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The differential geometry of a principal bundle' is a phrase which usually refers either to the study of an individual connection in the bundle and its associated invariants-curvature, holonomy and the topological invariants expressible in terms of these-or to the gauge theory of the bundle, that is, the study of moduli spaces formed by quotienting spaces of connections over the action of the gauge group. The book of Cordero, Dodson and de Leon, however, is concerned with a different class of questions; namely, given a metric, or a connection, or some other geometric structure, on the base manifold M of a principal bundle P(M, G), characterize and study those structures on P of the same (or related) type, which project to the given structure on M. In the case of the frame bundles of the title, there are a number of lifting, or prolongation, processes by which such structures on P may be obtained and, as one would expect, the properties of these prolonged structures are expressible in terms of those of the given structure and, perhaps, some bundle invariants. Though this area of geometry has never produced results as important as those which have arisen from the aforementioned areas, it has nonetheless been the subject of a sizeable body of work over the last three decades, and deserves a reasoned account in book form.The frame bundle F{M) of a manifold M is the manifold of all the vector space bases in all the tangent spaces at the various points of M. The bulk of the book under review concerns the prolongation of structures from M to F(M); thus the prolongation of general G-structures is considered in Chapter 2, of linear connections in Chapter 4, and of metrics in Chapter 7. Typical results here are (2.3.7) that a Gstructure is integrable if and only if its prolongation to F(M) is integrable, and (4.3.4) that a connection is torsion-free if and only if its prolongation is so. Chapter 9 deals with ' systems of connections' and gives an elegant reformulation, due to Garcia, of part of the Chern-Weil theorem on characteristic classes.The book is, with a few minor exceptions, reliable and, taken section by section, clearly written. There are, however, some global criticisms that might be made. Firstly, the book is very short on commentary. The Introduction consists of one brief page, and chapter introductions are minimal. The material treated here is by no means difficult, but one still expects some signposting and some defence of the subject. Secondly, the emphasis on frame bundles seems to the reviewer misjudged, or at least uneven. There are two distinct points here. Cordero, Dodson and de Leon construct F(M) in terms of jets and could therefore easily have considered frame bundles of arbitrary order. Indeed the main virtue of the jet calculus is that it allows the simultaneous consideration of differentiation processes of arbitrary order. The last of the ten chapters of the book deals with the second-order frame bundle F\M) and includes, for example, the result (10.8.4) that a (/-structure is integrable if and ...

Abstract. The second order tangent bundle T 2 M of a smooth manifold M consists of the equivalent classes of curves on M that agree up to their acceleration. It is known [1] that in the case of a finite n-dimensional manifold M , T 2 M becomes a vector bundle over M if and only if M is endowed with a linear connection. Here we extend this result to M modeled on an arbitrarily chosen Banach space and more generally to those Fréchet manifolds which can be obtained as projective limits of Banach manifolds. The result may have application in the study of infinite-dimensional dynamical systems.

Summary The two‐dimensional variability of areal density is of interest in the characterization of flat fibrous networks such as paper and non‐woven textiles and the corresponding one‐dimensional case is pertinent to the study of textile yarns. An idealization of such materials to the random case has proved useful in many studies. This paper gives an analytical description of the spatial variability of random networks made from uniform straight fibres. This is based on the derivation of the isotropic point autocorrelation function for the structure, and an analysis of the variance and covariance among finite inspection cells in complete sampling schemes.

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