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Global Fr\'echet regression is addressed from the observation of a strictlystationary bivariate curve process, evaluated in a finite--dimensional compactdifferentiable Riemannian manifold, with bounded positive smooth sectionalcurvature. The involved univariate curve processes respectively define thefunctional response and regressor, having the same Fr\'echet functional mean.The supports of the marginal probability measures of the regressor and responseprocesses are assumed to be contained in a ball, whose radius ensures theinjectivity of the exponential map. This map has time--varying origin at thecommon marginal Fr\'echet functional mean. A weighted Fr\'echet mean approachis adopted in the definition of the theoretical loss function. The regularizedFr\'echet weights are computed, in the time--varying tangent space from thelog--mapped regressors. Under these assumptions, and some Lipschitz regularitysample path conditions, when a unique minimizer exists, the uniformweak--consistency of the empirical Fr\'echet curve predictor is obtained, undermean--square ergodicity of the log--mapped regressor process in the first twomoments. A simulated example in the sphere illustrates the finite sample sizeperformance of the proposed Fr\'echet predictor. Predictions in time of thespherical coordinates of the magnetic field vector are obtained from thetime--varying geocentric latitude and longitude of the satellite NASA's MAGSATspacecraft in the real--data example analyzed.

Global Fr\'echet regression from time correlated bivariate curve data in manifolds