  In this paper, we prove the concavity of the Renyi entropy power for
nonlinear diffusion equation (NLDE) associated with the Laplacian and the
Witten Laplacian on compact Riemannian manifolds with non-negative Ricci
curvature or $CD(0,m)$-condition and on compact manifolds equipped with time
dependent metrics and potentials. Our results can be regarded as natural
extensions of a result due to Savar\'e and Toscani \cite{ST} on the concavity
of the Renyi entropy for NLDE on Euclidean spaces. Moreover, we prove that the
rigidity models for the Renyi entropy power are the Einstein or quasi-Einstein
manifolds and a special $(K,m)$-Ricci flow with Hessian solitons. Inspired by
Lu-Ni-Vazquez-Villani \cite{LNVV}, we prove the Aronson-Benilan estimates for
NLDE on compact Riemannian manifolds with $CD(0,m)$-condition. We also prove
the NIW formula which indicates an intrinsic relationship between the second
order derivative of the Renyi entropy power $N_p$, the $p$-th Fisher
information $I_p$ and the time derivative of the $W$-entropy associated with
NLDE. Finally, we prove the entropy isoperimetric inequality for the Renyi
entropy power and the Gagliardo-Nirenberg-Sobolev inequality on complete
Riemannian manifolds with non-negative Ricci curvature or $CD(0, m)$-condition
and maximal volume growth condition.
