  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 $\mathbb{R}\times N$ for some compact manifold
$N$.
