The tag has no wiki summary.

learn more… | top users | synonyms

12
votes
1answer
431 views

Local triviality of principal bundles

Suppose I define a principal $G$-bundle as a map $\pi: P \to M$ with a smooth right action of $G$ on $P$ that acts freely and transitively on the fibers of $\pi$. Does it follow that $P$ is locally ...
10
votes
1answer
223 views

Why base point makes a huge difference?

While preparing a talk, I was tempted to "prove" the following relationship: $$\text{Prin}_{G}(X)\cong [X,BG]\cong [B[\pi_{1}(X),BG]\cong [\Omega B[\pi_{1}(X)],\Omega BG]\cong [\pi_{1}(X),G]$$ Here ...
9
votes
3answers
854 views

Texts on Principal Bundles, Characteristic Classes, Intro to 4-manifolds / Gauge Theory

I am looking for a textbook that might serve as an introduction to principal bundles, curvature forms and characteristic classes, and perhaps towards 4-manifolds and gauge theory. Currently, the only ...
8
votes
1answer
195 views

Equivalence of Definitions of Principal $G$-bundle

I've finally gotten around to learning about principal $G$-bundles. In the literature, I've encountered (more than) four different definitions. Since I'm still a beginner, it's unclear to me whether ...
6
votes
1answer
82 views

Flat connection with non-trivial holonomy? I cannot get it

maybe this is a dumb question, but I cannot understand how a principal $G$-bundle can have non-trivial holonomy with a flat connection. Maybe I'm missing something, but doesn't Ambrose-Singer theorem ...
6
votes
1answer
95 views

Formal Definition of Yang Mills Lagrangian

I have a question regarding the Lagrangian in non abelian gauge theory. Say, $G$ is the gauge group and $\mathfrak g$ the associated Lie algebra. The Lagrangian is often written as $$ \mathcal ...
6
votes
0answers
169 views

Orbit space of a free, proper G-action principal bundle

Let $G$ be a topological group and let $r \colon E \times G \to E$ be a continuous right-action on a topological space $X$. If $p\colon E \to B$ is a continuous map into a topological space $B$ such ...
5
votes
1answer
58 views

When can a connection be lifted?

Let $P \rightarrow X$ be a principal $G$-bundle, and $P' \rightarrow X'$ be a principal $G'$-bundle. Let $(f',f'')$ be a morphism from $P'$ to $P$, i.e., a pair of maps $f': P' \rightarrow P$ and ...
5
votes
1answer
122 views

How does a left group action on the fiber of a principal bundle induce a right action on the total space?

Suppose I define a "principal $G$-bundle" as follows: A principal $G$-bundle is a fiber bundle $F \to P \overset{\pi}{\to} X$ with a left group action of $G$ on $F$ that is free and transitive, ...
5
votes
1answer
117 views

One-form on quotient manifold

Let $M$ be a smooth manifold with tangent bundle $TM$ and cotangent bundle $TM^*$ and $\psi\in TM^*$ a one-form. We denote the quotient manifold of $M$ by the free and proper $G$-action $\varphi$ as ...
5
votes
2answers
99 views

Principal $\mathrm{SL}_n$-bundles

It seems to be well-known that a principal $\mathrm{SL}_n$-bundle on a scheme or manifold $X$ is the same as a vector bundle of rank $n$ whose determinant is a trivial line bundle. One direction is ...
5
votes
0answers
60 views

Exact sequence of tangent spaces of principal $G$-bundles

Let $P$ be a smooth manifold, $G$ a Lie group, $\alpha:P\times G\to P$ a smooth action and $p:P\to P/G$ a smooth principal $G$-bundle. Then, we have the sequence $$ G \xrightarrow{\alpha_a} P ...
5
votes
1answer
105 views

Explicit description of flat connections under pullback on principal bundles over Riemann surfaces

I'm trying to find a proof/reference for a statement that I've seen quoted in some way or the other, but without reference. The setting: let $P\longrightarrow M$ be a flat principal $G$-bundle over ...
4
votes
1answer
146 views

Principle G bundles v.s. Flat G connection

What is the difference between Principle G bundles v.s. Flat G connection? I heard that for a discrete group $G$ (in physics, or a finite group $G$ in math), the principle G bundles is the same ...
4
votes
1answer
55 views

“Bundle of metrics” on a principal bundle?

