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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Proof check: a riemannian 2-manifold is orientable if its unit tagent bundle is principal

For a two-dimensional smooth manifold $M$ with a Riemannian metric, we can consider the unit tangent bundle $S\to M$, which is a circle bundle over $M$. I am trying to show that if $S\to M$ is a "...
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Connection one form and well-definedness of everything

I'm trying to learn differential geometry from Isham's book and I want to check if what I'm saying makes sense. Let $G \rightarrow P \rightarrow \mathcal{M}$ be a principle bundle. Then the connection ...
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Constructing a connection $1$-form from local forms.

I am following Section $10.1.3$ of Geometry, Topology and Physics by Nakahara, and have ran in to an issue regarding local connection forms. Consider a principal $G$-bundle, $P(M,G)$, and an open ...
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Explicit example of an Ehresmann connection on the trivial bundle $\mathbb{R}^4 \times S^1$

$\mathbb{R}^4 \times S^1$ is of course a principal $U(1)$-bundle and trivial. However, I cannot find an explicit example of the Ehresmann connection one-form on this bundle in terms of the coordinates ...
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Lie derivative on principle bundle

I'm a bit confused about Lie derivative on principal bundle $P(M,G)$. Let $g(t)$ be the flow generated by vector fields $Y$ and $X$ also vector field on $P$. According to the definition, $$\mathcal{L}...
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Equivalent definitions of almost quaternionic structures

I came across this thread today when I was reading about almost quaternionic structures. I was wondering if there exists an argument similar to the answer to the thread above that can show that the $...
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Principal bundle over Riemann surfaces

I have come across a paper which does not make explicit the notation they're following and I am a little bit confused. The paper in question is the following one https://www.kurims.kyoto-u.ac.jp/~...
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Why is the group of gauge transformations $\mathcal{G}$ on the frame bundle isomorphic to $\text{Diff}(M)$?

Let $LM \to M$ be the frame bundle on pseudo-Riemannian manifold $M$ and suppose that there exists a Lorentzian metric tensor $g \in \Gamma \big(T^0_2(M) \big)$ on $M$. Since the frame bundle is a ...
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Double “orthogonal” group structure on a space.

I am trying to find an appropriate setting for describing the following observation. When studying ${\Bbb R}^3$ we observe a group operation $\omega\colon ({\Bbb R}^+,\cdot) \times {\Bbb R}^3 \to {\...
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Reduction of structure group $GL_n$ of the endomorphism bundle to the centraliser.

I was reading this proof from http://www.numdam.org/item?id=AST_1982__96__1_0 where of structure group of the bundle $EndE$ which is taken to be $GL_n$ is reduced to the centraliser of $GL_n$. The ...
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Slight confusion about projection operator in principal bundle

Let $P(M,G)$ be a principal bundle with base space $M$ and structure group $G$. Let $\Gamma$ be a connection on $P$ so that the horizontal subspaces are denoted by $\Gamma_u$ for $u\in P$. Let $\pi:P\...
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Connection on isomorphic principal bundle

I’m trying to find a natural induced connection $1$-form on the isomorphic principal bundle, more explicitly, given a principal bundle isomorphism $(P:P_1\rightarrow P_2$, $\Phi:G\rightarrow H)$ ...
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Relationship among $BG, K(G,1)$ and Čech cohomology

(1) If $G$ is an abelian group. We know Čech cohomology equals singular cohomology, $\check{H^1}(X,G)=H^1_{sing}(X,G)$. In addition, $\check{H^1}(X,G)$ classifies the isomorphism classes of principal $...
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Quotient by the action of a group commuting with sequential colimits

Let $G$ be a group and suppose $X_1 \to X_2 \to X_3 \to \cdots$ is a sequence of topological $G$-spaces and continuous $G$-maps. The colimit of this sequence inherits then a $G$-action. Under what ...
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Canonical endomorphism induced by a section of adjoint bundle

I want to check the following statement: Suppose $G\rightarrow P \rightarrow M$ be a principal bundle, and $E=P\times_{\rho} V$ be an associated vector bundle defined by representation $\rho$, then a ...
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What are examples of nontrivial principal fiber bundles?

