The ridge lines on a surface can be defined either via contact of the surface with spheres, or via extrema of principal curvatures along lines of curvature. Certain subsets of ridge lines called crest lines have been singled out by some authors for medical imaging applications. There is a related concept of sub-parabolic line on a surface, also defined via extrema of principal curvatures. In this paper we study in detail the structure of the ridge lines, crest lines and sub-parabolic lines on a generic surface, and on a surface which is evolving in a generic (one-parameter) family. The mathematical details of this study are in Bruce et al.
Bruce, J. W., Giblin, P. J., & Tari, F. (1996). Ridges, crests and sub-parabolic lines of evolving surfaces. International Journal of Computer Vision, 18(3), 195-210. https://doi.org/10.1007/BF00123141