I've come across the term "bundle of metrics" on a principal bundle. In particular, my setting is that for $N\longrightarrow M$ a universal cover of a compact Riemann surface, $P\longrightarrow M$ a ...
4
votes
1answer
352 views

Intuition of Chern-Weil theory

Let $ P \rightarrow M$ be a $G$-principal bundle. The lie algebra of $G$ is $\frak{g}$ and $P$ has connection form $\omega \in H^1(P,\frak{g})$ and curvature form $\Omega \in H^2(P,\frak{g})$. We ...
4
votes
1answer
95 views

Problems understanding the construction of Hitchin moduli space in his paper “The self-duality equations on a riemann surface”

First, if this post must be broken up in separate questions, please tell me so. I thought it would be better if I simply posed my questions in one thread, as they are directly related to each other. ...
4
votes
1answer
131 views

Classifying map

Let $\xi=(E,p,B)$ a principal $G$-bundle and $\eta=(P,\pi,Q)$ a real vector bundle such that $\operatorname{rank}(\eta)=n$. We can consider a classificant space $BG$. What is the classifying map $f:X ...
4
votes
1answer
26 views

Lifting Riemannian metrics on principal bundles

Given a principal bundle $\pi:M\rightarrow M/G$, there are natural maps $$\pi_{\mathcal{F}}:\mathcal{F}(M)^G\rightarrow\mathcal{F}(M/G)$$ ...
4
votes
0answers
68 views

Connections on principal bundles and vector bundles

In Donaldson and Kronheimer's book on the geometry of four manifolds, a brief review of connections on principal bundles is given. Three equivalent definition are stated: 1) Via horizontal subspaces, ...
4
votes
1answer
68 views

Canonical connection on $CP^n$

I have heard something along the lines of "There is a canonical $U(1)$ connection on $CP^n$" and I am trying to understand what that means. First I suppose that the sentence refers to a line bundle ...
4
votes
0answers
179 views

Group actions and associated bundles

Let $P$ be a principal $G$-bundle over $B$, and let $G$ act on some space $F$ (feel free to work in your favorite category of spaces, if this helps). Then $\text{Aut}{P}$ (aka the group of gauge ...
3
votes
1answer
69 views

Naive question: what good are characteristic classes of principal bundles?

I recently read a development of characteristic classes on principal bundles through curvature forms and the Chern–Weil homomorphism. Unfortunately, this exposition concluded without listing any ...
3
votes
1answer
98 views

Learning Fibre Bundle from “Topology and Geometry” by Bredon

Bredon defines bundle projection in the following way: $\bf13.1.$ Definition. Let $X,B$ and $F$ be Hausdorff spaces and $p:X\to B$ a map. Then $p$ is called a bundle projection with fiber $F$, if ...
3
votes
1answer
84 views

When does the difference between a vector bundle and the associated frame bundle matter?

In the comments to this question How a principal bundle and the associated vector bundle determine each other, it was remarked that while there is a bijective correspondence between rank $n$ vector ...
3
votes
0answers
38 views

How are the components of a connection on a homogenous space related to the Mauer-Cartan form?

I am finding it hard to understand in what way the Mauer-Cartan form $\omega_G$ of a Lie group $G$ can be used to define a connection on a bundle $G \to G/H$ in the same way that parallel transport of ...
3
votes
0answers
64 views

What is the commutator of a horizontal and vector field for a connection on a Fiber bundle?

I would be tempted to rephrase my question as : why do people seem to care only about the curvature of a connection on fiber bundles ? Indeed, the curvature gives the vertical part of the commutator ...
2
votes
2answers
94 views

Is the $G$-action on a principal $G$-bundle proper?

Let $G$ be a Lie group. If $G$ acts properly and freely on a manifold $P$, then it is well-known that $P \to P/G$ form a principal $G$-bundle. I would like to know the converse: namely Question: if ...
2
votes
1answer
202 views

How a principal bundle and the associated vector bundle determine each other

It seems to me that given a vector bundle, the associated principal bundle is univocally determined. In fact one has to construct a principal bundle given the base, the fibre (the group $G$ in which ...
2
votes
3answers
100 views

$M \times N$ orientable if and only if $M, N$ orientable

For two manifolds $M$ and $N$ I'm trying to prove that $M \times N$ is orientable if and only if $M$ and $N$ are orientable. My attempt so far: $\impliedby)$ Assume $M, N$ are orientable. Then ...
2
votes
1answer
134 views

If there exists a global section then the principal bundle is trivial - problem with smoothness

Let $\pi \colon P \to B$ be a principal $G$-bundle and let $s \colon B \to P$ be it's smooth section. In order to show that $P \simeq B \times G$ I define the map $\varphi \colon P \to B \times G$ by ...
2
votes
1answer
73 views

Characteristic classes not defined on vector bundles

If you read the definition on Wikipedia, you'll see that they allow characteristic classes to be defined on general principal $G$-bundles (vector bundles being subsumed in this general case by looking ...
2
votes
2answers
73 views

If a connection on a principal $G$-bundle restricts to an $H$-subbundle, must its holonomy lie in $H$?

