  Let $ (M,g) $ be a smooth compact Riemannian manifold of dimension $ N \geq 3
$. Given $p_0 \in M$, $\lambda \in \mathcal{R}$ and $\sigma \in (0,2]$, we
study existence and non existence of minimizers of the following quotient:
\begin{equation}\label{Paper Equation} \mu_{\lambda,\sigma}=\inf_{u \in
H^1(M)\setminus \lbrace0\rbrace} \frac{\displaystyle\int_M |\nabla u|^2 dv_g
-\lambda \int_M u^2 dv_g }{\biggl(\displaystyle\int_M \rho^{-\sigma}
|u|^{2^*(\sigma)} dv_g\biggl)^{2/2^*(\sigma)}}, \end{equation} where
$\rho(.):=dist(p_0,.)$ denoted the geodesic distance from $p \in M$ to $p_0$.
In particular for $\sigma=2$, we provide sufficient and necessary conditions of
existence of minimizers in terms of $\lambda$. For $\sigma\in (0,2)$ we prove
existence of minimizers under scalar curvature pinching.
