  In [ABF19] the authors define three projections of Rd-valued stochastic
differential equations (SDEs) onto submanifolds: the Stratonovich, Ito-vector
and Ito-jet projections. In this paper, after a brief survey of SDEs on
manifolds, we begin by giving these projections a natural, coordinate-free
description, each in terms of a specific representation of manifold-valued
SDEs. We proceed by deriving formulae for the three projections in ambient
$\mathbb R^d$-coordinates. We use these to show that the Ito-vector and Ito-jet
projections satisfy respectively a weak and mean-square optimality criterion
for small t: this is achieved by solving constrained optimisation problems.
These results confirm, but do not rely on the approach taken in [ABF19], which
is formulated in terms of weak and strong Ito-Taylor expansions. In the final
section we exhibit examples showing how the three projections can differ, and
explore alternative notions of optimality.
