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ABSTRACT: Over the decades, non-Euclidean data have been emerged in various fields including neuroimaging, human kinetics study, and statistics. Parametric dimension reduction methods have been actively developed for the non-Euclidean data analysis. However, the methods are not enough to capture the curvilinear structure of data if the structure is complicated. To this end, I present new nonparametric dimension reduction methods for data observed on manifold, resulting in more flexible fits. To be more specific, first, I generalize a notion of principal curve into arbitrary dimensional spheres in the extrinsic and intrinsic ways where the principal curve is considered as a nonlinear generalization of principal component analysis (PCA). Secondly, the method is robustified by using absolute and Huber loss instead square loss. Lastly, for an effective initial estimate of complex structured data lying on manifold, a learning procedure is proposed. Theories including consistency and convergence rate of the procedure by means of statistical learning theory are further established on generic Riemannian manifolds. The consequences on the real seismological data and real Human motion capture data as well as simulated data on 2-sphere, 4-sphere show the promising characteristics of the proposed approaches.