On manifolds with nonnegative Ricci curvature and the infimum of volume growth order <2

We prove two rigidity theorems for open (complete and noncompact) n-manifolds M with nonnegative Ricci curvature and the infimum of volume growth order <2. The first theorem asserts that the Riemannian universal cover of M has Euclidean volume growth if and only if M is flat with an n-1 dimensional soul. The second theorem asserts that there exists a nonconstant linear growth harmonic function on M if and only if M is isometric to the metric product R N for some compact manifold N.