Having in mind applications to commutator estimates, which are of fundamental importance in PDEs for instance, we provide a new characterization of BMO space in terms of Carleson measures on Riemannian manifolds or some Lie groups. Under natural assumptions on the geometry of the ambient space, we can derive an equivalence of a family of BMO semi-norms with Carleson measures on the lift of the manifold to a cylindrical extension by the real line. The result is based on two key tools: a generalized harmonic extension of the BMO function on the lift and a local T(b) theorem to prove a square function estimate. This is the first step in a program to get sharp commutator estimates without using paradifferential calculus (in the sense of
Bony). The euclidean case has been developed by Lenzmann and Schikorra. We will give an application of our main result to a geometric version of
a famous result on Jacobians by Coifman-Meyer-Lions-Semmes. This is joint work with D. Brazke and A. Schikorra.