Copulae on products of compact Riemannian manifolds

One standard way of considering a probability distribution on the unit ncube, [0, 1], due to Sklar (1959) [A. Sklar, Fonctions de répartition à n dimensions et leur marges, Publ. Inst. Statist. Univ. Paris 8 (1959) 229231], is to decompose it into its marginal distributions and a copula, i.e. a probability distribution on [0, 1] with uniform marginals. The definition of copula was extended by Jones et al. (2014) [M.C. Jones, A. Pewsey, S. Kato, On a class of circulas: copulas for circular distributions, Ann. Inst. Statist. Math., to appear] to probability distributions on products of circles. This paper defines a copula as a probability distribution on a product of compact Riemannian manifolds that has uniform marginals. Basic properties of such copulae are established. Two fairly general constructions of copulae on products of compact homogeneous manifolds are given; one is based on convolution in the isometry group, the other using equivariant functions from compact Riemannian manifolds to their spaces of square integrable functions. Examples illustrate the use of copulae to analyse bivariate spherical data and bivariate rotational data.