A geometric interpretation of Hamilton’s Harnack inequality for the Ricci flow

In the paper of Li and Yau [LY], a differential Harnack inequality was proved for the heat equation on a Riemannian manifold. Their technique was based upon the maximum principal which made possible the extension to geometric evolution equations. In particular, Richard Hamilton proved differential Harnack inequalities for the mean curvature flow [H1] and the Ricci flow [H2]. He also extended the result of Li-Yau and proved a matrix Harnack inequality for the heat equation [H3]. Recently, Ben Andrews [A] has proved differential Harnack inequalities for very general curvature flows of hypersurfaces, including anisotropic flows. The purpose of this paper is to give a geometric interpretation of Hamilton’s Harnack inequality for the Ricci flow. We shall show that the Harnack quantity is in fact the curvature of a torsion-free connection compatible with a degenerate metric on space-time. More precisely, let ( M, g( t )) be a solution to the Ricci flow