Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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 $G$.

0
votes
0answers
31 views

Transition maps of principal bundle are smooth

Let $P(M,G)$ be a principal bundle. We choose an open covering $\{U_\alpha\}$ of $M$ and trivializations $\psi_\alpha:\pi^{-1}(U_\alpha)\rightarrow U_\alpha\times G$ defined as $\psi_\alpha(u)= (\...
2
votes
1answer
43 views

Looking at the connection 1-form on a principal G-bundle in coordinates

I'm reading this paper, and I'm confused about something. Let $A$ be a connection on a principal $G$-bundle $P$ over $\mathbb{R}^4$, and $F(A)$ its curvature. Let $\mathrm{ad}(P)=P\times_G\mathfrak{g}...
1
vote
1answer
43 views

G-principal bundle and homotopy retract

Suppose that $f:X\rightarrow Y$ a continuous map between (connected) CW-complexes such that there exists a continuous map $g:Y\rightarrow X$ with the property that $g\circ f$ is homotopy equivalent to ...
1
vote
1answer
58 views

Showing the Hopf fibration has no global sections

Let's consider a principle $U(1)$-bundle over $S^2$ with the transition function $g_{\infty 0} = z/|z|$ (it is known as the Hopf fibration). There is a simple topological argument showing that this ...
0
votes
0answers
24 views

Showing simple transivity of action on fibre

Let $(P,X,\pi, G)$ be a principal $G$-bundle (definition below). I want to show (a) $\pi(u\cdot g)= \pi (u)$ (b) If $\pi(u)=\pi(v)$ then, there exists unique $g\in G$ such that $u=v\cdot g$. (a) ...
1
vote
0answers
24 views

Why is a connection on the bundle $SO(M)$ metric compatible?

If we have an orientable manifold $M$ with a metric $g$ and signature $(r, s)$, we can define the principal-$SO(r, s)$ bundle $SO(M)$, the bundle of orthonormal frames of $TM$. This is a subset of the ...
0
votes
0answers
24 views

Transitivity of the action in definining principal bundles

In the wikipedia article https://en.wikipedia.org/wiki/Principal_bundle the definition of a principal $G-$bundle $\pi:P\rightarrow X$ demands that the action of $G$ on $P$ to be free and transitive. ...
1
vote
0answers
34 views

Correspondence between flat connections and fundamental group representations

Let $M$ be a manifold. Two stackexchange posts state that there is a correspondence $$ \{ (P,A): P \text{ a $G$-bundle}, A \text{ flat connection} \} \leftrightarrow \{ \text{morphisms } f:\pi_1(M) \...
0
votes
1answer
85 views

Universal Bundle — Understand the basic definition

I read two versions of discussions on universal bundles. I could not really see how the two definitions are really the same. From Wiki. The universal bundle in the theory of fiber bundles with ...
3
votes
0answers
40 views

Two notions of classifying space

Why is the categorical classifying space for a group G, i.e., geometric realization of the nerve of G(as a category of One object), the same as the topological classifying space for principle G ...
2
votes
0answers
43 views

Line bundles associated to principal circle bundles

Let $\pi: P \rightarrow B$ be a principal circle bundle over $B$ and $\rho: S^1 \times \mathbb{C} \rightarrow \mathbb{C}$ an effective left action. Then, one can associate to the bundle $\pi$ a ...
4
votes
1answer
81 views

Transitive action of $H^2(M;\Bbb Z)$ on $Spin^c$ structures over $M$

I’ve a problem understanding why the action of the second cohomology group (integer coefficients) of an oriented smooth manifold $M$ is free and transitive on the set of $Spin^c$. I’m following these ...
2
votes
1answer
56 views

If $H\leq G$, then $BH \to BG$ is a fiber bundle with fiber $G/H$

Suppose $G$ is a topological group and $H\leq G$ is a closed subgroup. The inclusion $H\to G$ induces a map on classifying spaces $BH\to BG$. I've seen in some sources that $BH\to BG$ is actually a ...
0
votes
0answers
22 views

Classification of principal bundles and of regular coverings

Let $G$ be a discrete group and let $X$ be a good space with fundamental group $\pi_1$. We know the following things: The connected principal $G$-bundles over $X$ are exactly the regular coverings of ...
2
votes
0answers
63 views

From sheaf torsors to geometric bundles on schemes

$\DeclareMathOperator{\Spec}{Spec}$ $\DeclareMathOperator{\Sym}{Sym}$ $\newcommand{\func}{\mathcal{O}}$ $\newcommand{\M}{\mathcal{M}}$ It is well known that the notions of locally free $\func_X$-...
2
votes
0answers
52 views

How to arrive at frame bundle objects from chart calculation on the base manifold

In this lecture by Fredric Schuller it is said that the wave-function is not a wave-function. He attempts to find an appropriate coordinate independent derivative using "chart calculations" on the ...
3
votes
2answers
105 views

How to recover the covariant derivative from the pull back from that on the principal bundle

I am watching these lecture series by Fredric Schuller. Covariant derivatives - Lec 25 - Frederic Schuller @minute 01:10:11 When we arrive at the covariant derivative from the principal bundle $P$ ...
0
votes
1answer
66 views

Exterior Covariant Derivative - Proof of Structure Equation for general r-form

Let $(P,M,\pi,G)$ be a principal bundle and $\omega \in \Omega^{1}(P,\mathfrak{g})$ a principal connection. Given a representation $\rho : G \to GL(V)$ and an equivariant form $\eta \in \Omega^{r}(P,V)...
1
vote
0answers
44 views

Mathematical challenge for unification of gravity and electromagnetism in classical theory?

