This book presents in a systematic manner a general nonparametric theory of statistics on manifolds with emphasis on manifolds of shapes, and with applications to diverse fields of science and engineering. There are many areas of significant application of statistics on manifolds. For example, directional statistics (statistics on the sphere S2) are used to study shifts in the Earth's magnetic poles over geological time, which have an important bearing on the subject of tectonics. Applications in morphometrics involve classification of biological species and subspecies. There are many important applications to medical diagnostics, image analysis (including scene recognition), and machine vision (e.g., robotics). We take a fresh look here in analyzing existing data pertaining to a number of such applications. It is our goal to lay the groundwork for other future applications of this exciting emerging field of nonparametric statistics.

Landmark-based shape spaces were first introduced by D. G. Kendall more than three decades ago, and pioneering statistical work on shapes with applications to morphometrics was carried out by F. Bookstein around the same time. Statistics on spheres, or directional statistics, arose even earlier, and a very substantial statistical literature on directional statistics exists, including a seminal 1953 paper by R. A. Fisher and books by Watson (1983), Mardia and Jupp (2000), Fisher et al. (1987), and others. For statistics on shape spaces, important parametric models have been developed by Kent, Dryden, Mardia, and others, and a comprehensive treatment of the literature may be found in the book by Dryden and Mardia (1998).