  We give a proof of Gaussian upper bound for the heat kernel coupled with the
Ricci ow. Previous proofs by Lei Ni [5] use Harnack inequality and doubling
volume property, also the recent proof by Zhang and Cao [6] uses Sobolev type
inequality that is conserved along Ricci ow. We will use a horizontal coupling
of curve [1] Arnaudon Thalmaier, C., in order to generalize Harnack inequality
with power -for inhomogeneous heat equation - introduced by F.Y Wang. In the
case of Ricci ow, we will derive on-diagonal bound of the Heat kernel along
Ricci ow (and also for the usual Heat kernel on complete Manifold).
