Holography, a covariant c function, and the geometry of the renormalization group

We propose a covariant geometrical expression for the c function for theories which admit dual gravitational descriptions. We state a c theorem with respect to this quantity and prove it. We apply the expression to a class of geometries, from domain walls in gauged supergravities, to extremal and near extremal $\mathrm{D}p$-branes, and the AdS Schwarzschild black hole. In all cases, we find agreement with expectations.