There has been growing interest in statistical analysis on random objects taking values in a non-Euclidean metric space. One important class of such objects consists of data on manifolds. This article is concerned with inference on the Fréchet mean and related population objects on manifolds. We develop the concept of nonparametric likelihood for manifolds and propose general inference methods by adapting the theory of empirical likelihood. In addition to the basic asymptotic properties, such as Wilks’ theorem of the empirical likelihood statistic, we present several generalizations of the proposed methodology: two-sample testing, inference on the Fréchet variance and local Fréchet regression, quasi Bayesian inference, and estimation
of the Fréchet mean set. Simulation and real data examples illustrate the usefulness of the proposed methodology and advantage against the conventional Wald test.