0
votes
0answers
151 views

Does every real vector bundle admit a metric?

Let $ E \longrightarrow B $ be a vector bundle. I know that if B is paracompact, the bundle admits a metric. My question is the following: is this true for any B? I also know that a reduction of ...
1
vote
2answers
1k views

Vector bundles vs principal $G$-bundles

It is well known that a (real) vector bundle $\pi : E\to B$ over a topological space (or manifold) $B$ is a fibre bundle whose fibres $$F=\pi^{-1}(x), \ \ \ x\in B $$ over any $x\in B$, are ...
2
votes
1answer
225 views

family of torsors and family of vector bundles

Suppose $X$ and $Y$ are smooth connected schemes over a field $k=\bar{k}$, $f: X\times_kY\to X$ is the first projection. You may assume $Y$ is proper if you like, my question is if $P\to X\times_kY$ ...
8
votes
2answers
878 views

Nice example of a topologically trivial bundle with nontrivial connection

So, I've been trying to understand what exactly an anomaly is, and how they arise in physics. Apparently an anomalous theory is some theory whose action is given by a section of some bundle (rather ...
2
votes
0answers
653 views

Vector bundles on some non-projective surfaces

Let $X$ be a smooth projective curve over a field $k$ and let $L$ be a line bundle on $X$. I will denote by $S$ the total space of $L$ -- this is a smooth surface over $k$ containing $X$ (as the zero ...
3
votes
6answers
1k views

Reference request: Moduli spaces of bundles over singular curves

I would like to know some reference (articles, books...) about any kind of moduli spaces of any of the following objects: vector bundles torsion-free sheaves principal bundles parabolic bundles ...
3
votes
1answer
677 views

Principal bundles and associated vector bundles, the case of the complex projective space (1,0)-forms

As can be guessed from some of my previous questions, I'm trying to understand, at the moment, the relationship between principal and their associated vector bundles. To this end I've been looking at ...
1
vote
4answers
352 views

Local Triviality of an Associated Bundle

I was reading this question link text and can't seem to see why, if $\pi: P \to B$ is a principle $G$-bundle and $$\rho:G \to GL_n(\mathbb{C})$$ is a representation of $G$, then the total space $P ...