Let X be an irreducible smooth
projective curve over an algebraically closed field k of characteristic p, with p > 5.
Let G be a connected reductive algebraic group over k. Let H be a Levi
factor of some parabolic subgroup of G and χ a character of H. Given a
reduction EH of the structure group of a G-bundle EG to H, let Eχ be the line
bundle over X associated to EH for the character χ. If G does not contain
any SL(n)∕Z as a simple factor, where Z is a subgroup of the center of
SL(n), we prove that a G-bundle EG over X admits a connection if and only
if for every such triple (H,χ,EH), the degree of the line bundle Eχ is a
multiple of p. If G has a factor of the form SL(n)∕Z, then this result is
valid if n is not a multiple of p. If G is a classical group but not of the form
SL(n)∕Z, then this criterion for the existence of connection remains valid even if
p ≥ 3.
