  By using an explicit Bellman function, we prove a bilinear embedding theorem
for the Laplacian associated with a weighted Riemannian manifold $(M,\mu_\phi)$
having the Bakry-Emery curvature bounded from below. The embedding, acting on
the cartesian product of $L^p(M,\mu_\phi)$ and $L^q(T^*M,\mu_\phi)$,
$1/p+1/q=1$, involves estimates which are independent of the dimension of the
manifold and linear in $p$. As a consequence we obtain linear dimension-free
estimates of the $L^p$ norms of the corresponding shifted Riesz transform. All
our proofs are analytic.
