Differential Geometry of Frame Bundles

Abstract

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 ...

On the geometry of orthonormal frame bundles

“…We present an alternative in our Proposition 2.5. The initial properties ofg on OM have been studied also (in a few pages) by Mok in [10] and presented later in the survey [5]. To the authors' knowledge, the present paper is the first systematic study of curvature of this metric.…”

“…After K. P. Mok, the Sasaki-Mok metric on LM was studied systematically by L. A. Cordero and M. de León in [3] and [4]. See also the monograph [5] written jointly with C. T. J. Dodson. The present authors investigated in [9] a more general family of natural metrics on LM .…”

“…We shall see in Section 2 that this metricg coincides with the metric treated by B. O'Neill in [11] as an example of the Riemannian submersions. This reference obviously remained unnoticed by the authors of [5] and [10]. O'Neill has also calculated the sectional curvatures of this metric, in a very compact and elegant form, but his formulas are not too convenient for geometric applications.…”

Isomorphism classes for Banach vector bundle structures of second tangents

“…This is a different representation of an object similar to that introduced by Narasimhan and Ramanan [14,15] for G-bundles, also allowing a proof of Weil's theorem (cf. [10,9,2]). …”

G1-Structures of second order

“…Definition 3 .1. Any infinitesimal connection ( [18], [5]) defined on the principal bundle E1 (Vm, G2 rt ) is calk;d a Gi-connection .…”