In the vector-field guided path-following problem, a sufficiently smooth vector field is designed such that its integral curves converge to and move along a one-dimensional geometric desired path. The existence of singular points where the vector field vanishes creates a topological obstruction to global convergence to the desired path and some associated topological analysis has been conducted in 1. In this paper, we strengthen the result in 1, Theorem 2 by showing that the domain of attraction of the desired path, which is a compact asymptotically stable one-dimensional embedded submanifold of an <inline-formula><tex-math notation="LaTeX">n</tex-math></inline-formula>-dimensional ambient manifold <inline-formula><tex-math notation="LaTeX">{\mathcal{M}}</tex-math></inline-formula>, is homeomorphic to <inline-formula><tex-math notation="LaTeX">\mathbb {R}^{n-1} \times \mathbb {S}^{1}</tex-math></inline-formula>, and not just homotopy equivalent to <inline-formula><tex-math notation="LaTeX">\mathbb {S}^{1}</tex-math></inline-formula> as shown in 1, Theorem 2. This result is extended for a <inline-formula><tex-math notation="LaTeX">k</tex-math></inline-formula>-dimensional compact manifold for <inline-formula><tex-math notation="LaTeX">k \ge 2</tex-math></inline-formula>.