It is well known that arteries are subject to residual stress. In earlier studies, the residual stress in the arterial ring relieved by a radial cut was considered in stress analysis. However, it has been found that axial strips sectioned from arteries also curled into arcs, showing that the axial residual stresses were relieved from the arterial walls. The combined relief of circumferential and axial residual stresses must be considered to accurately analyze stress and strain distributions under physiological loading conditions. In the present study, a mathematical model of a stress-free configuration of artery was proposed using Riemannian geometry. Stress analysis for arterial walls under unloaded and physiologically loaded conditions was performed using exponential strain energy functions for porcine and human common carotid arteries. In the porcine artery, the circumferential stress distribution under physiological loading became uniform compared with that without axial residual strain, whereas a gradient of axial stress distribution increased through the wall thickness. This behavior showed almost the same pattern that was observed in a recent study in which approximate analysis accounting for circumferential and axial residual strains was performed, whereas the circumferential and axial stresses increased from the inner surface to the outer surface under a physiological condition in the human common carotid artery of a two-layer model based on data of other recent studies. In both analyses, Riemannian geometry was appropriate to define the stress-free configurations of the arterial walls with both circumferential and axial residual strains.