Riemannian Simultaneous Inference for Tangent Vector Field Regression

This paper addresses nonparametric regression for tangent vector fields on Riemannian manifolds without boundary. The proposed kernel estimator first performs parallel transport to align nearby responses to the target tangent space and then computes a volume-corrected local average. The work establishes theoretical results including uniform second-order bias, finite-bandwidth covariance, and stochastic convergence rates for simultaneous statistical inference.