  Let (M,g) be a compact Riemannian manifold of dimension n \geq 2. In this
work we prove the validity of the optimal L^p-Riemannian Gagliardo-Nirenberg
inequality for 1 < p \leq 2. Our proof relies strongly on a new distance lemma
which. In particular, we extend L^p-Euclidean Gagliardo-Nirenberg inequalities
due to Del Pino and Dolbeault and the optimal L^2-Riemannian
Gagliardo-Nirenberg inequality due to Broutteland in a unified framework.
