  Recently Shiing-Shen Chern suggested that the six dimensional sphere
$\mathbb{S}^6$ has no complex structure. Here we explore the relations between
his arguments and Yang-Mills theories. In particular, we propose that Chern's
approach is widely applicable to investigate connections between the geometry
of manifolds and the structure of gauge theories. We also discuss several
examples of manifolds, both with and without a complex structure.
