  We provide a clarification of the description of Langevin diffusions on
Riemannian manifolds and of the measure underlying the invariant density. As a
result we propose a new position-dependent Metropolis-adjusted Langevin
algorithm (MALA) based upon a Langevin diffusion in $\mathbb{R}^d$ which has
the required invariant density with respect to Lebesgue measure. We show that
our diffusion and the diffusion upon which a previously-proposed
position-dependent MALA is based are equivalent in some cases but are distinct
in general. A simulation study illustrates the gain in efficiency provided by
the new position-dependent MALA.
