Intrinsic comparisons for data is critical in many problems. One of the major challenges of intrinsic point clouds registration is the possible ambiguity due to the infinite-dimensional self-isometric group. In this talk, I will discuss our work on nonrigid point clouds registration using a rotation invariant sliced Wasserstein distance on the LB eigenmaps. After that, I will also discuss an extension of this method for mapping non-isometric shapes using LB eigensystem via conformal deformation. Both problems are related to orthogonality constrained optimization problems.