I am looking for an example of $G$ principal fiber bundle over a topological space with $G$ a topological group such that is not trivial. In particular, I would be glad to have an example where G is $...
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G-principal bundles on formal disc

How can I prove that the $G$-fiber principal bundles on $\operatorname{Spec} \mathbb{C}[[t]]$ and $\operatorname{Spec} \mathbb{C}((t))$ are trivial when $G$ is an algebraic linear group on $\mathbb{C}$...
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Two definitions of principal fiber

I have two definition of $G$-principal fiber bundle when $G$ is a linear algebraic group complex. Let be $X$ a complex variety. A principal fiber bundle on $X$ is a couple ($\xi,q$) where $\xi$ is a ...
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Local coordinates of one form on a principal bundle

I am reading Natural and Gauge Natural Formalism for Classical Field Theory by Lorenzo Fatibene and I am really confused by his definition of a connection in local coordinates. Let's say we have a ...
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Confusion on Nerve of a Category and Segal's model of Classifying spaces

If we have a topological category and the underlying category forgetting the topological structure, are the nerves same. They should be, is what my guess is from the definition of nerve of a category. ...
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Thom space functor preserves homotopy

Let $X$ be a paracompact space and consider two homotopic maps $f,g: X\to B$, where $E\to B$ is a real vector bundle over a paracompact space $B$. We know that these maps have isomorphic pullback ...
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What are the R - families of connections

I've the definition of a connection on a vector bundle and on a G-principal bundle. But reading I have found this statement if R is a $\mathbb{C}-$algebra take the R-families of connections over the ...
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Tensor product vector bundle - bijection from vector valued functions to sections defined by local sections

Let $G_i\to P_i\to M$ be principal fiber bundles with representations $\rho_i\colon G_i\to\mathrm{GL}(V_i)$ and associated vector bundles $E_i\to M$. Given local sections $s_i\colon U\to P_i$, I ...
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Path lifting in principal bundles

Let $\pi : P \rightarrow M$ be a principal $G$-bundle and let $\gamma : [0,1] \rightarrow M$ be a smooth path. I define a lifting of $\gamma$ on $P$ in the following way: if $\{U_\alpha\}_\alpha$ is a ...
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Trivality of principal bundles [closed]

Let $\pi : P \rightarrow M$ be a principal G-bundle. There are topological properties of M and G that imply the triviality of the principal bundle?
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Universal property of Classifying spaces encoded in $1$-cocycle of $EG \to BG$

Let $G$ be a topological group (for sake of simplicity I think it suffice to assume that $G$ is finite group. It is well known that for any homotopy class of a map $\phi: X \to BG$ (with $X$ nice '...
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Connection on principal $G$-bundle v.s. connection on associated vector bundle

I'm confused with the relation between these two. I'll summarize what I know so far and explain my questions. So for a principal $G$-bundle $P$, we can define a principal $G$-connection on it, we can ...
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How to define integrals for (U(1) valued) compact variables on a circle S1

My question is how to properly define integrals of some compact variables (functions) that are defined on a circle. Let me explain my puzzle with one concrete example (which comes from a physics ...
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Construction of a Covering Space by 'Twisting'

I have a question about the explanantion of the idea behind the classiyfing spaces $BG$ with respect (topological) group $G$. In wikipedia is stated that The classifying property required of $BG$ in ...
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With a local anomaly, is the determinant of the Dirac operator still a section of a complex line bundle?

In the literature about anomalies in quantum field theory, the determinant of the Dirac operator plays an important role. The Dirac operator may depend on some background data, and the subject of ...
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Principal bundle map over different bases, and pullback

It is known that a morphism between principal $G$-bundles over the same base must be an isomorphism of principal bundles. Can I ask: is it true that a morphism of principal $G$-bundles over different ...
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A horizontal bundle provides a connection one-form for a principal $G$-bundle

Let's have a principal $G$ bundle $P \xleftarrow{\triangleleft G} P \xrightarrow{\pi} M$. At some point $p \in P$, let's consider the tangent space $T_p P$. We define the vertical tangent space as $...
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Relation between connection and connection form

In the smooth setting, we define a connection in a principal $G$-bundle $(P, \pi, M)$ as a smooth assignment to each point $p \in P$ of a subspace $H_p P$ of $T_p P$ such that the following two ...
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Transitive action of $S^1$ on $SO(3)$

I'm working to understand why is $\pi : SO(3) \rightarrow S^2$ wich associates to a matrix $A \in SO(3)$ it's first column, is an $S^1$- principal bundle, with the right action of $S^1$ on SO(3) ...
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The 1-1 correspondence between sections of a principal G-bundle P and isomorphisms between the trivial principal G-bundle and P

Exercise: Show that a principal bundle admits a (global) section of and only if it is trivializable. In more detail: show that if $\sigma$ is a section of the principal G-bundle $\pi : P \rightarrow M$...
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Variational calculus on Lie algebra valued one forms - Chern-simons theory

The action in the Chern-Simons theory is given as: $$S=\frac{k}{4\pi}\int_M \text{Tr}(A \: \wedge \:dA + \frac{2}{3}A \: \wedge \:A \: \wedge \: A). $$ The wikepida page gives the euler lagrange ...
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morphism between two princial bundle over the same mainfold?

