  Gaussian processes are widely employed as versatile modeling and predictive
tools in spatial statistics, functional data analysis, computer modeling and in
diverse applications of machine learning. Such processes have been widely
studied over Euclidean spaces, where they are constructed using specified
covariance functions or covariograms. These functions specify valid stochastic
processes that can be used to model complex dependencies in spatial statistics
and other machine learning contexts. Valid (positive definite) covariance
functions have been extensively studied for Gaussian processes on Euclidean
spaces. Such investigations have focused, among other aspects, on the
identifiability and consistency of covariance parameters as well as the problem
of spatial interpolation and prediction within the fixed-domain or infill
paradigm of asymptotic inference. This manuscript undertakes analogous
theoretical developments for Gaussian processes constructed over Riemannian
manifolds. We begin by establishing formal notions and conditions for the
equivalence of two Gaussian random measures on compact manifolds. We build upon
recently introduced Mat\'ern covariograms on compact Riemannian manifold,
derive the microergodic parameter and formally establish the consistency of
maximum likelihood estimators and the asymptotic optimality of the best linear
unbiased predictor (BLUP). The circle and sphere are studied as two specific
examples of compact Riemannian manifolds with numerical experiments that
illustrate the theory.
