This book does an excellent job of explaining... geometrical ideas from a statistical point of view, using statistical examples as opposed to, for example, dynamical examples to illustrate the geometric concepts. It should prove most helpful in enlarging the typical statistician's working geometric vocabulary.
-Short Book Reviews of the ISI
It is particularly good on the... problem of when to use coordinates and indices, and when to use the more geometric coordinate-free methods. Examples of this are its treatments of the affine connection and of the string theory of Barndorff-Nielsen, both of which are excellent. The first of these concepts is probably the most sophisticated that has been used in statistical applications. It can be viewed in many different ways, not all of which are necessarily useful to the statistician; the approach taken in the book is both relevant and clear.
-Bulletin of London Mathematical Society