  A path-following control algorithm enables a system's trajectories under its
guidance to converge to and evolve along a given geometric desired path. There
exist various such algorithms, but many of them can only guarantee local
convergence to the desired path in its neighborhood. In contrast, the control
algorithms using a well-designed guiding vector field can ensure almost global
convergence of trajectories to the desired path; here, "almost" means that in
some cases, a measure-zero set of trajectories converge to the singular set
where the vector field becomes zero (with all other trajectories converging to
the desired path). In this paper, we first generalize the guiding vector field
from the Euclidean space to a general smooth Riemannian manifold. This
generalization can deal with path-following in some abstract configuration
space (such as robot arm joint space). Then we show several theoretical results
from a topological viewpoint. Specifically, we are motivated by the observation
that singular points of the guiding vector field exist in many examples where
the desired path is homeomorphic to the unit circle, but it is unknown whether
the existence of singular points always holds in general (i.e., is inherent in
the topology of the desired path). In the $n$-dimensional Euclidean space, we
provide an affirmative answer, and conclude that it is not possible to
guarantee global convergence to desired paths that are homeomorphic to the unit
circle. Furthermore, we show that there always exist \emph{non-path-converging
trajectories} (i.e., trajectories that do not converge to the desired path)
starting from the boundary of a ball containing the desired path in an
$n$-dimensional Euclidean space where $n \ge 3$. Examples are provided to
illustrate the theoretical results.
