Skip to main content

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 $G$.

Filter by
Sorted by
Tagged with
0 votes
1 answer
34 views

Vector bundle associated to the universal cover $\mathbb{R}\to S^1$

It's a well known fact that, given a principal $G$-bundle (where $G$ is a Lie subgroup of $\text{GL}(r,\mathbb{R})$) $$\pi_P:P\to X$$ there is an associated vector bundle $$\pi_E:E(P):=(P\times \...
Kandinskij's user avatar
  • 3,549
1 vote
1 answer
40 views

Why is the curvature a horizontal form?

Let $\pi:P\to M$ be a principal $G$-bundle, let $\mathfrak{g}$ be the Lie algebra of $G$ and let $\omega\in\Omega^1(P,\mathfrak{g})$ be a connection form, i.e. $$R_g^*\omega=\text{Ad}_{g^{-1}}\omega\ \...
Armando Patrizio's user avatar
3 votes
0 answers
42 views

Is there a simple description of the total space of a principal S^1 bundle over a compact surface?

It is known that principal $S^1$-bundles over a compact surface $\Sigma_g$ are classified by their Chern classes in $H^2(\Sigma_g, \mathbb{Z}) \cong \mathbb{Z}$. When the Chern number is zero, the ...
Rei Henigman's user avatar
  • 1,379
1 vote
0 answers
14 views

Why is the lift of a foliate vector field to the transverse bundle a foliate vector field w.r.t. the lifted foliation?

Let $M$ be a manifold with a foliation $F$ of codimension $q$, and let $p: B \rightarrow M$ be the transverse frame bundle (which over each point of $M$ consists of frames in the quotient of the ...
rosecabbage's user avatar
  • 1,665
0 votes
0 answers
16 views

Connection on $U(1)$-bundle vs. 1-form

My original idea of a $\mathfrak{u}(1)$-valued connection $\omega$ is that it's simply a normal $1$-form. But in the middle of page 3 of https://www.arxiv.org/abs/math/0511710 he says "a ...
user615345's user avatar
2 votes
1 answer
30 views

What is the action of $O(k)$ on $V_k(\mathbb R^n)$ making it a principal bundle?

Let $V_k(\mathbb R^n)$ be the Stiefel manifold of ordered $k$-tuples of vectors in $\mathbb R^n$. I have seen in many places that $V_k(\mathbb R^n)$ is an $O(k)$ principal bundle over the Grassmanian ...
Chris's user avatar
  • 3,177
1 vote
0 answers
50 views

Derivation of a spin connection in general relativity

On my journey to understand the mathematical structure behind general relativity, I came across the concept of a spin connection. which (I understand) is a connection defined in a spinor bundle. The ...
Tomás's user avatar
  • 191
2 votes
1 answer
37 views

Spinor bundles and spin structures

I want to check my understanding of the relationship between spinor bundles and spin structures. So, as far as I understand, given a principal $\operatorname{SO}(n)$-bundle $\pi : P_{SO} \rightarrow M$...
Tomás's user avatar
  • 191
2 votes
0 answers
66 views

Is the Hopf bundle $SU(2)/U(1)$ or the dual?

The Hopf bundle is defined as the quotient map of $S^3\subset \mathbb{C}^2$ by the action of $U(1)$ by multiplication $(z_1,z_2)\mapsto (e^{i\varphi}z_1,e^{i\varphi}z_2)$. It's known that it is the ...
Alex Bogatskiy's user avatar
0 votes
0 answers
37 views

Why $K_\nabla$ is the same as $K_{\omega_S}$?

Let $G\subset GL_n(\mathbb{R})$ be a Lie subgroup and let $M$ be a manifold. Consider $S$ a $G-$structure on $M$ and $\nabla$ a connection compatible with $S$. We know that if $E=E(P,V)$ is the vector ...
Armando Patrizio's user avatar
0 votes
0 answers
46 views

Ehresmann connections on general G-bundles?

There are two definitions of connections on bundles in terms of horizontal bundles: In the case of a principal $G$-bundle $P$, a connection is a subbundle $HP<TP$ such that $HP\oplus VP=TP$ and it ...
Alex Bogatskiy's user avatar
0 votes
1 answer
37 views

Why are Klein geometries flat?

