Geometric structures arising from kernel density estimation on Riemannian manifolds

Estimating the kernel density function of a random vector taking values on Riemannian manifolds is considered. We make use of the concept of exponential map in order to define the kernel density estimator. We study the asymptotic behavior of the kernel estimator which contains geometric quantities (i.e. the curvature tensor and its covariant derivatives). Under a Hölder class of functions defined on a Riemannian manifold with some global losses, the L2-minimax rate and its relative efficiency are obtained.