  \small{In 2004, Del Pino and Dolbeault \cite{DPDo} and Gentil \cite{G}
investigated, independently, best constants and extremals associated to sharp
Euclidean $L^p$-entropy inequalities. In this work, we present some important
advances in the Riemannian context. Namely, let $(M,g)$ be a compact Riemannian
manifold of dimension $n \geq 3$. For $1 < p \leq 2$, we prove that the sharp
Riemannian $L^p$-entropy inequality
  \[\int_M |u|^p \log(|u|^p) dv_g \leq \frac{n}{p} \log ({\cal A}_{opt} \int_M
|\nabla u|_g^p dv_g + {\cal B}) \]
  \n holds on all functions $u \in H^{1,p}(M)$ such that $||u||_{L^p(M)} = 1$.
Moreover, we show that the first best Riemannian constant ${\cal A}_{opt}$ is
equal to the corresponding Euclidean one. Our approach is inspired on the
Bakry, Coulhon, Ledoux and Sallof-Coste's idea \cite{Ba} of getting Euclidean
entropy inequalities as a limit case of suitable Gagliardo-Nirenberg
inequalities. It is conjectured that the above inequality sometimes fails for
$p > 2$.}