Let $P \to M$ be a principal $G$-bundle, equipped with a principal connection $D$. Let $Q \subset P$ be a principal subbundle with fiber $H$, where $H \leq G$ is a (let's say closed and connected) ...
2
votes
2answers
176 views

Principal bundle automorphism generating global gauge transformations

Consider a principal $G$-bundle $P$ with connection form $\omega$. An automorphism $f$ of $P$ is by definition a (smooth) $G$-equivariant map: $f(p \cdot g) =f(p) \cdot g$ for all $p\in P$ and $g\in ...
2
votes
1answer
124 views

Connections and covariant derivatives

Let $A$ be a connection on a principal $G$-bundle $P$, let $\chi :G\rightarrow GL(V)$ be a representation of $G$, and let $E:=P\times _\chi V$ be the associated gauge bundle. Then, there is a ...
2
votes
2answers
100 views

Manifold non-orientable iff. frame bundle is connected

Let $M$ be a connected smooth manifold and $L(M):=\bigcup_{x\in M}L_xM$ its frame bundle where $L_xM:=\{(v_1,\dots,v_n):\{v_1,\dots,v_n\}\text{ is a basis of }T_xM\}$. $M$ is non-orientable iff. ...
2
votes
1answer
112 views

Show that an Ehresmann connection on a principal G bundle is equivalent to a Lie Algebra Valued one form.

Let $E$ be a smooth principal $G$-bundle on M. The vertical bundle $V$ is defined as $V=\ker(d\pi:TE\to \pi^*TM)$. An Ehresmann connection on $E$ is a smooth subbundle $H$ of $TE$ (also called the ...
2
votes
1answer
40 views

This result holds in general or just for vector bundles?

If $(P,\pi, M)$ is a principal $G$-bundle, then given a left $G$-space $F$, using the $G$-product we can create a new bundle $(P_F, \pi_F, M)$ that is said to be associated to the first. Also, if we ...
2
votes
1answer
49 views

On the structure of a vector bundle

Let $P \rightarrow X$ be a principal $G$-bundle, $\rho: G\rightarrow GL(V)$ and $\sigma: G\rightarrow GL(W)$ be two finite dimensional linear representations of $G$. Let $E=P\times_\rho V$ and ...
2
votes
1answer
79 views

a question about space of smooth sections

Let $\Gamma(M,L) $ be the space of smooth sections, then why $\Gamma(M,L) $ is isomorphic to $A=\{f:L^{\times}\to \mathbb{C}; f(cz)=c^{-1}f(z), c\in \mathbb{C}-\{0\} , z\in L^{\times}\}$ . Here ...
2
votes
2answers
104 views

Universal property of universal bundles.

A classifying space for a group $G$ is a topological space $BG$ with a principle $G$-bundle $p : EG \to BG$ where $EG$ is contractile, so that $BG = EG/G$. A classifying space is universal in the ...
2
votes
0answers
18 views

Contraction of second exterior covariant derivative with metric

Let $G \hookrightarrow P \to M$ be a principal $G$ bundle, $P \times_\rho V$ be a vector bundle associated to representation $\rho$ of $G$ on $V$. If $\omega$ is a connection $1$-form on $P$ then we ...
2
votes
0answers
55 views

Reference for principal bundles and related concepts

I am looking for a good reference for fibre bundles, Ehresmann connections, principal $G$-bundles and principal Ehresmann connections (the $G$-equivariant version of Ehresmann connections). Could ...
2
votes
0answers
48 views

How to decompose connections on the complexified orthonormal frame bundle?

Let $E\rightarrow M$ be an orientable vector bundle of rank n equipped with some Riemannian metric, $P:=F_{SO(n)}(E)$ the orthonormal frame bundle. I say that $P^{c}:=F_{SO(n)}(E)\times_{SO(n)} ...
2
votes
0answers
62 views

Clarification on some notation and “assumptions” in page 143-144 of the book “Quantum Fields and Strings: A Course for Mathematicians, Volume 1”

I was trying to read the chapter $1$ (at page $143$) of this book Quantum Fields and Strings: A Course for Mathematicians, Volume 1 that is supposed to be an introduction to modern quantum field ...
2
votes
0answers
45 views

Heisenberg manifold

I am interested in the Heisenberg manifold, which is the quotient of the real Heisenberg group by the discrete Heisenberg (sub)group. It is a 3 -manifold which may be viewed as a circle bundle over ...
2
votes
0answers
96 views

Local trivializations for orthonormal frame bundle

Let $(E,\pi, M)$ be a real vector bundle of Rank $N$. Then one can define its frame bundle $GL(E)$ as follows: $GL(E)_x:=\{\text{ordered bases of }E_x\}$ (for $x\in M$). $GL(E):=\bigcup_{x\in M} ...
1
vote
1answer
64 views

Principal Bundle- Definition cum Exercise from “Geometry and Topology” by Bredon

The definition of fiber bundle can be found from here: Definition of Fiber Bundle Then Bredon defines Principal bundle in the exercise as follows: I am not able to show how K acts naturally on ...
1
vote
1answer
55 views

Difference between various type of bundles having a group as fibre

I am trying to understand the difference between these three objects: 1- a fiber bundle in which the fiber is a group $G$ 2- a fiber bundle in which the fiber is a group $G$ and the structure group ...
1
vote
1answer
148 views

Principal stable $SL(2)$-bundles on a genus $2$ compact Riemann surface.

Let $X$ be a compact Riemann surface with genus $2$. Can you give me examples of stable principal $SL(2)$-bundles on $X$?