A Klein geometry can be seen as a principal $H$-bundle of the form $G\to G/H$. It is known (cf. Sharpe and Chern's book) that Klein geometries are examples of flat Cartan geometries because their ...
Alex Bogatskiy's user avatar
2 votes
0 answers
47 views

The equivalence relation when constructing the associated bundle

When constructing from a typical fibre $F$, an action of a Lie group $G$ on that fibre and a $G$-principal bundle $P$ over some base space $M$ the associated bundle $P[F]$, one does this by ...
Jannik Pitt's user avatar
  • 2,050
1 vote
1 answer
31 views

Coframe on coset space via connection one form

Suppose $G$ is a Lie group and $H$ is a Lie subgroup. Then, consider the principal $H$-bundle, $\pi:G\longrightarrow G/H$ where $G/H$ is coset manifold and $\pi$ is the canonical projection. Then, we ...
lolabol's user avatar
  • 11
1 vote
1 answer
42 views

Why specifying values on a local section is enough to determine the local values of a tensorial form on a principal $G$-bundle?

Let $\pi: E \rightarrow B$ be a principal $G$-bundle. I have an initial context from which my question comes and I will explain how; I believe what I need is more general, but I may be wrong, and this ...
rosecabbage's user avatar
  • 1,665
4 votes
1 answer
51 views

Self duality of a connection is invariant under a gauge transformation

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$, and let $P\to M$ be a (smooth) principal $G$-bundle over an oriented Riemannian smooth 4-manifold $(M,g)$. Let $E=P\times_{\text{Ad}}\mathfrak{g}...
blancket's user avatar
  • 1,850
0 votes
1 answer
57 views

Understanding the definition of Principal bundle (Kobayashi and Nomizu)

In Foundations of Differential Geometry by Kobayashi and Nomizu they give the definition for "A (differentiable) principal fibre bundle over $M$ with group $G$". $M$ is a (differentiable) ...
Jagerber48's user avatar
  • 1,441
1 vote
1 answer
38 views

Covariant derivative induced by pullback connection under an automorphism

Let $G$ be a Lie group and $\pi:P\to M$ a smooth principal $G$-bundle. Let $\omega$ be a connection on $P$; it is a $\mathfrak{g}$-valued 1-form on $P$ where $\mathfrak{g}$ is the Lie algebra of $G$. (...
blancket's user avatar
  • 1,850
0 votes
0 answers
37 views

Definition of the Principal Bundle for smooth manifolds?

The fiber bundle is defined as A fiber bundle is defined as the tuple $(E, B, \pi)$ where $\pi: E \to B$ is a continuous surjective map from topological space $E$ to topological space $B$. ...
Jagerber48's user avatar
  • 1,441
1 vote
0 answers
26 views

Is this is valid/standard definition of a principal bundle?

I'm having a hard time understanding principal bundles. It seems there are a lot of definitions around which is making it even more confusing for me. Is this a valid definition of a principal bundle: ...
Jagerber48's user avatar
  • 1,441
0 votes
1 answer
48 views

Difference between $G$-principal bundle and fiber bundle with fiber $G$?

I'm having a hard time understanding principal bundles. I understand that a fiber bundle can be defined $(E, B, \pi, F)$ where $E, B, F$ are topological spaces and $\pi: E\to B$ is a surjection. For ...
Jagerber48's user avatar
  • 1,441
2 votes
1 answer
119 views

All actions of $SO(3)$ on $S^2$

There are two obvious (smooth left) actions of $\mathrm{SO}(3)$ on $S^2$. There is the standard action by which $\mathrm{SO}(3)$ acts by 3D rotations on the standard embedding of $S^2$ in $\mathbb{R}^...
Forgetful Functor's user avatar
0 votes
1 answer
30 views

About equivariant vector valued forms on principal bundle

Let $\pi: M \to E$ be a $G$-equivariant vector bundle and let us adopt the notation $$C^{\infty}(M,E)^{G} = \{\tilde{s} \in C^{\infty}(M,E) | \ \forall g \in G \ \tilde{s} \cdot g = \tilde{s}\}$$ ...
Integral fan's user avatar
2 votes
1 answer
120 views

Euler class of a principal $SO(2)$-bundle over a lens space

Note that for a manifold $X$, isomorphism classes of principal $SO(2)$-bundles over $X$ are classified by their Euler classes in $H^2(Z;\Bbb Z)$. Now consider a lens space $L(p,q)$; we have $H^2(L(p,q)...
blancket's user avatar
  • 1,850
0 votes
0 answers
42 views

