THE RIEMANNIAN STRUCTURE OF EUCLIDEAN SHAPE SPACES - A NOVEL ENVIRONMENT FOR STATISTICS

The Riemannian metric structure of the shape space SIGMA(m)k for k lab elled points in R(m) was given by Kendall for the atypically simple si tuations in which m = 1 or 2 and k greater-than-or-equal-to 2. Here we deal with the general case (m greater-than-or-equal-to 1, k greater-t han-or-equal-to 2) by using the properties of Riemannian submersions a nd warped products as studied by O'Neill. The approach is via the asso ciated size-and-shape space that is the warped product of the shape sp ace and the half-line R+ (carrying size), the warping function being e qual to the square of the size. When combined with parallel studies by Le of the corresponding global geodesic geometry, the results obtaine d here determine the environment in which shape-statistical calculatio ns have to be acted out. Finally three different applications are disc ussed that illustrate the theory and its use in practice.