  Continuous-time projected dynamical systems are an elementary class of
discontinuous dynamical systems with trajectories that remain in a feasible
domain by means of projecting outward-pointing vector fields. They are
essential when modeling physical saturation in control systems, constraints of
motion, as well as studying projection-based numerical optimization algorithms.
Motivated by the emerging application of feedback-based continuous-time
optimization schemes that rely on the physical system to enforce nonlinear hard
constraints, we study the fundamental properties of these dynamics on general
locally-Euclidean sets. Among others, we propose the use of Krasovskii
solutions, show their existence on nonconvex, irregular subsets of
low-regularity Riemannian manifolds, and investigate how they relate to
conventional Carath\'eodory solutions. Furthermore, we establish conditions for
uniqueness, thereby introducing a generalized definition of prox-regularity
which is suitable for non-flat domains. Finally, we use these results to study
the stability and convergence of projected gradient flows as an illustrative
application of our framework. We provide simple counter-examples for our main
results to illustrate the necessity of our already weak assumptions.
