  In this paper, motivated by the works of Bakry et. al in finding sharp Li-Yau
type gradient estimate for positive solutions of the heat equation on complete
Riemannian manifolds with nonzero Ricci curvature lower bound, we first
introduce a general form of Li-Yau type gradient estimate and show that the
validity of such an estimate for any positive solutions of the heat equation
reduces to the validity of the estimate for the heat kernel of the Riemannian
manifold. Then, a sharp Li-Yau type gradient estimate on the three dimensional
hyperbolic space is obtained by using the explicit expression of the heat
kernel and some optimal Li-Yau type gradient estimates on general hyperbolic
spaces are obtained.

Differential Geometry