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This work considers the problem of producing sanitized differentially private estimates of data which lives on a Riemannian manifold. We present extensions of the Laplace mechanism and the K-Norm gradient mechanism onto manifolds via the Riemannian metric of the manifold. We consider the case of releasing the Fr\’echet mean of data on a general manifold and show that if one ignores the structure of the data and rely solely on an ambient space, there is a decrease in utility of the sanitized summary statistic. We illustrate the framework through simulations on the sphere and symmetric positive definite matrices as well as an application to corpus callosum data represented on Kendall’s shape space.