Existence of principal G-bundle given an associated vector bundle

I am wondering if the following is true. Let $G$ be a Lie group, $V$ a vector space, $\rho$ a representation of $G$ on V, and $\pi: E\rightarrow M$ a vector bundle with fibre $V$. Does there exist a ...
Flo's user avatar
  • 11
4 votes
1 answer
90 views

Smoothness of horizontal bundle defined by connection one-form

Let $G$ denote a Lie group and $\mathfrak{g}$ its Lie algebra. $P$ is a smooth principal bundle. Given a smooth $\mathfrak{g}$-valued one-form $\omega_p: TP_p \to \mathfrak{g}$, which fulfils the ...
anonymous250's user avatar
0 votes
0 answers
12 views

Momentum space from principal bundle

I wish to consider a principal bundle which has a fibre $F$ of spatial $d$-dimensional momenta over a base manifold of $\mathbb{R}$. So, naturally the momenta $k^i$ is the generator of the structure ...
Dr. user44690's user avatar
0 votes
0 answers
103 views

A numerable G-principal bundle $E \rightarrow B$ is universal iff E is contractible

I am having some trouble understanding aspects of the proof that shows that a numerable $G$-principal bundle $E\rightarrow B$ is a universal $G$-bundle if $E$ is contractible. The proof starts off by ...
Topological Sigma Grindset's user avatar
1 vote
1 answer
91 views

Is the shear map of a smooth principal bundle always a diffeomorphism?

For a Lie group $G$, let $\pi \colon P \rightarrow M$ be a principal $G$ bundle over a smooth manifold $M$. There is a canonical shear map $ S \colon P \times G \rightarrow P \times_{M} P$ defined as $...
mathematics student's user avatar
4 votes
0 answers
145 views

Geometry of electrodynamics

In this question, I'd like to go over the physics - math dictionary occurring in the geometric structure (Principal bundle/spin bundles etc.) of Maxwell electrodynamics and the Dirac field. Consider ...
Integral fan's user avatar
1 vote
0 answers
106 views

Vanishing of connection matrices for flat principal $G$-bundle

Background Recall that for a real vector bundle, there is a well known integrability theorem. Theorem. Suppose there is a vector bundle $E$ with fiber $\mathbb R^n$. If $A$ is a flat connection on $E$,...
Mohith Nagaraju's user avatar
0 votes
1 answer
99 views

Equating two definitions of principal fiber bundles

I am following these lectures on principal fiber bundles. Here, a principal fiber bundle is defined as a fiber bundle of which total space $P$ has a right free action of some Lie group and which is ...
Lourenco Entrudo's user avatar
1 vote
1 answer
42 views

Principal G-bundle induces a homeomorphism between orbit space and base

I am reading Chapter 14 of the algebraic topology book by Tammo tom Dieck. However, it is not going smooth since the very beginning. I will give the definitions first: Def (Principal G-bundle). Let $G$...
Mizutsuki's user avatar
  • 494
1 vote
0 answers
64 views

3-connection on nontrivial 3-manifold

I'm studying Chern-Simons theory on topological nontrivial 3-manifold (I come from a physics background, so I'm new to some mathematical concepts). If the first homology group $H_1(M)$ is nontrivial ...
polology's user avatar
2 votes
0 answers
36 views

Does the monodromy action of a fiber bundle lie in the bundle's structure group?

Suppose we have a (topological) fiber bundle $p:E\to B$ with fiber $F$ and structure group $G$. Since $G$ acts on $F$ by homeomorphisms, it induces an action on the (integral) homology $H_*(F)$, i.e., ...
Steve's user avatar
  • 207
2 votes
0 answers
62 views

Are principal fibrations the same as group bundles?

I am reading Hatcher's algebraic topology, where he defines a fibration $F\to E\to B$ to be principal if up to choices of homotopy equivalences it can be written as $\Omega B'\to F'\to E'\to B'$ (with ...
DevVorb's user avatar
  • 1,433
0 votes
0 answers
32 views

Finding the group bundle associated to a $\mathbb{Z}[\pi_1]$ module

I am trying to understand twisted coefficients better, because although I technically know the definitions both using modules and group bundles, my intution of the subject doesn't really exist yet. To ...
DevVorb's user avatar
  • 1,433
3 votes
1 answer
168 views

Explicit formula for the principal connection 1-form induced by a Cartan connection

Let $P \subseteq G$ be a closed Lie subgroup. Suppose that a principal $P$-bundle $\mathcal{P} \to M$ is equipped with a Cartan connection $\omega: T\mathcal{P} \to \mathfrak{g}$. Then the extended ...
ಠ_ಠ's user avatar
  • 10.7k
0 votes
0 answers
28 views

Exponentiate additive transition functions for $\mathbb{A}^1$-bundles.

