  Let (M,g) be a smooth compact Riemannian manifold of dimension n \geq 2, 1 <
p < n and 1 \leq q < r < p^\ast = \frac{np}{n-p} be real parameters. This paper
concerns to the validity of the optimal Gagliardo-Nirenberg inequality
  (\int_M |u|^r\; dv_g)^{\frac{\tau}{r \theta}} \leq (A_{opt} (\int_M |\nabla_g
u|^p\; dv_g)^{\frac{\tau}{p}} + B_{opt} (\int_M |u|^p\; dv_g)^{\frac{\tau}{p}})
(int_M |u|^q\; dv_g)^{\frac{\tau(1 - \theta)}{\theta q}} \; .
  This kind of inequality is studied in Chen and Sun (Nonlinear Analysis 72
(2010), pp. 3159-3172) where the authors established its validity when 2 < p <
r < p^\ast and (implicitly) \tau = 2. Here we solve the case p \geq r and
introduce one more parameter 1 \leq \tau \leq \min\{p,2\}. Moreover, we prove
the existence of extremal function for the optimal inequality above.
