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3

It follows from the answer to your methane molecule question that the first two Stiefel-Whitney classes are trivial, because the corresponding cohomology groups are trivial. Then $w_3$, which is the mod $2$ reduction of $W_3=\beta w_2$, is also trivial. As in my comment, the triviality of the Stiefel-Whitney classes implies triviality of the bundle $\xi$ ...

5

By a theorem of Serre, see Theorem 1 in this paper, vector bundles over smooth affine curves over fields are direct sum of a line bundle and a trivial bundle. Then the classification problem is given by the line bundle classification. In particular, both questions have positive answers over fields. For smooth affine curves over a discrete valuation ring, ...

1

Let me answer my last question (last comment), if we give a $\rho:G\rightarrow G$ an automorphism of $G$, let us denote by $$E^\rho=E\times^GG$$ where the action of $G$ on itself is given by $\rho$, then generally $E^\rho$ is not isomorphique to $E$. It is when $\rho$ is an interior automorphism. For exemple taking $\rho(g)=^tg^{-1}$, $G=GL_r$, then one can ...

6

A way to rephrase QY's answer is to say that a vector bundle E is flat if there is a local system L such that $L \otimes_{\mathbb{C}} C_X \simeq E$ where $C_X$ is the trivial vector bundle (or, said differently, $C_X$ is the bundle whose sections on an open U are given by $C_X(U)$ - the continuous functions $f\colon U \to \mathbb C$).

16

A flat vector bundle over a topological space is a bundle whose transition functions can be taken to be locally constant; equivalently, over a path-connected space, it's the same data as a principal $G$-bundle ($G = GL_n(\mathbb{R})$ or $GL_n(\mathbb{C})$ as appropriate) where $G$ is given the discrete topology. Over a reasonable space $X$ this is the same ...

16

According to Kamber-Tondeur (1967), a principal $G$-bundle over a space $X$ is flat if it is induced from the universal covering bundle of $X$ by a homomorphism $\pi_1X\to G$. In the differentiable case this is equivalent to the existence of a connection with curvature zero [15, Lemma 1]. (...) A vector bundle is called flat, if its associated ...

3

I think that it is also possible to give a positive answer to the question using purely topological arguments, namely Ostrand's theorem: for every paracompact space $X$ of dimension $n$ and every open covering $\mathscr U$ of $X$, there exists $n+1$ families $\mathscr V_1,\dotsc, \mathscr V_{n+1}$ of disjoint open sets whose union is a covering that refines ...

9

I assume you meant for the fiberwise rank of $E$ to be constant, say $r>0$ (or at least uniformly bounded above). The answer is "yes" when the Stein space has finite dimension (equivalently, when its analytic irreducible components, all of which must be equidimensional via consideration of their connected normalizations, have uniformly bounded dimension). ...

9

In the complex analytic world*, every vector bundle which is globally generated by a finite set of global sections has a finite trivialising cover: Let $X$ be a complex space and let $E$ be a locally free (hence coherent) $\mathcal{O}_X$-module of constant rank $r$, generated by the global sections $s_1,s_2,\dots s_d$. For each subset $I=\{i_1,\dots i_r\}$ ...

1

I believe you have answered this yourself. Principal bundles are trivial iff they admit a global smooth section. Vector bundles always admit a global smooth section (zero section). Therefore vector principal G-bundles are always trivial. The only available examples of vector G-bundles are thus of the form M x G, where G is both a vector space and a Lie ...

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