  In algorithms for solving optimization problems constrained to a smooth
manifold, retractions are a well-established tool to ensure that the iterates
stay on the manifold. More recently, it has been demonstrated that retractions
are a useful concept for other computational tasks on manifold as well,
including interpolation tasks. In this work, we consider the application of
retractions to the numerical integration of differential equations on
fixed-rank matrix manifolds. This is closely related to dynamical low-rank
approximation (DLRA) techniques. In fact, any retraction leads to a numerical
integrator and, vice versa, certain DLRA techniques bear a direct relation with
retractions. As an example for the latter, we introduce a new retraction,
called KLS retraction, that is derived from the so-called unconventional
integrator for DLRA. We also illustrate how retractions can be used to recover
known DLRA techniques and to design new ones. In particular, this work
introduces two novel numerical integration schemes that apply to differential
equations on general manifolds: the accelerated forward Euler (AFE) method and
the Projected Ralston--Hermite (PRH) method. Both methods build on retractions
by using them as a tool for approximating curves on manifolds. The two methods
are proven to have local truncation error of order three. Numerical experiments
on classical DLRA examples highlight the advantages and shortcomings of these
new methods.
