Cortical surface registration or matching facilitates atlasing, cortical morphology-function comparison and statistical analysis. Methods that geodesically shoot surfaces into one another, as currents or varifolds, provide an elegant mathematical framework for generic surface matching and dynamic local features estimation, such as deformation momenta. However, conventional current and varifold matching methods only use the normals of the surface to measure its geometry and guide the warping process, which overlooks the importance of the direction in the convoluted cortical sulcal and gyral folds. To cope with the stated limitation, we decompose each cortical surface into its normal and tangent varifold representations, by integrating principal curvature direction field into the varifold matching framework, thus providing rich information for the direction of cortical folding and better characterization of the cortical geometry. To include more informative cortical geometric features in the matching process, we adaptively place control points based on the surface topography, hence the deformation is controlled by points lying on gyral crests (or “hills”) and sulcal fundi (or “valleys”) of the cortical surface, which are the most reliable and important topographic and anatomical landmarks on the cortex. We applied our method for registering the developing cortical surfaces in 12 infants from 0 to 6 months of age. Both of these variants significantly improved the matching accuracy in terms of closeness to the target surface and the precision of alignment with regional anatomical boundaries, when compared with several state-of-the- art methods: (1) diffeomorphic spectral matching, (2) current-based surface matching and (3) original varifold-based surface matching.