I am trying to better understand the mathematical foundations of a possible reconciliation between quantum field theory and gravity in general relativity. However, before the application of the ...
0
votes
1answer
31 views

Question regarding global sections of principal bundles over a smooth oriented surface

I was reading a proof of the following theorem by A. Ramanathan : Let $X$ be a smooth connected oriented surface and $G$ be a connected topological group. Then there is a bijection between the set of ...
1
vote
1answer
29 views

Pullback of local sections over the total space

Let $\pi\colon P\to X$ be some $G$-principal bundle; $\{U_\alpha\}$ a cover of $X$; and $\{s_\alpha\colon U_\alpha\to\pi^{-1}(U_\alpha)\}$ a collection of local sections. Claim: The pullback $\pi^*P$ ...
2
votes
0answers
50 views

Spherical coordinates on the 2-sphere and k-forms

Consider the 2-sphere $S^2$inside $\mathbb R^3$, and let \begin{align} S:(0,\infty)\times(0,\pi)\times(0,2\pi)&\to\mathbb R^3\setminus\{(0,0,0)\} \\ (r,\phi,\theta)&\mapsto(r\sin\phi\cos\theta,...
3
votes
0answers
36 views

Space of principal connections is affine modelled on $\Lambda^1(M;\mathfrak{g})$?

I'm working within the jet-formulation espoused by Saunders in "The Geometry of Jet Bundles" and am struggling to prove the stated result. I would like to stay in this context and understand the ...
0
votes
0answers
31 views

Proof of a lemma concerning pull-backs of bundles

One has the following definitions: Let $f: M \rightarrow N$ be a smooth map between manifolds and let $\pi: P \rightarrow N$ be a $G$-principal bundle over $N$. The $pull-back$ of $P$ by $f$ is ...
2
votes
0answers
28 views

How to interpret $\nabla_i = \frac{\partial}{\partial x_i} +A_i$ in terms of a connection $A$ on a principal bundle.

I'm reading Hitchin's paper Self-duality Equations on a Riemann Surface (Hitchin, self duality) In the first chapter on pages 63/64 he considers a principal $G$-bundle $P$ over $\mathbb R^4$ and a ...
3
votes
1answer
29 views

Reading request: induced representations & fiber bundles

To my understanding, the theory of induced representations of a group $G$ can be formulated in terms of vector bundles associated with the group considered as a principal bundle ${G \to G/H }$. I'm ...
2
votes
0answers
37 views

Recovering a principal connection from its monodromy

Given a principal bundle $P \to M$ with structure group $G$ ($M$ and $G$ are connected), it is well known that one can recover the data of $P$ and a flat connection $\Gamma \in \Omega^{1}(P, \mathfrak{...
0
votes
0answers
26 views

Is $G$ action on a principal bundle a smooth submersion?

Let $\pi : P \rightarrow M$ be a principal $G$ bundle. Suppose $R_{g}: P \rightarrow P$ is a smooth map induced by the smooth right $G$ action on $P$. Is $ R_{g} $ a smooth submersion for all $g \in ...
0
votes
1answer
15 views

Lift of principal bundle over a compact Lie group

Suppose $\alpha:M\to N$ is a homomorphism of compact Lie groups and $\eta=(E,B,N)$ is an $N-$principal bundle over a compact manifold. Give a condition of $\alpha$ such that $\eta$ can lift as an $M-$...
4
votes
0answers
76 views

Is a principal bundle of a principal bundle still principal?

Let $\left(P_1,\pi_1,M,G_1\right)$ and $\left(P_2,\pi_2,P_1,G_2\right)$ be two principal bundles, where $M$, $P_1$ and $P_2$ are differential manifolds, and $G_1$ and $G_2$ are Lie groups. With these ...
1
vote
0answers
31 views

Is a certain subset of a group acting on a Hausdorff space closed?

Let $G$ be a topological group acting continuously on a Hausdorff space $X$. For $K\subset X $ compact, define \begin{equation} G_K=\{ g\in G : g K\cap K \neq \varnothing \}.\end{equation} Is $G_K$ ...
0
votes
1answer
28 views

Question concerning the relation between the reduction of a $G$-principal bundle to some subgroup $H$ of $G$ and Čech cocycles.

