We present a diffeomorphic approach for constructing intrinsic shape atlases of sulci on the human cortex. Sulci are represented as square-root velocity functions of continuous open curves in R3, and their shapes are studied as functional representations of an infinite-dimensional sphere. This spherical manifold has some advantageous properties – it is equipped with a Riemannian metric on the tangent space and facilitates computational analyses and correspondences between sulcal shapes. Sulcal shape mapping is achieved by computing geodesics in the quotient space of shapes modulo scales, translations, rigid rotations and reparameterizations. The resulting sulcal shape atlas preserves important local geometry inherently present in the sample population. The sulcal shape atlas is integrated in a cortical registration framework and exhibits better geometric matching compared to the conventional euclidean method. We demonstrate experimental results for sulcal shape mapping, cortical surface registration, and sulcal classification for two different surface extraction protocols for separate subject populations.