The rigidity of eigenfunctions' gradient estimates

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).