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

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℝ</m:mi> <m:mi>k</m:mi> </m:msup> </m:math> {Rᵏ} factor. In particular, we obtain two rigidity results for open n -manifolds M with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>Ric</m:mi> <m:mi>M</m:mi> </m:msub> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {RicM≥ 0} and the infimum of volume growth order <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi/> <m:mo><</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> {<2} . The first result asserts that there exists a nonconstant linear growth harmonic function on M if and only if M is isometric to the metric product <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ℝ</m:mi> <m:mo>×</m:mo> <m:mi>N</m:mi> </m:mrow> </m:math> {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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {n-1} dimensional soul.