  In this paper, we propose a new splitting algorithm for dynamical low-rank
approximation motivated by the fibre bundle structure of the set of fixed rank
matrices. We first introduce a geometric description of the set of fixed rank
matrices which relies on a natural parametrization of matrices. More precisely,
it is endowed with the structure of analytic principal bundle, with an explicit
description of local charts. For matrix differential equations, we introduce a
first order numerical integrator working in local coordinates. The resulting
algorithm can be interpreted as a particular splitting of the projection
operator onto the tangent space of the low-rank matrix manifold. It is proven
to be exact in some particular case. Numerical experiments confirm this result
and illustrate the robustness of the proposed algorithm.
