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

Abstract We study open (complete and noncompact) n -manifolds M with nonnegative Ricci curvature, under the condition that any asymptotic cone of M splits off an ℝ k R^{k} factor. In particular, we obtain two rigidity results for open n -manifolds M with Ric M ≥ 0 Ric₌ 0 and the infimum of volume growth order 2 ℝ × N {R N for some compact manifold N. The second asserts that the Riemannian universal cover of M has Euclidean volume growth if and only if M is flat with an n - 1 n-1 dimensional soul.