuitive way to see these identities hold? Let $M$ be a smooth $n$-manifold, $F(M)$ a frame bundle of $M$,...
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Principal bundle structure of $S^1\to S^1$ given by $z\to z^2$ map

Let $S^1$ be unit circle on $C^\star$(punctured complex plane). Consider $S^1\to S^1$ by $z\to z^2$ map. Identify $Z_2$ action as $e:z\to z$ and $g:z\to -z$ action on $S^1$ of domain. Let $U=S^1-\{1\}$...
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2 dimensional torus bundles over $\mathbb{S}^2$

With my limited knowledge of bundles, it seems that the isomorphism classes of principal torus bundles are in one-to-one correspondence with the homotopy classes of maps $[\mathbb{S}^2,\mathbb{C}P^\...
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Exterior covariant derivative of connection 1-forms

I defined the connection 1-form on a principal bundle $\mathcal{P}=(P,M,\pi;G)$ as the $\mathfrak{g}$-valued 1-form $ \bar{\omega_{p}}=\left( \theta^{A}_{(L)}(p)+\mathrm{Ad}^{A}_{B}(g^{-1})\omega^{B}...
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The definition of principal G-bundle

I've encountered many different definitons of principal $G$ bundle from Morita's Geometry of differential forms , Hamilton's Mathematical gauge theory , Kobayashi and Nomizu's Foundations of ...
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Vector valued forms and right equivariance

Let $\pi:P\to M$ be principal $G$-bundle, $V$ real vector space and $\rho:G\to Aut(V)$ representation of group G. Following definition is from Differential Geometry: Connections, Curvature, and ...
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does the curvature fixes a connection on the principal bundle?

Consider a connection $d$ on a principle bundle $E\to M$. It gives gives you a curvature by the usual $$ d^2s=Fs \quad \forall s\in \Gamma(E) $$ I was wondering if the connection is completely ...
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May a Trivial Principal Bundle Admit a Non-Trivial Reduction of the Structure Group?

Let $\pi\colon P \to M$ be a principal bundle with structure group $G$ with respect to the Lie group right action $\triangleleft\colon P \times G \to P$. To my knowledge, a reduction of the structure ...
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Identify line bundles on $\mathbb{P}^n$

The base field is $\mathbb{C}$ throughout the question. Consider the projective space $\mathbb{P}^n$. It corresponds to a principal $\mathbb{C}^*$-bundle $$ \mathbb{C}^* \to \mathbb{C}^{n+1}-0 \to \...
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Understanding a question on vector bundles

I have the following question on vector bundles which I am struggling to understand: Consider the tangent bundle $TS^2$ with an inner product on the fibres, and let $Y\subset TS^2$ consist of all ...
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Why is the curvature of the connection $\bar{z_1}dz_1 + \bar{z_2}dz_2$ on the Hopf fibration not exact?

Let $\pi : S^3 \to S^2$ be the Hopf fibration, where we take $S^3 \subset \mathbb{C}^2$, $S^2 = \mathbb{C}\mathbb{P}^1$, and $\pi(z_1, z_2) = [z_1 : z_2]$. This is a principal $U(1)$-bundle. The form $...
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Horizontal forms on principal $\mathbb{S}^1-$bundle

Assume that $M$ is a principal $\mathbb{S}^1-$bundle, i.e. an $\mathbb{S}^1-$manifold with the property that the action $M\times \mathbb{S}^1\to M$ is free (no fixed points) and the orbits $\mathcal{O}...
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Grassmann manifold

Grassmann manifold $G_{k,n}$ is the set of k-dimensional subspaces of $\mathbb{R^n}$. Let’s consider the set of k-frames $V_{k,n}$. I want to show that $$ G \to V_{k,n} \to^{\pi} G_{k,n} $$ can be ...
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spin structure on $\Gamma \backslash S^n$

Let $Γ \subset SO(n + 1)$ be a finite group acting freely on the unit sphere $S^n \subset \mathbb{R}^{n+1}$. Show that the quotient $\Gamma \backslash S^n$ admits a spin structure if and only if there ...
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Principal bundles of Lie groups in a short exact sequence

I have asked this question on physics stack exchange already but I feel it’s more of a math question, so I’m posting it here. Consider a short exact sequence of Lie groups $$1 \rightarrow G \...
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Maurer-Cartan form defines a connection on a Lie Group considered as a principal $G$ bundle.

Given a Lie group $G$ we can consider it as a principal $G$ bundle over any element in $G$. Consider the Maurer-Cartan form $\omega$ defined by the left translation: for $X\in T_gG$ we have $\omega(X)...
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Transition functions of $S^3$ as a principal bundle over $\mathbb{C}P^1$.

Consider he action of $U(1)$ on $S^3$ by $(z_1,z_2)\cdot e^{i\phi}\mapsto(z_1e^{i\phi},z_2e^{i\phi})$. This action is clearly free and proper. Therefore we can consider $S^3$ as a principal bundle ...
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Frame bundle of associative vector bundle returns the origional prinicipal bundle.

Background and definitions: For a vector bundle $E$ let $F(E)$ denote its frame bundle. For a principal bundle $P$, and representation $(\rho,GL(n,\mathbb{R}))$ let $E(P,\mathbb R^n)$ denote the ...
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Immersed principal subbundles

In Kobayashi and Nomizu's Foundations of Differential Geometry vol. 1 the authors define immersed principal subbundles (actually, they call them imbeddings) of a principal $G$-bundle $\pi: P \to M$ as ...
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Associated vector bundles: Given a vector bundle $E$ show that $F(E)\times \mathbb{R}^n/GL(n,\mathbb{R})\cong E$

I am trying to understand the map between vector bundles and principal fibre bundles and would like to work out the following example explicitly. Moreover in this example I am only currently looking ...
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When is a connection on the adjoint bundle induced by a principal connection?

Let $P\rightarrow M$ be a principal $G$-bundle with a connection 1-form $\omega$. In a local trivialisation $\tau_U \colon U\rightarrow P_U$ ($U \subset M$) we can pull the connection back to the ...
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Isomorphism between space of differential forms and space of basic forms on pricipal $S^1$-bundle and independence of cohomology class

Let $M$ be be a principal $S^1$-bundle, i.e. a smooth manifold with a smooth $S^1$-action (multiplication to the right) that has no fixed points such that the orbits through every point $p \in M$, ...
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Differential operators of a principle G-bundle

I understand for a pair of smooth vector bundles $E$ and $F$ over a smooth manifold $M$ it makes sense to talk about the differential operators between the section spaces of $E$ and of $F$. The ...
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Thin homotopy invariance of parallel transport

I have seen many claims that parallel transport in thin homotopy invariant, but I cannot seem to prove it. My conventions are: Let $\pi\colon E\to B$ be a principle $G$ bundle with connection $\omega$...
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Are classifying spaces of $GL$, $SL$, $O$ the same?

In the theory of principal bundle classification, we have the Milnor's construction $EG$ for a topological group $G$ and a classifying space $BG$. We know the classifying space for $GL_n(\mathbb{C})$ ...
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Definition of principal $G$-bundle might be missing details or have implicit assumptions on actions on each trivializing open set

My book is Connections, Curvature, and Characteristic Classes by Loring W. Tu (I'll call this Volume 3), a sequel to both Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott (...

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