Tagged Questions

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 $G$.

711 views

260 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 ...
97 views

Does a left group action on a principal bundle induce an action on associated vector bundles?

Let $G\hookrightarrow P\xrightarrow{\pi}M$ be a principal $G$-bundle with right action $\cdot$ and suppose we are also given a left action $\rho: U\times P\rightarrow P$ of some group $U$ on $P$. ...
344 views

Classify sphere bundles over a sphere

Problem (1) Classify all $S^1$ bundles over the base manifold $S^2$. (2) Do the same question for $S^2$ bundles. Moreover, does there exist a universal method to solve this kind of problem?...
404 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 $g\in G$...
77 views

Applications of Principal Bundle Construction: Vague Question

I recently read the principal $G$-bundle construction on a smooth manifold $M$, where $G$ is a Lie group. To understand them better, I am looking for some applications. Can the principal $G$-...
31 views

Trouble proving identity - Gauge theory/Maurer-Carton one-form/Adjoint representation

The Identity I am trying to prove is the one in this already asked question how to show that ${ad}_{g_{\alpha\beta}} \circ g_{\alpha\beta}^{\star}\theta=-g_{\beta\alpha}^{\star}\theta$? The author ...
172 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 $X$?
Proof of $G\rightarrow G/H$ is a Principal H bundle
Let $G$ be a Lie group and let $H$ be a closed subgroup (not necessarily normal). Then $G$ is a principal $H$-bundle over the (left) coset space $G/H$. I could proof that the fibers are all ...
Let $f \,: X \to Y$ be a map between connected spaces. Let: $$X^{(k)} \to \ldots \to X^{(0)} \approx X$$ and $$Y^{(k)} \to \ldots \to Y^{(0)} \approx Y$$ be whitehead towers for $X$ and $Y$. What ...