  We introduce a class of time dependent random fields on compact Riemannian
monifolds. These are represented by time-changed Brownian motions. These
processes are time-changed diffusion, or the stochastic solution to the
equation involving the Laplace-Beltrami operator and a time-fractional
derivative of order $\beta\in (0,1)$. The time dependent random fields we
present in this work can therefore be realized through composition and can be
viewed as random fields on randomly varying manifolds.
