Consider the scaling invariant, sharp log entropy (functional)
introduced by Weissler on noncompact manifolds with nonnegative Ricci
curvature. It can also be regarded as a sharpened version of
Perelman's W entropy in the stationary case. We prove that it has a
minimizer if and only if the manifold is isometric to $\R^n$.
Using this result, it is proven that a class of noncompact manifolds
with nonnegative Ricci curvature is isometric to $\R^n$. Comparing
with some well known flatness results in on asymptotically flat
manifolds and asymptotically locally Euclidean (ALE) manifolds, their
decay or integral condition on the curvature tensor is replaced by the
condition that the metric converges to the Euclidean one in C1 sense
at infinity. No second order condition on the metric is needed.