  Let $\mathcal{M}$ be a Riemannian $n$-manifold with a metric such that the
manifold is Ahlfors-regular. We also assume either non-negative Ricci
curvature, or that the Ricci curvature is bounded from below together with a
bound on the gradient of the heat kernel. We characterize BMO-functions $u:
\mathcal{M} \to \mathbb{R}$ by a Carleson measure condition of their
$\sigma$-harmonic extension $U: \mathcal{M} \times (0,\infty) \to \mathbb{R}$.
We make crucial use of a $T(b)$ theorem proved by Hofmann, Mitrea, Mitrea, and
Morris.
  As an application we show that the famous theorem of
Coifman--Lions--Meyer--Semmes holds in this class of manifolds: Jacobians of
$W^{1,n}$-maps from $\mathcal{M}$ to $\mathbb{R}^n$ can be estimated against
BMO-functions, which now follows from the arguments for commutators recently
proposed by Lenzmann and the second-named author using only harmonic
extensions, integration by parts, and trace space characterizations.
