  We develop an advanced approach for extending Gaussian Differential Privacy
(GDP) to general Riemannian manifolds. The concept of GDP stands out as a
prominent privacy definition that strongly warrants extension to manifold
settings, due to its central limit properties. By harnessing the power of the
renowned Bishop-Gromov theorem in geometric analysis, we propose a Riemannian
Gaussian distribution that integrates the Riemannian distance, allowing us to
achieve GDP in Riemannian manifolds with bounded Ricci curvature. To the best
of our knowledge, this work marks the first instance of extending the GDP
framework to accommodate general Riemannian manifolds, encompassing curved
spaces, and circumventing the reliance on tangent space summaries. We provide a
simple algorithm to evaluate the privacy budget $\mu$ on any one-dimensional
manifold and introduce a versatile Markov Chain Monte Carlo (MCMC)-based
algorithm to calculate $\mu$ on any Riemannian manifold with constant
curvature. Through simulations on one of the most prevalent manifolds in
statistics, the unit sphere $S^d$, we demonstrate the superior utility of our
Riemannian Gaussian mechanism in comparison to the previously proposed
Riemannian Laplace mechanism for implementing GDP.