Let $ \pi : P \rightarrow B$ and $\pi' ; Q \rightarrow B$ be two principal G- bundles. Why this is true: If $f : P \rightarrow Q $ is a morphism of the pricipal G-bundles P and Q ( i.e. $f(p.g)= f(p)...
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Trivialising homeomorphisms for a principal $G$-bundle as $G$-space morphisms

Let $P$ by a principal $G$-fibre bundle over a locally-compact Hausdorff space $X$. Denote by $$ h: U \times G \to P|_{U} $$ a trivialising homeomorphism for a trivialising open set $U \subseteq X$. ...
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Is it a G-bundle?

Remark 3.3 in Behrend and Dhillon's paper "On the motivic class of the stack of bundles" says $\mu(BP) \mu(G) =\mu(G/P) $ for all parabolic subgroups of G, where this requires the torsor ...
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If the bundle of orthonormal frames has a continuos/smooth global section will the bundle of spin frames also have one?

Let $(M,g)$ be a semi-Riemannian manifold with metric of signature $(p,q)$. I believe the signature of the metric is not relevant for this discussion so I leave it arbitrary (corrections to this are ...
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Unimportance of the metric for spin structures

I have some pedantic confusions when it comes to spin structures. Let $B$ be a nice space (like a CW complex) and let $\xi$ be an oriented vector bundle on $\xi$. If I choose a metric on $\xi$, then ...
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Dot product of functions on cosets

Let the measures of locally compact groups $\,K < G\,$ be $\, dk\,$ and $\, dg\,$, correspondingly. For a Hilbert space $\mathbb{V}$ equipped with a dot product $\,\langle~,~\rangle\,$, introduce ...
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Extension of a principal bundle

Let $G$ be a Lie group and $M$ a smooth manifold. The universal cover $\tilde M$ is a principal $\pi_1(M)$-bundle over $M$. Question: Why does any homomorphism $\phi:\pi_1(M)\to G$ induces a principal ...
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Bijection between quotient space of principal smooth and topological bundle.

I was reading the notes on bundles "Differential Topology of Fiber Bundles" by Karl-Hermann Neeb and on the final pages (138-139) "Homotopy theory of bundles" the author author ...
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Vector Bundles, Principal Bundles and Bundle of Frames

I have the following question. I have seen many times in the literature the following construction and I don't understand it too well, so I hope some of you can actually help me. For the sake of ...
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classifying space induces a equivalence of categories between PBun$_G(M)$ and $\Pi(M,BG)$ for finite groups $G$

Let $G$ be a finite group, $BG$ its classifying space and M a manifold. Then it is mentioned in https://arxiv.org/abs/1705.05171 (Remark 2.3 d) that there is an equivalence of categories $$ \Pi (M,BG) ...
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Principal $G$-bundle maps and sections of Associated Bundles

These days, I am making my way towards the classification result about homotopy classes of maps from a CW-complex $Y$ to $BG$ and isomorphism classes of principal $G$-bundles over $Y$, where $G$ is a ...
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From when did people started assuming a principal bundle is locally trivial?

Question is as in the title: From when did people started assuming a principal bundle is locally trivial? I am asking this because, Dale Husemoller in his book "Fibre bundles" (1966) ...
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Local trivializations and transition law of Atiyah Lie algebroid

Let $P\rightarrow M$ be a $G$-principal bundle and denote by $\mathrm{at}(P):=TP/G$ the Atiyah Lie algebroid over $M$. I want to understand how $\mathrm{at}(P)$ is a locally trivial Lie algebroid with ...
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Properties of connection principal bundle

I am working on a proof of Cartan's formula $$ DA(u,v) = dA(u,v) + [A(u),A(v)], $$ where $A$ is a connection one form on a principal bundle and $u,v$ are two vector fields. $D$ denotes exterior ...

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