  We are interested in the Caffarelli-Kohn-Nirenberg inequality (CKN in short),
introduced by these authors in 1984. We explain why the CKN inequality can be
viewed as a Sobolev inequality on a weighted Riemannian manifold. More
precisely, we prove that the CKN inequality can be interpreted in this way on
three different and equivalent models, obtained as weighted versions of the
standard Euclidean space, round sphere and hyperbolic space. This result can be
viewed as an extension of conformal invariance to the weighted setting. Since
the spherical CKN model we introduce has finite measure, the $\Gamma$-calculus
introduced by Bakry and Emery provides an easy way to prove the Sobolev
inequalities. This method allows us to recover the optimality of the region of
parameters describing symmetry-breaking of minimizers of the CKN inequality,
introduced by Felli and Schneider and proved by Dolbeault, Esteban and Loss in
2016. Finally, we develop the notion of n-conformal invariants, exhibiting a
way to extend the notion of scalar curvature to weighted manifolds such as the
CKN models.
