The Riemannian Structure of Euclidean Shape Spaces: A Novel Environment for Statistics

The Riemannian metric structure of the shape space ∑ᵏₘ for k labelled points in Rᵐ was given by Kendall for the atypically simple situations in which $m = 1$ or 2 and k ≥ 2. Here we deal with the general case (m ≥ 1, k ≥ 2) by using the properties of Riemannian submersions and warped products as studied by O'Neill. The approach is via the associated size-and-shape space that is the warped product of the shape space and the half-line R₊ (carrying size), the warping function being equal to the square of the size. When combined with parallel studies by Le of the corresponding global geodesic geometry, the results obtained here determine the environment in which shape-statistical calculations have to be acted out. Finally three different applications are discussed that illustrate the theory and its use in practice.