  Data taking value on a Riemannian manifold and observed over a complex
spatial domain are becoming more frequent in applications, e.g. in
environmental sciences and in geoscience. The analysis of these data needs to
rely on local models to account for the non stationarity of the generating
random process, the non linearity of the manifold and the complex topology of
the domain. In this paper, we propose to use a random domain decomposition
approach to estimate an ensemble of local models and then to aggregate the
predictions of the local models through Fr\'{e}chet averaging. The algorithm is
introduced in complete generality and is valid for data belonging to any smooth
Riemannian manifold but it is then described in details for the case of the
manifold of positive definite matrices, the hypersphere and the Cholesky
manifold. The predictive performance of the method are explored via simulation
studies for covariance matrices and correlation matrices, where the Cholesky
manifold geometry is used. Finally, the method is illustrated on an
environmental dataset observed over the Chesapeake Bay (USA).
