A Riemannian distance is defined which is appropriate for clustering multivariate data. This distance requires that data is first fitted with a differentiable density model allowing the definition of an appropriate Riemannian metric. A tractable approximation is developed for the case of a Gaussian mixture model and the distance is tested on artificial data, demonstrating an ability to deal with differing length scales and linearly inseparable data clusters. Further work is required to investigate performance on larger data sets. 1 Introduction Finding clusters in multivariate data is a difficult task in general, perhaps not least because the very definition of a cluster is ambiguous. A number of clustering algorithms have been developed in the literature which aim to minimise some sort of partitioning error measure (see, for example, [1]). Some notion of distance is required for these algorithms and typically the Euclidean distance is used. Recently, Tipping introduced a novel Rieman...