The Gagliardo–Nirenberg interpolation inequality on a bounded domain is of the form $$|D^j u|_{L^p} \leq C_1|D^m u|_{L^r}^\alpha|u|_{L^q}^{1-\alpha} + C_2|u|_{L^s}$$ where are there restrictions on the constants above except for $s$ which can be arbitrary.
Does this inequality hold on a compact manifold without boundary?
Due to the presence of the final term I am inclined to say yes but I haven't found a reference.