  In this paper, we prove the concavity of the Shannon entropy power for the
heat equation associated with the Laplacian or the Witten Laplacian on complete
Riemannian manifolds with suitable curvature-dimension condition and on compact
super Ricci flows. Under suitable curvature-dimension condition, we prove that
the rigidity models of the Shannon entropy power are Einstein or quasi Einstein
manifolds with Hessian solitons. Moreover, we prove the convexity of the
Shannon entropy power for the conjugate heat equation introduced by G. Perelman
on Ricci flow and that the corresponding rigidity models are the shrinking
Ricci solitons. As an application, we prove the entropy isoperimetric
inequality on complete Riemannian manifolds with non-negative (Bakry-Emery)
Ricci curvature and the maximal volume growth condition.
