Inspired by Perelman's work on the entropy formula for the Ricci flow, we will first present some Harnack inequalities and the $W$-entropy formulas  for the heat equation of the Witten Laplacian on Riemannian manifolds with suitable curvature-dimension condition as well as Perelman's super Ricci flow. Then we introduce the $W$-entropy and prove its monotonicity along the geodesic flow on the Wasserstein space over Riemannian manifolds. We find that these two $W$-entropy formulas  have the same feature. This leads us to introduce a  deformation of geometric flows on the Wasserstein space over Riemannian manifolds, which interpolates the geodesic flow on the  Wasserstein space and the heat equation of the Witten Laplacian on the  underlying manifold. We prove an entropy-energy formula along the deformation of geometric flows with a parameter $c$. Some rigidity theorems will be derived. Joint work with Songzi Li (Fudan Univ and Touloue Univ).