  On compact Riemannian manifolds with non-negative Ricci curvature and smooth
(possibly empty), convex (or mean convex) boundary, if the sharp Li-Yau type
gradient estimate of an Neumann (or Dirichlet) eigenfunction holds at some
non-critical points of the eigenfunction; we show that the manifold is
isometric to the product of one lower dimension manifold and a round circle (or
a line segment).
