  Consider the scaling invariant, sharp log entropy (functional) introduced by
Weissler \cite{W:1} on noncompact manifolds with nonnegative Ricci curvature.
It can also be regarded as a sharpened version of Perelman's W entropy
\cite{P:1} 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 the well
known flatness results in \cite{An:1}, \cite{Ba:1} and \cite{BKN:1} 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
$C^1$ sense at infinity. No second order condition on the metric is needed.