Consider an smooth complex elliptic curve $E$ glued from two affine curves ($p\in\mathbb{C}\setminus0$) $$C_{(x,y)}: y^2=x^3+px\\C_{(s,t)}: t^2=ps^3+s$$ via the coordinate change $s=1/x,t=y/x^2$. It ...
Display Name's user avatar
  • 1,375
3 votes
1 answer
250 views

Induced connection 1-form on orthogonal frame bundle by the Levi-Civita connection on tangent bundle

I recently posted a similar question but I have deleted that one and replaced it with this one, which is hopefully more focused. On page 317 of this paper by Pu-Young Kim and Joon-Sik Park they say ...
CBBAM's user avatar
  • 6,009
0 votes
1 answer
142 views

Understanding Associated Frame bundle terminology $u e_i = e_j^i X_j$

I am self-studying https://sayanmuk.github.io/StochasticAnalysisManifolds.pdf and I am struggling with the definitions on page 37-38 Let us now see how the connection $\nabla$ manifests itself on ...
Pedro Gomes's user avatar
  • 3,931
0 votes
0 answers
59 views

Valid interpretation of Higher order frame bundles and their jet groups?

I've been trying to develop an intuition for higher order frame bundles to help me understand them and this is what I've come up with. Criticisms welcome, as I'm not sure it's valid? NOTE: Always I ...
R. Rankin's user avatar
  • 338
5 votes
1 answer
326 views

Why does the bar construction model the classifying space in both topology and AG?

For a topological group $G$, we can construct the classifying space $BG$ as the geometric realization of the nerve of $G$. I have seen a very similar assertion in the context of algebraic geometry: ...
Hyunbok Wi's user avatar
1 vote
1 answer
130 views

A question related to frame bundle of a vector bundle

I am currently reading up on principal fiber bundles from a set of lecture notes on the subject, and I am trying to make sense of frame bundle of a vector bundle. Consider a vector bundle $p:E\to M$ ...
neophyte's user avatar
  • 520
1 vote
0 answers
70 views

Doubts on definition of spin structure

In Friedrich's book "Dirac operators in Riemannian geometry" there is this definition for spin structure. Let $X$ be a CW-complex and $(Q, \pi, X, SO(n))$ a $SO(n)$-principal bundle on $X$. ...
Wukong's user avatar
  • 11
0 votes
0 answers
27 views

Some other perspective of reduction of structure group

$(E,p,M)$ is a fiber bundle. Suppose there exists a local trivialization $LT_1$ of $E$ such that on the overlapping areas the transition can be realized by left action of a topological group $G$ (...
jw_'s user avatar
  • 509
1 vote
0 answers
21 views

$F \subset E$ as a sub-vector bundle of E, Show that it is a $GL_p$-reduction of $Fr(E)$

Suppose $F \subset E$ is a sub-bundle of E,where one uses "frames of $E$ adapted to $F$", meaning that the first $p$ components of the frame are a frame of $F$ (where we denoted by $r$ and $...
Z.Y.H's user avatar
  • 141
0 votes
1 answer
82 views

Formal definition of a $U(1)$ connection

Let $\pi:P \rightarrow M$ be a $U(1)$ principal bundle. I often see people refer to a "$U(1)$ connection" but I cannot find a formal definition of this term. The closest I got was this ...
CBBAM's user avatar
  • 6,009
2 votes
0 answers
53 views

Is the quotient manifold theorem true if we don't require the action to be smooth?

Let $M$ be a topological manifold with a continuous, free, and proper action of a topological group $G$ on $M$. Is $M/G$ a topological manifold? Is $M \to M/G$ a locally trivial principal bundle? If ...
Math's user avatar
  • 279
2 votes
1 answer
114 views

Differential form taking values in a vector bundle?

Let $\pi: P \rightarrow M$ be a $G$-principal bundle. Define $\Omega^k_{\text{hor}}(P, \mathfrak{g})^{\text{Ad}}$ to be the set of $k$-forms $\omega$ taking values in the Lie algebra $\mathfrak{g}$ ...
CBBAM's user avatar
  • 6,009

1
2 3 4 5
15