  A gauge condition is presented here to quantize non-Abelian gauge theory on
the manifold $R\otimes S^{1}\otimes S^{1}\otimes S^{1}$, which is free from the
Gribov ambiguity. Perturbative calculations in the new gauge behave like the
axial gauge in ultraviolet region, while infrared behaviours of the
perturbative series are quite nontrivial. The new gauge condition, which reads
$n\cdot\partial n\cdot A=0$, may not satisfy the requirement that
$A^{\mu}(\infty)=0$ in conventional perturbative calculations. However, such
contradiction is not harmful for gauge theories constructed on the manifold
$R\otimes S^{1}\otimes S^{1}\otimes S^{1}$.
