Abstract: This paper discusses the first eigenvalue on a compact Riemann manifold with the negative lower bound Ricci curvature. Let M be a compact Riemann manifold with the Ricci curvature≥-R, R=const.≥0 and d is the diameter of M. Our main result is that the first eigenvalue λ₁ of M satisfies λ₁≥π²/d²-0.518.R.

Key words: Ricci , curvature;first , eigenvalue