Let $G$ be a Lie group and $M$ be a smooth manifold. I have to show that a $G$-principal bundle admits a reduction to some subgroup $H$ of $G$ if and only if there exists a Čech cocycle $\{(U_{\...
4
votes
1answer
145 views

How does one introduce characteristic classes

How do you introduce or how are you introduced to characteristic classes. I am assuming the student is comfortable with principal bundles and connections on principal bundles.
1
vote
0answers
53 views

References: Equivalence between local systems and vector bundles (with flat connections)

I am working on a topic related to the Riemann-Hilbert Correspondence, and I was wondering if there is a reference that explicitly goes over the following equivalence of categories (when $M$ is a ...
0
votes
1answer
111 views

References for endomorphism bundle and adjoint bundle

I am trying to understand what are endomorphism bundle(of a vector bundle) and adjoint bundle(of a principal bundle) but could not find any references on google. Searching Adjoint bundle gives https:...
0
votes
0answers
25 views

Matrix representation of connections and curvatures on a principal(vector) bundle

Let $E\rightarrow B$ be a vector bundle and $\nabla$ be a connection on this vector bundle. I read some where that this connection and it’s curvature form can be represented by a matrix. I do not ...
1
vote
1answer
77 views

Associated principal bundle for a given vector bundle

The book I am referring is Foundations of Differential geometry by Kobayashi and Nomizu. In second volume, chapter on Characteristic classes, they write the following. Let $E$ be a complex vector ...
2
votes
0answers
27 views

Proving that a form is horizontal in the Chern Weil method proof

Follow up to this question. $\omega$ is a one parameter family of connections on a principal bundle of frames $E$, associated to a complex vector bundle $V\to M$. In the proof of the Chern Weil ...
1
vote
1answer
106 views

Different definitions of irreducible $\mathrm{SU}(2)$ connections

Let $Y$ be Poincaré's integral homology 3-sphere. Let $\pi:P\to Y$ be a (necessarily trivial) $\mathrm{SU}(2)$-principal bundle over $Y$. Fix $x_0\in Y$ and $p_0\in P_x:=\pi^{-1}(x_0)$. The ...
2
votes
1answer
38 views

Killing field associate to an element in the Lie Algebra

Given a principal $G$ bundle $E\to M$, where $G$ is a Lie group, I was told that for every $u \in \mathfrak{g}$ (the Lie algebra of $G$) we can define a $G$-invariant vector field $X_u$ over $E$. I'm ...
2
votes
0answers
78 views

Holonomy group and irreducible $\mathrm{SU}(2)$-connections

Let $P\to M$ be a $\mathrm{SU}(2)$-principal bundle over a closed connected manifold $M$ and let $A$ be a flat connexion form on $P$. Fixing $x\in M$ we have a homomorphism $$ \mathrm{Hol}_x(A) : \...
1
vote
1answer
72 views

Prove that a “tensor product” principal $G$-bundle coincides with a “pullback” via topos morphism

From Moerdijk, Classifying spaces and classifying topoi, page 22. Consider a right $G$-set $S$ with the discrete topology. Let $E$ be a principal $G$-bundle over the topological space $X$. One can ...
2
votes
2answers
51 views

Terminal object for Prin(X,G) (principal $G$-bundles)

Consider the category $Prin(X,G)$ of principal $G$-bundles over a topological space $X$ ($G$ is a group). My question is: does this category have a terminal object? I would say that the category of $...
0
votes
1answer
38 views

Holonomy bundle is a covering space

I have trouble understanding following content, this is from Foundations of Differential geometry by Kobayashi and Nomizu in section on Flat connections. Let $\Gamma$ be a flat connection in ...
2
votes
0answers
42 views

Smooth Principal Bundle from continuous transition functions?

Setting: Let $M$ be a smooth manifold and $\{U_\alpha\}_{\alpha \in \mathcal{I}}$ a locally finite covering of open subsets. Furthermore let $G$ be a smooth Lie group. Now assume we are given a family ...
6
votes
1answer
125 views

Lie algebra of a horizontal vector field and a fundamental vector field

Let $G$ be a Lie group, $\mathfrak{g}$ be its Lie algebra, $(P,M,G)$ be a principal $G$ bundle with connection $\omega$. By a horizontal vector field on $P$, we mean a vector field $X:P\rightarrow TP$...
2
votes
1answer
61 views

Universal covering as a principal $G(\tilde{X})$-bundle

Let $X$ be a manifold and consider the universal covering $$p:\tilde{X}\longrightarrow X$$ I know that this has a structure as a principal $G(\tilde{X})$-bundle but I can't manage to correctly define ...
1
vote
1answer
50 views

Vertical vectors on a principal bundle

Let $$\pi:P\longrightarrow M$$ be a principal $G$-bundle, and consider, for each $p\in P$, the map $$\begin{matrix}l_p:&G&\longrightarrow&P\\&g&\longmapsto &p\cdot g\end{matrix}...
2
votes
1answer
110 views

Local Form of Covariant Derivative Induced from a Connection one-form

Let $P\rightarrow M$ be a Principal G-Bundle with $E=P\times_\rho V$ the associated vector bundle with $\rho$ a representation of $G$ on $GL(V)$. Also let $\omega$ be a connection one-form on $P$ i.e. ...
2
votes
0answers
53 views

What is meant by an Orthogonal $1$-form

In this article, titled Holonomy Groups in Riemannian Geometry, on pg 37, the following line appears (before equation 2.5.3) Let $\theta$ be a horizontal, orthogonal $1$-form on $\mathcal F_{GL}$ ...