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A gauge fixing condition is presented here for non-Abelian gauge theory on the manifold R⊗S1⊗S1⊗S1. It is proved that the new gauge fixing condition is continuous and free from the Gribov ambiguity. While perturbative calculations based on the new gauge condition behave like those based on the axial gauge in ultraviolet region, infrared behaviours of the perturbative series under the new gauge fixing condition are quite nontrivial. The new gauge condition, which reads n⋅∂n⋅A=0, may not satisfy the boundary condition Aμ(∞)=0 as required by conventional perturbative calculations for gauge theories on the manifold S4. However, such contradiction is not harmful for the theory considered here.