Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the stability of flat complex vector bundles over parallelizable
  manifolds   We investigate the flat holomorphic vector bundles over compact complex
parallelizable manifolds $G / \Gamma$, where $G$ is a complex connected Lie
group and $\Gamma$ is a cocompact lattice in it. The main result proved here is
a structure theorem for flat holomorphic vector bundles $E_\rho$ associated to
any irreducible representation $\rho : \Gamma \rightarrow \text{GL}(r,{\mathbb
C})$. More precisely, we prove that $E_{\rho}$ is holomorphically isomorphic to
a vector bundle of the form $E^{\oplus n}$, where $E$ is a stable vector
bundle. All the rational Chern classes of $E$ vanish, in particular, its degree
is zero.
  We deduce a stability result for flat holomorphic vector bundles $E_{\rho}$
of rank 2 over $G/ \Gamma$. If an irreducible representation $\rho :
\Gamma\rightarrow \text{GL}(2, \mathbb {C})$ satisfies the conditionmthat the
induced homomorphism $\Gamma\rightarrow {\rm PGL}(2, {\mathbb C})$ does not
extend to a homomorphism from $G$, then $E_{\rho}$ is proved to be stable.

35
The categories of this paper are: mtrl-th
9
The categories of this paper are: nlin
23
The categories of this paper are: solv-int
17
The categories of this paper are: q-fin
20
The categories of this paper are: acc-phys
25
The categories of this paper are: patt-sol
11
The categories of this paper are: hep-lat
26
The categories of this paper are: bayes-an
12
The categories of this paper are: nucl-th
15
The categories of this paper are: nucl-ex
34
The categories of this paper are: chem-ph
24
The categories of this paper are: dg-ga
32
The categories of this paper are: atom-ph
14
The categories of this paper are: math-ph
33
The categories of this paper are: plasm-ph
8
The categories of this paper are: hep-ex