Skeleton curve-guided five-axis sweep scanning is an emerging surface inspection technique with marvelous inspection efficiency. The precondition of applying such a technique to a complex surface is to extract the skeleton curves of a two-manifold surface associated with partitioning the surface into compact surfaces. However, the work on extracting skeletons of two-manifold triangle mesh surfaces is scarce, as existing skeletonization methods mostly focus on either 2D planar shapes or 3D solid shapes. In this paper, we present a new approach to extract the skeletons on a two-manifold triangle mesh with boundaries. The skeletons are formed in the most intuitive manner of wave front propagation which is based on computing the initial value problem of geodesics on a triangle mesh. The step-length of wave front propagation is adaptively controlled to guarantee the appropriate density of skeleton points for having good connectivity. Experiments show that, as a direct application of the proposed skeleton generation and its associated surface partitioning result, the five-axis sweep scanning path of complex free-form surface can be generated conveniently. Experimental results also validate that the computation of our proposed approach is simpler and faster than the state-of-the-art geodesic Voronoi diagram (GVD) method when most of the two-manifold triangle mesh is nearly planar. Additionally, the augmented parameter used for tracing the wave front’s geodesic propagation information is helpful for the skeleton-based surface partition, which is necessary for skeleton curve-guided five-axis sweep scanning.