Sharp Sobolev inequalities in the presence of a twist

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n \ge 3$. Let also $A$ be a smooth symmetrical positive $(0,2)$-tensor field in $M$. By the Sobolev embedding theorem, we can write that there exist $K,B>0$ such that for any $u \in H_1^2(M)$,

$\left(\int_M\vert u\vert^{2^\star}dv_g\right)^{2/2^\star}\le K \int_MA_x(\nabla u, \nabla u)dv_g + B\int_Mu^2dv_g$

where $H_1^2(M)$ is the standard Sobolev space of functions in $L^2$ with one derivative in $L^2$. We investigate in this paper the value of the sharp $K$ in the equation above, the validity of the corresponding sharp inequality, and the existence of extremal functions for the saturated version of the sharp inequality.